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Zero Pixel Directional Boundary by Vector Transform | 1 INTRODUCTION . Boundaries are considered to be one of the important interpretable visual cues that can describe both the low-level image characteristics as well as high-level semantics of the image . Human vision uses occluding contours and boundaries to interpret unseen or seen objects and classes . Because boundaries are an important visual cue for image understanding , they are also exploited as priors in several vision tasks ( Zhu et al. , 2020 ; Kim et al. , 2021 ; Hatamizadeh et al. , 2019 ; Revaud et al. , 2015 ; Cashman & Fitzgibbon , 2012 ) . Some key works on contours ( Cootes et al. , 2001 ; Matthews & Baker , 2004 ; Kass et al. , 1988 ) have greatly impacted early research in computer vision . Although the advent of deep learning and end-to-end learning has somewhat shifted the focus away from interpretable visual cues , boundary discovery still remains an important task in computer vision . Supervised deep learning has greatly transformed problems such as object detection and segmentation ( Redmon et al. , 2016 ; Chen et al. , 2017 ; Cheng et al. , 2020 ) by redefining the problem ( Kirillov et al. , 2019 ) , using high-quality datasets ( Cordts et al. , 2016 ; Neuhold et al. , 2017 ) and better network architectures ( Cheng et al. , 2020 ; 2021 ; Wang et al. , 2021b ) . Boundary detection , however , has seen a rather modest share of such progress . Although , modern deeply learned methods ( Xie & Tu , 2015 ; Liu et al. , 2017 ; Maninis et al. , 2017 ) provide better accuracy and the possibility to learn only the high-level boundaries , a particularly elusive goal in learned boundary detection has been the so-called crisp boundaries ( Isola et al. , 2014 ; Wang et al. , 2018 ; Deng et al. , 2018 ) . The formulation of boundary detection as a binary segmentation task naturally introduces class imbalance , which makes detecting pixel thin sharp boundaries extremely difficult . Arguably a majority of recent methods in boundary detection are proposed in order to tackle the aforementioned issue . Many methods address the lack of ‘ crispness ’ by fusing high-resolution features with the middleand high-level features ( Xie & Tu , 2015 ; Liu et al. , 2017 ) . Such a strategy has been successful in other dense prediction tasks ( Ronneberger et al. , 2015 ) as well . Others propose different loss functions ( Kokkinos , 2016 ; Deng et al. , 2018 ; Kervadec et al. , 2019 ) to address class imbalance . Despite the improvements , we identify two issues regarding crisp boundary detection . The first is that the evaluation protocol ( Martin et al. , 2004 ) does not necessarily encourage crisp detection , as the quantification is done after Non-Maximal Suppression ( NMS ) . Such an evaluation may be misleading when the network outputs need to be used at training time for other high-level tasks , e.g. , segmentation ( Kim et al. , 2021 ) . Second , the current losses ( Kokkinos , 2016 ; Xie & Tu , 2015 ; Ma et al. , 2021 ) push for edges as crisp as the ground-truth rather than as crisp as possible . This is particularly harmful since many boundary detection datasets ( Arbelaez et al. , 2010 ; Silberman et al. , 2012 ) contain ambiguous boundaries or inconsistently thick boundaries . In this paper , we take a different perspective on boundary detection . Boundaries are formed where visual features change , popularly referred to as the differential representation ( Boykov et al. , 2006 ; Kervadec et al. , 2019 ) . Such an assumption treats the boundaries as a set of 1-D surfaces embedded in a continuous 2D space , implying they do not have any thickness . Many previous energyminimization based approaches ( Chan & Zhu , 2005 ; Chan & Vese , 2001 ; Paragios et al. , 2002 ; Boykov et al. , 2006 ; Ma & Manjunath , 2000 ) and a few current methods ( Kervadec et al. , 2019 ) tackle boundaries in a similar way . Level-set methods ( Chan & Zhu , 2005 ; Boykov et al. , 2006 ; Ma & Manjunath , 2000 ) consider boundaries as the level-set of a continuous function of the image . Specifically , ( Ma & Manjunath , 2000 ) defines the energy function related to the distance and direction of the boundary at each pixel and extracts the directional normals at the boundary using such an energy . Inspired by such works and also recent works on 3D implicit functions ( Tancik et al. , 2020 ; Sitzmann et al. , 2020 ; Mildenhall et al. , 2020 ) , we represent boundaries via a field of unit vectors defined at each pixel , pointing towards the closest boundary surface . The proposed vector field representation naturally solves class imbalance . In distance transforms , vector fields are considered incomplete euclidean transforms ( Osher & Sethian , 1988 ) , equal to the Jacobian of the signed distance field . The vector field we use is in fact the Jacobian of the positive distance field . In contrast to distance fields , it provides high sensitivity at the boundaries and is easily localizable . We demonstrate the equivalence of the normal field to the surface contour representation using the level set of the normal field ’ s divergence , providing infinitely sharp boundaries . Owing to the zero-thickness , we refer to our result as the zero-pixel boundary . Our method is virtually hyper-parameter free at training and test time , and can provide zero-pixel thick boundaries at training time . In order to evaluate the boundaries using the surface interpretation , we also advocate the use of surface distances including the average symmetric surface distance ( assd ) metric that is less prone to class imbalance and variable boundary thickness in the ground-truth ( Kervadec et al. , 2019 ) . Such metrics are very popular in biomedical image segmentation ( Yeghiazaryan & Voiculescu , 2018 ) . We show significant improvements in all metrics using our boundary representation when compared to the various combinations of Dice ( Dice , 1945 ) and cross-entropy losses in several large datasets . 2 RELATED WORK . There is a rich history of boundary detection in computer vision . Previous work on boundaries showed a diversity of definitions and approaches . We differentiate them based on two different interpretations of boundaries : i.e. , i ) a boundary is a separation between two or more image regions with different visual features and ii ) a boundary is a thin group of pixels belonging to a specific class . It should be noted that most modern methods fall under the second category . Perhaps the most notable representatives of the first strand are the energy-based segmentation methods ( Chan & Vese , 2001 ; Comaniciu & Meer , 2002 ; Boykov et al. , 2006 ; Grady , 2006 ; Ma & Manjunath , 2000 ) . These methods relied on hand-crafted features and various optimization strategies to compute the low-level boundaries . In particular , Ma & Manjunath ( 2000 ) compute the normal vectors of the low-level boundaries from an energy function , without looking into an equivalent learnable representation . Graph-based segmentation methods ( Shi & Malik , 2000 ; Felzenszwalb & Huttenlocher , 2004 ; Cheng et al. , 2016 ) construct a graph from the image and cut the graph to obtain non-overlapping regions whose boundaries are viewed as image boundaries . A few deep learning methods followed a similar approach ( Wang et al. , 2021a ; Kervadec et al. , 2019 ) . Despite the advantage of the definition , current representations in this category are hard to adapt to generic boundaries and a compact multi-boundary representation with good performance remains lacking . A larger corpus of work utilizes pixel-wise image features to decide whether pixels belong to a ‘ boundary ’ class . They form our category ii ) methods . Early methods utilize various filter operators to detect discontinuities in image intensities or colors ( Canny , 1986 ; Sobel , 1972 ) . Learning based methods substantially boost the edge detection performance by classifying handcrafted features ( Konishi et al. , 2003 ; Martin et al. , 2004 ; Arbelaez et al. , 2010 ; Dollár & Zitnick , 2013 ; Hallman & Fowlkes , 2015 ) . Modern deep neural network ( DNN ) methods have further improved this field by learning powerful feature representations , particularly high-level semantic information ( Shen et al. , 2015 ; Bertasius et al. , 2015 ; Xie & Tu , 2015 ; Liu et al. , 2017 ; Wang et al. , 2018 ; Deng et al. , 2018 ; He et al. , 2019 ) . Yang et al . ( 2016 ) leveraged the powerful deep features to detect only object boundaries . Others try to simultaneously detect edges and predict the semantic class of each edge point , so-called semantic edge detection ( Hariharan et al. , 2011 ; Yu et al. , 2017 ; Liu et al. , 2018 ; Yu et al. , 2018 ) . On the other hand , classifying pixels as boundary class introduces class imbalance during training . A common counter-strategy is to use a weighted cross-entropy loss giving the non-boundary pixels a small weight and the boundary class a large weight ( Xie & Tu , 2015 ; Liu et al. , 2017 ; He et al. , 2019 ) . Yet , despite an improvement over regular cross-entropy , it does not solve the problem . To thin the boundaries , Non-Maximal Suppression ( NMS ) is usually adopted . Such methods may be harmful when directly integrated with higher-level tasks such as segmentation ( Kim et al. , 2021 ) . The Dice loss ( Dice , 1945 ) was thus advocated to generate crisp boundaries before NMS ( Deng et al. , 2018 ) , but it still produces several pixel thick boundaries and suffers more from missed predictions . Variations of the Dice loss ( Shit et al. , 2021 ) have been proposed to counter the missed detections . However , the right approach still depends on the downstream tasks ( Zhu et al. , 2020 ; Kim et al. , 2021 ; Hatamizadeh et al. , 2019 ; Shit et al. , 2021 ) and in either case a careful selection of training as well as testing hyper-parameters is required . We provide an alternative approach , motivated by the class imbalance , while having no sensitive hyper-parameter . 3 BOUNDARY TRANSFORM AND REPRESENTATION . In this section , we first discuss the surface representation of boundaries as a normal vector field transform and prove its relevant properties . Our boundary representation is inspired by recent work on implicit neural 3D surface representations ( Park et al. , 2019 ; Mescheder et al. , 2019 ) , energybased methods on edge detection ( Ma & Manjunath , 2000 ; Boykov et al. , 2006 ) and distance transforms ( Osher & Sethian , 1988 ) . In 3D surface representations , a Signed Distance Function ( SDF ) ( Osher & Sethian , 1988 ) or occupancy map ( Mescheder et al. , 2019 ) is used as representation . We instead propose a unit vector field from every point to the closest boundary . This choice is motivated by the high sensitivity and richer boundary context provided by the unit vector field , as shown in our experimental results in § 5 . Fig . 1 shows our vector field representation of the ground-truth , with predictions using a standard quiver plot on a sub-sampled set of points . We assume a continuous boundary image domain Ω ⊂ R2 with the set of boundary points { x′ } = Π ⊂ Ω . Each point is denoted as a 2-vector x = ( x , y ) ∈ R2 . In order to encode boundary properties on the whole image , we compute a signed x and y distance field separately and finally encode only the direction . The result is a unit vector field that represents the boundary . We can express our boundary representation by the following transform for any point x ∈ Ω : fx ( x ) = inf x′∈Π dx ( x , x ′ ) , fy ( x ) = inf x′∈Π dy ( x , x ′ ) , v ( x ) = f ( x ) ‖f ( x ) ‖2 , if ‖f ( x ) ‖2 6= 0 , otherwise n , n = lim dx→0+ f ( x ) ‖f ( x ) ‖2 . ( 1 ) Equation ( 1 ) defines the transform as a function v ( x ) : Ω→ R2 going from the boundary Π to a field representation . Here , dx and dy are operators that give the signed x and y difference , respectively . Similarly , fx and fy are the x and y components of the field f. Note that we choose the field vector arbitrarily among the two possible values at the boundary using the positive dx distance . We note the following properties of the vector field v. Property 3.1 The vector field v ( x ) is equal to the unit normal field at the boundary . Proof This is a well known result ( Osher & Fedkiw , 2003 ) and can be proved easily ( see equation ( 2.4 ) in the reference ) . The fact that we forcefully choose one normal over its negative directional normal at the boundary points does not affect the statement . Property 3.2 Given a vector field representation v ( x ) of a boundary , one can obtain the binary boundary representation by considering the following transform : g ( x ) = div v ( x ) . ( 2 ) The original boundary set Π can then be found by taking the zero level set of g ( x ) + 2 , i.e. , Π = L0 ( g + 2 ) . ( 3 ) Proof In the infinitesimal neighborhood of the boundary points , using property 3.1 , the vector field is normal to the boundary , provided that good approximate normals can be obtained from equation ( 1 ) . As the neighborhood size approaches zero , the tangential vector components approach zero around a point for a continuous boundary segment . Thus , around such an infinitesimal neighborhood , the normal fields pointing in opposite direction will subtract perfectly , creating a divergence flow of -2 and around 0 or positive away from boundaries . Strictly speaking the result holds only for piece-wise smooth surfaces ( Osher & Fedkiw , 2003 ) , with lower than -2 divergence possible at discontinuous surface points . Property 3.3 The relation is one-to-one between the binary boundary representation and the proposed vector field representation in a continuous domain . Proof This property is the result of equation ( 1 ) , for the forward transform and equation ( 3 ) for the inverse transform , providing a one-to-one relation . Note that the vector field transform as defined in equation ( 1 ) has to correct for two different kinds of indeterminate states . The first is on the boundary , that is solved by using the right hand limit so that one of the two opposite directions is chosen consistently . The second is when the infimum operation in equation ( 1 ) produces two or more closest points , corrected by choosing any one of the points for the infimum . The vector fields around such points flip directions creating a positive divergence as shown in Fig . 2 . More discussions are provided in §5 about the latter , which are in fact helpful for deciding superpixel centers . The above properties and their proofs are crucial for the validity of the proposed boundary representation and also to go from one representation to another for inference and visualization . Vector Transform and the Distance Transform . In essence , the normalized vector field proposed in equation ( 1 ) is another representation of the distance transform . Let φ ( x ) ∈ R+ define the distance transform , then the vector field v ( x ) in equation ( 1 ) can be obtained by the following partial derivatives ( Osher & Sethian , 1988 ; Osher & Fedkiw , 2003 ) : v ( x ) = −∇φ ( x ) . ( 4 ) One can optimize a given network by minimizing the loss on the distance transform ( DT ) or SDF ( Dapogny & Frey , 2012 ; Caliva et al. , 2019 ; Park et al. , 2019 ) instead of using the loss on the normalized vector field . Compared to the binary mask , the Vector transform ( VT ) , DT and SDF have an added advantage that they are sensitive to small topological changes . SDF on the other hand , does not support overlapping and open surfaces and is not easily adaptable to the image boundary problem . However , there are several reasons which make DT unsuitable for learning boundaries . During training , when the distance field is close to 0 i.e. , around the boundaries , any loss on the DT or SDF loses importance . Apart from the convergence problems of DT with gradient descent Osher & Fedkiw ( 2003 ) , DT is also hard to localize by thresholding under noise compared to the SDF and VT . In SDF , localizing the surface amounts to finding the zero crossings and in VT , the divergence measure in equation ( 2 ) provides an extremely sharp contrast making equation ( 3 ) trivial to solve . These differences in the thresholding problem can be seen in Fig . 5 in Appendix B and the experimental results . Additionally , despite reducing the class imbalance compared to binary boundary prediction , DT has an implicit bias to the weighted average of its typical range . On the other hand , a normalized vector field from VT equation ( 1 ) is sensitive to the topology similar to a distance field while also being localizable and sensitive at the boundaries , as shown in Fig . 1 . | The paper proposes a method to detect boundaries in an image by interpreting boundaries as 1-D surfaces and formulating a one-to-one vector transformation function. The paper provides theoretical justification of the vector transformation representation. Experimental results show the effectiveness on publicly available datasets. | SP:b0d6a0501228f45c6c9076c61d6fc0b3928a97aa |
Zero Pixel Directional Boundary by Vector Transform | 1 INTRODUCTION . Boundaries are considered to be one of the important interpretable visual cues that can describe both the low-level image characteristics as well as high-level semantics of the image . Human vision uses occluding contours and boundaries to interpret unseen or seen objects and classes . Because boundaries are an important visual cue for image understanding , they are also exploited as priors in several vision tasks ( Zhu et al. , 2020 ; Kim et al. , 2021 ; Hatamizadeh et al. , 2019 ; Revaud et al. , 2015 ; Cashman & Fitzgibbon , 2012 ) . Some key works on contours ( Cootes et al. , 2001 ; Matthews & Baker , 2004 ; Kass et al. , 1988 ) have greatly impacted early research in computer vision . Although the advent of deep learning and end-to-end learning has somewhat shifted the focus away from interpretable visual cues , boundary discovery still remains an important task in computer vision . Supervised deep learning has greatly transformed problems such as object detection and segmentation ( Redmon et al. , 2016 ; Chen et al. , 2017 ; Cheng et al. , 2020 ) by redefining the problem ( Kirillov et al. , 2019 ) , using high-quality datasets ( Cordts et al. , 2016 ; Neuhold et al. , 2017 ) and better network architectures ( Cheng et al. , 2020 ; 2021 ; Wang et al. , 2021b ) . Boundary detection , however , has seen a rather modest share of such progress . Although , modern deeply learned methods ( Xie & Tu , 2015 ; Liu et al. , 2017 ; Maninis et al. , 2017 ) provide better accuracy and the possibility to learn only the high-level boundaries , a particularly elusive goal in learned boundary detection has been the so-called crisp boundaries ( Isola et al. , 2014 ; Wang et al. , 2018 ; Deng et al. , 2018 ) . The formulation of boundary detection as a binary segmentation task naturally introduces class imbalance , which makes detecting pixel thin sharp boundaries extremely difficult . Arguably a majority of recent methods in boundary detection are proposed in order to tackle the aforementioned issue . Many methods address the lack of ‘ crispness ’ by fusing high-resolution features with the middleand high-level features ( Xie & Tu , 2015 ; Liu et al. , 2017 ) . Such a strategy has been successful in other dense prediction tasks ( Ronneberger et al. , 2015 ) as well . Others propose different loss functions ( Kokkinos , 2016 ; Deng et al. , 2018 ; Kervadec et al. , 2019 ) to address class imbalance . Despite the improvements , we identify two issues regarding crisp boundary detection . The first is that the evaluation protocol ( Martin et al. , 2004 ) does not necessarily encourage crisp detection , as the quantification is done after Non-Maximal Suppression ( NMS ) . Such an evaluation may be misleading when the network outputs need to be used at training time for other high-level tasks , e.g. , segmentation ( Kim et al. , 2021 ) . Second , the current losses ( Kokkinos , 2016 ; Xie & Tu , 2015 ; Ma et al. , 2021 ) push for edges as crisp as the ground-truth rather than as crisp as possible . This is particularly harmful since many boundary detection datasets ( Arbelaez et al. , 2010 ; Silberman et al. , 2012 ) contain ambiguous boundaries or inconsistently thick boundaries . In this paper , we take a different perspective on boundary detection . Boundaries are formed where visual features change , popularly referred to as the differential representation ( Boykov et al. , 2006 ; Kervadec et al. , 2019 ) . Such an assumption treats the boundaries as a set of 1-D surfaces embedded in a continuous 2D space , implying they do not have any thickness . Many previous energyminimization based approaches ( Chan & Zhu , 2005 ; Chan & Vese , 2001 ; Paragios et al. , 2002 ; Boykov et al. , 2006 ; Ma & Manjunath , 2000 ) and a few current methods ( Kervadec et al. , 2019 ) tackle boundaries in a similar way . Level-set methods ( Chan & Zhu , 2005 ; Boykov et al. , 2006 ; Ma & Manjunath , 2000 ) consider boundaries as the level-set of a continuous function of the image . Specifically , ( Ma & Manjunath , 2000 ) defines the energy function related to the distance and direction of the boundary at each pixel and extracts the directional normals at the boundary using such an energy . Inspired by such works and also recent works on 3D implicit functions ( Tancik et al. , 2020 ; Sitzmann et al. , 2020 ; Mildenhall et al. , 2020 ) , we represent boundaries via a field of unit vectors defined at each pixel , pointing towards the closest boundary surface . The proposed vector field representation naturally solves class imbalance . In distance transforms , vector fields are considered incomplete euclidean transforms ( Osher & Sethian , 1988 ) , equal to the Jacobian of the signed distance field . The vector field we use is in fact the Jacobian of the positive distance field . In contrast to distance fields , it provides high sensitivity at the boundaries and is easily localizable . We demonstrate the equivalence of the normal field to the surface contour representation using the level set of the normal field ’ s divergence , providing infinitely sharp boundaries . Owing to the zero-thickness , we refer to our result as the zero-pixel boundary . Our method is virtually hyper-parameter free at training and test time , and can provide zero-pixel thick boundaries at training time . In order to evaluate the boundaries using the surface interpretation , we also advocate the use of surface distances including the average symmetric surface distance ( assd ) metric that is less prone to class imbalance and variable boundary thickness in the ground-truth ( Kervadec et al. , 2019 ) . Such metrics are very popular in biomedical image segmentation ( Yeghiazaryan & Voiculescu , 2018 ) . We show significant improvements in all metrics using our boundary representation when compared to the various combinations of Dice ( Dice , 1945 ) and cross-entropy losses in several large datasets . 2 RELATED WORK . There is a rich history of boundary detection in computer vision . Previous work on boundaries showed a diversity of definitions and approaches . We differentiate them based on two different interpretations of boundaries : i.e. , i ) a boundary is a separation between two or more image regions with different visual features and ii ) a boundary is a thin group of pixels belonging to a specific class . It should be noted that most modern methods fall under the second category . Perhaps the most notable representatives of the first strand are the energy-based segmentation methods ( Chan & Vese , 2001 ; Comaniciu & Meer , 2002 ; Boykov et al. , 2006 ; Grady , 2006 ; Ma & Manjunath , 2000 ) . These methods relied on hand-crafted features and various optimization strategies to compute the low-level boundaries . In particular , Ma & Manjunath ( 2000 ) compute the normal vectors of the low-level boundaries from an energy function , without looking into an equivalent learnable representation . Graph-based segmentation methods ( Shi & Malik , 2000 ; Felzenszwalb & Huttenlocher , 2004 ; Cheng et al. , 2016 ) construct a graph from the image and cut the graph to obtain non-overlapping regions whose boundaries are viewed as image boundaries . A few deep learning methods followed a similar approach ( Wang et al. , 2021a ; Kervadec et al. , 2019 ) . Despite the advantage of the definition , current representations in this category are hard to adapt to generic boundaries and a compact multi-boundary representation with good performance remains lacking . A larger corpus of work utilizes pixel-wise image features to decide whether pixels belong to a ‘ boundary ’ class . They form our category ii ) methods . Early methods utilize various filter operators to detect discontinuities in image intensities or colors ( Canny , 1986 ; Sobel , 1972 ) . Learning based methods substantially boost the edge detection performance by classifying handcrafted features ( Konishi et al. , 2003 ; Martin et al. , 2004 ; Arbelaez et al. , 2010 ; Dollár & Zitnick , 2013 ; Hallman & Fowlkes , 2015 ) . Modern deep neural network ( DNN ) methods have further improved this field by learning powerful feature representations , particularly high-level semantic information ( Shen et al. , 2015 ; Bertasius et al. , 2015 ; Xie & Tu , 2015 ; Liu et al. , 2017 ; Wang et al. , 2018 ; Deng et al. , 2018 ; He et al. , 2019 ) . Yang et al . ( 2016 ) leveraged the powerful deep features to detect only object boundaries . Others try to simultaneously detect edges and predict the semantic class of each edge point , so-called semantic edge detection ( Hariharan et al. , 2011 ; Yu et al. , 2017 ; Liu et al. , 2018 ; Yu et al. , 2018 ) . On the other hand , classifying pixels as boundary class introduces class imbalance during training . A common counter-strategy is to use a weighted cross-entropy loss giving the non-boundary pixels a small weight and the boundary class a large weight ( Xie & Tu , 2015 ; Liu et al. , 2017 ; He et al. , 2019 ) . Yet , despite an improvement over regular cross-entropy , it does not solve the problem . To thin the boundaries , Non-Maximal Suppression ( NMS ) is usually adopted . Such methods may be harmful when directly integrated with higher-level tasks such as segmentation ( Kim et al. , 2021 ) . The Dice loss ( Dice , 1945 ) was thus advocated to generate crisp boundaries before NMS ( Deng et al. , 2018 ) , but it still produces several pixel thick boundaries and suffers more from missed predictions . Variations of the Dice loss ( Shit et al. , 2021 ) have been proposed to counter the missed detections . However , the right approach still depends on the downstream tasks ( Zhu et al. , 2020 ; Kim et al. , 2021 ; Hatamizadeh et al. , 2019 ; Shit et al. , 2021 ) and in either case a careful selection of training as well as testing hyper-parameters is required . We provide an alternative approach , motivated by the class imbalance , while having no sensitive hyper-parameter . 3 BOUNDARY TRANSFORM AND REPRESENTATION . In this section , we first discuss the surface representation of boundaries as a normal vector field transform and prove its relevant properties . Our boundary representation is inspired by recent work on implicit neural 3D surface representations ( Park et al. , 2019 ; Mescheder et al. , 2019 ) , energybased methods on edge detection ( Ma & Manjunath , 2000 ; Boykov et al. , 2006 ) and distance transforms ( Osher & Sethian , 1988 ) . In 3D surface representations , a Signed Distance Function ( SDF ) ( Osher & Sethian , 1988 ) or occupancy map ( Mescheder et al. , 2019 ) is used as representation . We instead propose a unit vector field from every point to the closest boundary . This choice is motivated by the high sensitivity and richer boundary context provided by the unit vector field , as shown in our experimental results in § 5 . Fig . 1 shows our vector field representation of the ground-truth , with predictions using a standard quiver plot on a sub-sampled set of points . We assume a continuous boundary image domain Ω ⊂ R2 with the set of boundary points { x′ } = Π ⊂ Ω . Each point is denoted as a 2-vector x = ( x , y ) ∈ R2 . In order to encode boundary properties on the whole image , we compute a signed x and y distance field separately and finally encode only the direction . The result is a unit vector field that represents the boundary . We can express our boundary representation by the following transform for any point x ∈ Ω : fx ( x ) = inf x′∈Π dx ( x , x ′ ) , fy ( x ) = inf x′∈Π dy ( x , x ′ ) , v ( x ) = f ( x ) ‖f ( x ) ‖2 , if ‖f ( x ) ‖2 6= 0 , otherwise n , n = lim dx→0+ f ( x ) ‖f ( x ) ‖2 . ( 1 ) Equation ( 1 ) defines the transform as a function v ( x ) : Ω→ R2 going from the boundary Π to a field representation . Here , dx and dy are operators that give the signed x and y difference , respectively . Similarly , fx and fy are the x and y components of the field f. Note that we choose the field vector arbitrarily among the two possible values at the boundary using the positive dx distance . We note the following properties of the vector field v. Property 3.1 The vector field v ( x ) is equal to the unit normal field at the boundary . Proof This is a well known result ( Osher & Fedkiw , 2003 ) and can be proved easily ( see equation ( 2.4 ) in the reference ) . The fact that we forcefully choose one normal over its negative directional normal at the boundary points does not affect the statement . Property 3.2 Given a vector field representation v ( x ) of a boundary , one can obtain the binary boundary representation by considering the following transform : g ( x ) = div v ( x ) . ( 2 ) The original boundary set Π can then be found by taking the zero level set of g ( x ) + 2 , i.e. , Π = L0 ( g + 2 ) . ( 3 ) Proof In the infinitesimal neighborhood of the boundary points , using property 3.1 , the vector field is normal to the boundary , provided that good approximate normals can be obtained from equation ( 1 ) . As the neighborhood size approaches zero , the tangential vector components approach zero around a point for a continuous boundary segment . Thus , around such an infinitesimal neighborhood , the normal fields pointing in opposite direction will subtract perfectly , creating a divergence flow of -2 and around 0 or positive away from boundaries . Strictly speaking the result holds only for piece-wise smooth surfaces ( Osher & Fedkiw , 2003 ) , with lower than -2 divergence possible at discontinuous surface points . Property 3.3 The relation is one-to-one between the binary boundary representation and the proposed vector field representation in a continuous domain . Proof This property is the result of equation ( 1 ) , for the forward transform and equation ( 3 ) for the inverse transform , providing a one-to-one relation . Note that the vector field transform as defined in equation ( 1 ) has to correct for two different kinds of indeterminate states . The first is on the boundary , that is solved by using the right hand limit so that one of the two opposite directions is chosen consistently . The second is when the infimum operation in equation ( 1 ) produces two or more closest points , corrected by choosing any one of the points for the infimum . The vector fields around such points flip directions creating a positive divergence as shown in Fig . 2 . More discussions are provided in §5 about the latter , which are in fact helpful for deciding superpixel centers . The above properties and their proofs are crucial for the validity of the proposed boundary representation and also to go from one representation to another for inference and visualization . Vector Transform and the Distance Transform . In essence , the normalized vector field proposed in equation ( 1 ) is another representation of the distance transform . Let φ ( x ) ∈ R+ define the distance transform , then the vector field v ( x ) in equation ( 1 ) can be obtained by the following partial derivatives ( Osher & Sethian , 1988 ; Osher & Fedkiw , 2003 ) : v ( x ) = −∇φ ( x ) . ( 4 ) One can optimize a given network by minimizing the loss on the distance transform ( DT ) or SDF ( Dapogny & Frey , 2012 ; Caliva et al. , 2019 ; Park et al. , 2019 ) instead of using the loss on the normalized vector field . Compared to the binary mask , the Vector transform ( VT ) , DT and SDF have an added advantage that they are sensitive to small topological changes . SDF on the other hand , does not support overlapping and open surfaces and is not easily adaptable to the image boundary problem . However , there are several reasons which make DT unsuitable for learning boundaries . During training , when the distance field is close to 0 i.e. , around the boundaries , any loss on the DT or SDF loses importance . Apart from the convergence problems of DT with gradient descent Osher & Fedkiw ( 2003 ) , DT is also hard to localize by thresholding under noise compared to the SDF and VT . In SDF , localizing the surface amounts to finding the zero crossings and in VT , the divergence measure in equation ( 2 ) provides an extremely sharp contrast making equation ( 3 ) trivial to solve . These differences in the thresholding problem can be seen in Fig . 5 in Appendix B and the experimental results . Additionally , despite reducing the class imbalance compared to binary boundary prediction , DT has an implicit bias to the weighted average of its typical range . On the other hand , a normalized vector field from VT equation ( 1 ) is sensitive to the topology similar to a distance field while also being localizable and sensitive at the boundaries , as shown in Fig . 1 . | This paper is about learning-based edge detection. The authors proposed a new loss function for end-to-end edge detection to overcome the label imbalance and edge thickness problems. The loss function is based on the vector transform field, which is closely related to distance transform. The final edge map can be easily obtained by thresholding the divergence of the predicted vector transform field. The authors compared the proposed VT loss with the dice loss, the combination of dice loss and cross entropy loss, weighted cross entropy, and the distance transform based loss, on Cityscapes, Mapillary Vistas, Synthia, and BSDS500 datasets. The experiment results show the promising performance of the proposed VT loss on the edge detection task. | SP:b0d6a0501228f45c6c9076c61d6fc0b3928a97aa |
Zero Pixel Directional Boundary by Vector Transform | 1 INTRODUCTION . Boundaries are considered to be one of the important interpretable visual cues that can describe both the low-level image characteristics as well as high-level semantics of the image . Human vision uses occluding contours and boundaries to interpret unseen or seen objects and classes . Because boundaries are an important visual cue for image understanding , they are also exploited as priors in several vision tasks ( Zhu et al. , 2020 ; Kim et al. , 2021 ; Hatamizadeh et al. , 2019 ; Revaud et al. , 2015 ; Cashman & Fitzgibbon , 2012 ) . Some key works on contours ( Cootes et al. , 2001 ; Matthews & Baker , 2004 ; Kass et al. , 1988 ) have greatly impacted early research in computer vision . Although the advent of deep learning and end-to-end learning has somewhat shifted the focus away from interpretable visual cues , boundary discovery still remains an important task in computer vision . Supervised deep learning has greatly transformed problems such as object detection and segmentation ( Redmon et al. , 2016 ; Chen et al. , 2017 ; Cheng et al. , 2020 ) by redefining the problem ( Kirillov et al. , 2019 ) , using high-quality datasets ( Cordts et al. , 2016 ; Neuhold et al. , 2017 ) and better network architectures ( Cheng et al. , 2020 ; 2021 ; Wang et al. , 2021b ) . Boundary detection , however , has seen a rather modest share of such progress . Although , modern deeply learned methods ( Xie & Tu , 2015 ; Liu et al. , 2017 ; Maninis et al. , 2017 ) provide better accuracy and the possibility to learn only the high-level boundaries , a particularly elusive goal in learned boundary detection has been the so-called crisp boundaries ( Isola et al. , 2014 ; Wang et al. , 2018 ; Deng et al. , 2018 ) . The formulation of boundary detection as a binary segmentation task naturally introduces class imbalance , which makes detecting pixel thin sharp boundaries extremely difficult . Arguably a majority of recent methods in boundary detection are proposed in order to tackle the aforementioned issue . Many methods address the lack of ‘ crispness ’ by fusing high-resolution features with the middleand high-level features ( Xie & Tu , 2015 ; Liu et al. , 2017 ) . Such a strategy has been successful in other dense prediction tasks ( Ronneberger et al. , 2015 ) as well . Others propose different loss functions ( Kokkinos , 2016 ; Deng et al. , 2018 ; Kervadec et al. , 2019 ) to address class imbalance . Despite the improvements , we identify two issues regarding crisp boundary detection . The first is that the evaluation protocol ( Martin et al. , 2004 ) does not necessarily encourage crisp detection , as the quantification is done after Non-Maximal Suppression ( NMS ) . Such an evaluation may be misleading when the network outputs need to be used at training time for other high-level tasks , e.g. , segmentation ( Kim et al. , 2021 ) . Second , the current losses ( Kokkinos , 2016 ; Xie & Tu , 2015 ; Ma et al. , 2021 ) push for edges as crisp as the ground-truth rather than as crisp as possible . This is particularly harmful since many boundary detection datasets ( Arbelaez et al. , 2010 ; Silberman et al. , 2012 ) contain ambiguous boundaries or inconsistently thick boundaries . In this paper , we take a different perspective on boundary detection . Boundaries are formed where visual features change , popularly referred to as the differential representation ( Boykov et al. , 2006 ; Kervadec et al. , 2019 ) . Such an assumption treats the boundaries as a set of 1-D surfaces embedded in a continuous 2D space , implying they do not have any thickness . Many previous energyminimization based approaches ( Chan & Zhu , 2005 ; Chan & Vese , 2001 ; Paragios et al. , 2002 ; Boykov et al. , 2006 ; Ma & Manjunath , 2000 ) and a few current methods ( Kervadec et al. , 2019 ) tackle boundaries in a similar way . Level-set methods ( Chan & Zhu , 2005 ; Boykov et al. , 2006 ; Ma & Manjunath , 2000 ) consider boundaries as the level-set of a continuous function of the image . Specifically , ( Ma & Manjunath , 2000 ) defines the energy function related to the distance and direction of the boundary at each pixel and extracts the directional normals at the boundary using such an energy . Inspired by such works and also recent works on 3D implicit functions ( Tancik et al. , 2020 ; Sitzmann et al. , 2020 ; Mildenhall et al. , 2020 ) , we represent boundaries via a field of unit vectors defined at each pixel , pointing towards the closest boundary surface . The proposed vector field representation naturally solves class imbalance . In distance transforms , vector fields are considered incomplete euclidean transforms ( Osher & Sethian , 1988 ) , equal to the Jacobian of the signed distance field . The vector field we use is in fact the Jacobian of the positive distance field . In contrast to distance fields , it provides high sensitivity at the boundaries and is easily localizable . We demonstrate the equivalence of the normal field to the surface contour representation using the level set of the normal field ’ s divergence , providing infinitely sharp boundaries . Owing to the zero-thickness , we refer to our result as the zero-pixel boundary . Our method is virtually hyper-parameter free at training and test time , and can provide zero-pixel thick boundaries at training time . In order to evaluate the boundaries using the surface interpretation , we also advocate the use of surface distances including the average symmetric surface distance ( assd ) metric that is less prone to class imbalance and variable boundary thickness in the ground-truth ( Kervadec et al. , 2019 ) . Such metrics are very popular in biomedical image segmentation ( Yeghiazaryan & Voiculescu , 2018 ) . We show significant improvements in all metrics using our boundary representation when compared to the various combinations of Dice ( Dice , 1945 ) and cross-entropy losses in several large datasets . 2 RELATED WORK . There is a rich history of boundary detection in computer vision . Previous work on boundaries showed a diversity of definitions and approaches . We differentiate them based on two different interpretations of boundaries : i.e. , i ) a boundary is a separation between two or more image regions with different visual features and ii ) a boundary is a thin group of pixels belonging to a specific class . It should be noted that most modern methods fall under the second category . Perhaps the most notable representatives of the first strand are the energy-based segmentation methods ( Chan & Vese , 2001 ; Comaniciu & Meer , 2002 ; Boykov et al. , 2006 ; Grady , 2006 ; Ma & Manjunath , 2000 ) . These methods relied on hand-crafted features and various optimization strategies to compute the low-level boundaries . In particular , Ma & Manjunath ( 2000 ) compute the normal vectors of the low-level boundaries from an energy function , without looking into an equivalent learnable representation . Graph-based segmentation methods ( Shi & Malik , 2000 ; Felzenszwalb & Huttenlocher , 2004 ; Cheng et al. , 2016 ) construct a graph from the image and cut the graph to obtain non-overlapping regions whose boundaries are viewed as image boundaries . A few deep learning methods followed a similar approach ( Wang et al. , 2021a ; Kervadec et al. , 2019 ) . Despite the advantage of the definition , current representations in this category are hard to adapt to generic boundaries and a compact multi-boundary representation with good performance remains lacking . A larger corpus of work utilizes pixel-wise image features to decide whether pixels belong to a ‘ boundary ’ class . They form our category ii ) methods . Early methods utilize various filter operators to detect discontinuities in image intensities or colors ( Canny , 1986 ; Sobel , 1972 ) . Learning based methods substantially boost the edge detection performance by classifying handcrafted features ( Konishi et al. , 2003 ; Martin et al. , 2004 ; Arbelaez et al. , 2010 ; Dollár & Zitnick , 2013 ; Hallman & Fowlkes , 2015 ) . Modern deep neural network ( DNN ) methods have further improved this field by learning powerful feature representations , particularly high-level semantic information ( Shen et al. , 2015 ; Bertasius et al. , 2015 ; Xie & Tu , 2015 ; Liu et al. , 2017 ; Wang et al. , 2018 ; Deng et al. , 2018 ; He et al. , 2019 ) . Yang et al . ( 2016 ) leveraged the powerful deep features to detect only object boundaries . Others try to simultaneously detect edges and predict the semantic class of each edge point , so-called semantic edge detection ( Hariharan et al. , 2011 ; Yu et al. , 2017 ; Liu et al. , 2018 ; Yu et al. , 2018 ) . On the other hand , classifying pixels as boundary class introduces class imbalance during training . A common counter-strategy is to use a weighted cross-entropy loss giving the non-boundary pixels a small weight and the boundary class a large weight ( Xie & Tu , 2015 ; Liu et al. , 2017 ; He et al. , 2019 ) . Yet , despite an improvement over regular cross-entropy , it does not solve the problem . To thin the boundaries , Non-Maximal Suppression ( NMS ) is usually adopted . Such methods may be harmful when directly integrated with higher-level tasks such as segmentation ( Kim et al. , 2021 ) . The Dice loss ( Dice , 1945 ) was thus advocated to generate crisp boundaries before NMS ( Deng et al. , 2018 ) , but it still produces several pixel thick boundaries and suffers more from missed predictions . Variations of the Dice loss ( Shit et al. , 2021 ) have been proposed to counter the missed detections . However , the right approach still depends on the downstream tasks ( Zhu et al. , 2020 ; Kim et al. , 2021 ; Hatamizadeh et al. , 2019 ; Shit et al. , 2021 ) and in either case a careful selection of training as well as testing hyper-parameters is required . We provide an alternative approach , motivated by the class imbalance , while having no sensitive hyper-parameter . 3 BOUNDARY TRANSFORM AND REPRESENTATION . In this section , we first discuss the surface representation of boundaries as a normal vector field transform and prove its relevant properties . Our boundary representation is inspired by recent work on implicit neural 3D surface representations ( Park et al. , 2019 ; Mescheder et al. , 2019 ) , energybased methods on edge detection ( Ma & Manjunath , 2000 ; Boykov et al. , 2006 ) and distance transforms ( Osher & Sethian , 1988 ) . In 3D surface representations , a Signed Distance Function ( SDF ) ( Osher & Sethian , 1988 ) or occupancy map ( Mescheder et al. , 2019 ) is used as representation . We instead propose a unit vector field from every point to the closest boundary . This choice is motivated by the high sensitivity and richer boundary context provided by the unit vector field , as shown in our experimental results in § 5 . Fig . 1 shows our vector field representation of the ground-truth , with predictions using a standard quiver plot on a sub-sampled set of points . We assume a continuous boundary image domain Ω ⊂ R2 with the set of boundary points { x′ } = Π ⊂ Ω . Each point is denoted as a 2-vector x = ( x , y ) ∈ R2 . In order to encode boundary properties on the whole image , we compute a signed x and y distance field separately and finally encode only the direction . The result is a unit vector field that represents the boundary . We can express our boundary representation by the following transform for any point x ∈ Ω : fx ( x ) = inf x′∈Π dx ( x , x ′ ) , fy ( x ) = inf x′∈Π dy ( x , x ′ ) , v ( x ) = f ( x ) ‖f ( x ) ‖2 , if ‖f ( x ) ‖2 6= 0 , otherwise n , n = lim dx→0+ f ( x ) ‖f ( x ) ‖2 . ( 1 ) Equation ( 1 ) defines the transform as a function v ( x ) : Ω→ R2 going from the boundary Π to a field representation . Here , dx and dy are operators that give the signed x and y difference , respectively . Similarly , fx and fy are the x and y components of the field f. Note that we choose the field vector arbitrarily among the two possible values at the boundary using the positive dx distance . We note the following properties of the vector field v. Property 3.1 The vector field v ( x ) is equal to the unit normal field at the boundary . Proof This is a well known result ( Osher & Fedkiw , 2003 ) and can be proved easily ( see equation ( 2.4 ) in the reference ) . The fact that we forcefully choose one normal over its negative directional normal at the boundary points does not affect the statement . Property 3.2 Given a vector field representation v ( x ) of a boundary , one can obtain the binary boundary representation by considering the following transform : g ( x ) = div v ( x ) . ( 2 ) The original boundary set Π can then be found by taking the zero level set of g ( x ) + 2 , i.e. , Π = L0 ( g + 2 ) . ( 3 ) Proof In the infinitesimal neighborhood of the boundary points , using property 3.1 , the vector field is normal to the boundary , provided that good approximate normals can be obtained from equation ( 1 ) . As the neighborhood size approaches zero , the tangential vector components approach zero around a point for a continuous boundary segment . Thus , around such an infinitesimal neighborhood , the normal fields pointing in opposite direction will subtract perfectly , creating a divergence flow of -2 and around 0 or positive away from boundaries . Strictly speaking the result holds only for piece-wise smooth surfaces ( Osher & Fedkiw , 2003 ) , with lower than -2 divergence possible at discontinuous surface points . Property 3.3 The relation is one-to-one between the binary boundary representation and the proposed vector field representation in a continuous domain . Proof This property is the result of equation ( 1 ) , for the forward transform and equation ( 3 ) for the inverse transform , providing a one-to-one relation . Note that the vector field transform as defined in equation ( 1 ) has to correct for two different kinds of indeterminate states . The first is on the boundary , that is solved by using the right hand limit so that one of the two opposite directions is chosen consistently . The second is when the infimum operation in equation ( 1 ) produces two or more closest points , corrected by choosing any one of the points for the infimum . The vector fields around such points flip directions creating a positive divergence as shown in Fig . 2 . More discussions are provided in §5 about the latter , which are in fact helpful for deciding superpixel centers . The above properties and their proofs are crucial for the validity of the proposed boundary representation and also to go from one representation to another for inference and visualization . Vector Transform and the Distance Transform . In essence , the normalized vector field proposed in equation ( 1 ) is another representation of the distance transform . Let φ ( x ) ∈ R+ define the distance transform , then the vector field v ( x ) in equation ( 1 ) can be obtained by the following partial derivatives ( Osher & Sethian , 1988 ; Osher & Fedkiw , 2003 ) : v ( x ) = −∇φ ( x ) . ( 4 ) One can optimize a given network by minimizing the loss on the distance transform ( DT ) or SDF ( Dapogny & Frey , 2012 ; Caliva et al. , 2019 ; Park et al. , 2019 ) instead of using the loss on the normalized vector field . Compared to the binary mask , the Vector transform ( VT ) , DT and SDF have an added advantage that they are sensitive to small topological changes . SDF on the other hand , does not support overlapping and open surfaces and is not easily adaptable to the image boundary problem . However , there are several reasons which make DT unsuitable for learning boundaries . During training , when the distance field is close to 0 i.e. , around the boundaries , any loss on the DT or SDF loses importance . Apart from the convergence problems of DT with gradient descent Osher & Fedkiw ( 2003 ) , DT is also hard to localize by thresholding under noise compared to the SDF and VT . In SDF , localizing the surface amounts to finding the zero crossings and in VT , the divergence measure in equation ( 2 ) provides an extremely sharp contrast making equation ( 3 ) trivial to solve . These differences in the thresholding problem can be seen in Fig . 5 in Appendix B and the experimental results . Additionally , despite reducing the class imbalance compared to binary boundary prediction , DT has an implicit bias to the weighted average of its typical range . On the other hand , a normalized vector field from VT equation ( 1 ) is sensitive to the topology similar to a distance field while also being localizable and sensitive at the boundaries , as shown in Fig . 1 . | This paper proposes treating boundaries as 1D surfaces using vector transformations in Equation (1). The paper then proves a 1-to-1 relationship between the vector field representation and a binary boundary map. The paper further learns the vector representation into an NN by using a variant of HR-Net that outputs a dense two-channel image with x and y component of the vector field and train it with Mean Sqaured Error loss. Since the predicted boundaries using this method can be infinitely thin, so the paper proposes using average surface distances to evalute the performance, and shows improved results of their method comparing against three other losses using the same NN architecture. | SP:b0d6a0501228f45c6c9076c61d6fc0b3928a97aa |
Relating transformers to models and neural representations of the hippocampal formation | 1 INTRODUCTION . The last ten years have seen dramatic developments using deep neural networks , from computer vision ( Krizhevsky et al. , 2012 ) to natural language processing and beyond ( Vaswani et al. , 2017 ) . During the same time , neuroscientists have used these tools to build models of the brain that explain neural recordings at a precision not seen before ( Yamins et al. , 2014 ; Banino et al. , 2018 ; Whittington et al. , 2020 ) . For example , representations from convolutional neural networks ( Lecun et al. , 1998 ) predict neurons in visual and inferior temporal cortex ( Yamins et al. , 2014 ; Khaligh-Razavi & Kriegeskorte , 2014 ) , representations from transformer neural networks ( Vaswani et al. , 2017 ) predict brain representations in language areas ( Schrimpf et al. , 2020 ) , and lastly recurrent neural networks ( Cueva & Wei , 2018 ; Banino et al. , 2018 ; Sorscher et al. , 2019 ) have been shown to recapitulate grid cells ( Hafting et al. , 2005 ) from medial entorhinal cortex . Being able to use models from machine learning to predict brain representations provides a deeper understanding into the mechanistic computations of the respective brain areas , and offers deeper insight into the nature of the models . As well as using off-the-shelf machine learning models , neuroscience has developed bespoke deep learning models ( mixing together recurrent networks with memory networks ) that learn neural representations that mimic the exquisite spatial representations found in hippocampus and entorhinal cortex ( Whittington et al. , 2020 ; Uria et al. , 2020 ) , including grid cells ( Hafting et al. , 2005 ) , band cells ( Krupic et al. , 2012 ) , and place cells ( O ’ Keefe & Dostrovsky , 1971 ) . However , since these models are bespoke , it is not clear whether they , and by implication the hippocampal architecture , are capable of the general purpose computations of the kind studied in machine learning . In this work we 1 ) show that transformers ( with a little twist ) recapitulate spatial representations found in the brain ; 2 ) show a close mathematical relationship of this transformer to current hippocampal models from neuroscience ( with a focus on Whittington et al . ( 2020 ) though the same is true for Uria et al . ( 2020 ) ) ; 3 ) offer a novel take on the computational role of the hippocampus , and an instantiation of hippocampal indexing theory ( Teyler & Rudy , 2007 ) ; 4 ) offer novel insights on the role of positional encodings in transformers . 5 ) discuss whether similar computational principles might apply to broader cognitive domains , such as language , either in the hippocampal formation or in neocortical circuits . Note , we are not saying the brain is closely related to transformers because it learns the same neural representations , instead we are saying the relationship is close because we have shown a mathe- matical relationship between transformers and carefully formulated neuroscience models of the hippocampal formation . This relationship helps us get a better understanding of hippocampal models , it also suggests a new mechanism for place cells that would not be possible without this mathematical relationship , and finally it tells us something formal about position encodings in transformers . 2 TRANSFORMERS . Transformer Neural Networks ( Vaswani et al. , 2017 ) are highly successful machine learning algorithms . Originally developed for language , transformers perform well on other tasks that can be posed sequentially , such as mathematical understanding , logic problems ( Brown et al. , 2020 ) , and image processing ( Dosovitskiy et al. , 2020 ) . Transformers accept a set of observations ; X = { x1 , x2 , x3 , · · · , xT } ( xt could be a word embedding or image patch etc ) , and aim to predict missing elements of that set . The missing elements could be in the future , i.e . xt > T , or could be a missing part of a sentence or image , i.e . { x1 = the , x2 = cat , x3 = sat , x4 = ? , x5 = the , x6 = mat } . Self-attention . The core mechanism of transformers is self-attention . Self-attention allows each element to ‘ attend ’ to all other elements , and update itself accordingly . In the example data-set above , the 4th element ( ? ) could attend to the 2nd ( cat ) , 3rd ( sat ) , and 6th ( mat ) to understand it should be on . Formally , to attend to another element each element ( xt is a row vector ) emits a query ( qt = xtWq ) and compares it to other elements keys ( kτ = xtWk ) . Each element is then updated using yt = ∑ τ κ ( qt , kτ ) vτ , where κ ( qt , kτ ) is kernel describing the similarity of qt to kτ and vτ is the value computed by each element vτ = xtWv . Intuitively , the similarity measure κ ( qt , kτ ) places more emphasis on the elements that are relevant for prediction ; in this example , the keys may contain information about whether the word is a noun , verb or adjective , while the query may ‘ ask ’ for any elements that are nouns or verbs - elements that match this criteria ( large κ ( qt , kτ ) , i.e . cat , sat , mat ) are ‘ attended ’ to and therefore contribute more to the output yt . Typically , the similarity measure is a softmax i.e . κ ( qt , kτ ) = e βqt·kτ∑ τ′ e βqt·kτ′ . These equations can be succinctly expressed in matrix form , with all elements updated simultaneously : yt = softmax ( qtK T √ dk ) V → Y = softmax ( QK T √ dk ) V ( 1 ) Here Q , K , V are matrices with rows filled by qt , kt , vt respectively , and the softmax is taken independently for each row . After this update , each yt is then sent through a deep network ( fθ ( · · · ) ) typically consisting of residual ( He et al. , 2016 ) and layer-normalisation ( Ba et al. , 2016b ) layers to produce zt = fθ ( yt ) . Z is the output of the transformer which can then be used for prediction , or sent through subsequent transformer blocks . Position encodings . Self-attention is permutation invariant and so tells you nothing about order of the inputs . Should the data be sequential ( i.e . meaning depends on the order of elements , such as in language , or navigation as we will see later ! ) , it is necessary to additionally encode the position/ where x is in the sequence . This is typically done by adding a ‘ position encoding ’ that uniquely identifies each time-step ( et - typically sines and cosines ) to each input : xt ← xt+et . Alternatively the position embedding can be appended i.e . ht = [ xt , et ] , with self attention then performed using ht as input . 3 TRANSFORMERS LEARN ENTORHINAL REPRESENTATIONS . Here we show that transformers ( with a small modification ) recapitulate spatial representations - grid and band cells - when trained on tasks that require abstract spatial knowledge . Spatial understanding task . The task ( more detail in Appendix ) is to predict upcoming sensory observations xt+1 conditioned on taking an action at while moving around spatial environments ( Figure 1a ) . For example , after seeing { ( x1 = cat , a1 = North ) , ( x2 = dog , a2 = East ) , ( x3 = frog , a3 = South ) , ( x4 = pig , a4 = West ) , ( x5 = ? , a5 = · · · ) } , the aim is to predict x5 = cat . For simplicity , we treat sensory observations as one-hot vectors , thus the prediction problem is a classification problem . e0 x0 e1 x1 e2 x2 e3 ? y3 q0 k0 v0 q1 k1 v1 q2 k2 v2 q3 iWv iWq iWk Transformer ( with causal attention mask ) Input with position encoding Query Key , Value A B D B C A A D B B C A D A D C D C Environment 1 Environment 2 Observations ( x ) Actions ( a ) A B A ? C AAD ? D Recurrent connections W ( a0 ) W ( a1 ) W ( a2 ) With linear activations With ReLu activationsReal grid cell rate maps Cell rate maps Autocorrelations Cell rate maps Autocorrelations a b dc e f Band cells Figure 1 : ( a ) Sequence prediction in spatial navigation tasks test abstract spatial understanding since some sensory predictions can only be done by knowing ( generalising ) certain rules e.g . North + East + South + West = 0 or Parent + Sibling + Niece = 0 . Note , we use sequences drawn from much larger graphs . ( b ) Transformer with recurrent position encodings . ( c ) Real grid cell rate-maps ( Hafting et al. , 2005 ) . ( d-f ) Learned position embedding rate-maps ( i.e . average activity at each spatial location ; plots are spatially smoothed ) . ( d-e ) Resembling grid cells with ( e ) linear activation or ( e ) ReLu activation post transition . ( f ) Resembling band cells ( Krupic et al. , 2012 ) . When faced with an unseen stimulus-action pair ( e.g . x4 = pig , a4 = West above ; an action you have never taken at that stimulus before ) , successful prediction requires more than just remembering specific sequences of stimulus-action pairs ; knowledge of the rules of space must be known ; i.e . North + East + South + West = 0 allows prediction of x5 = cat . Crucially , such rules generalise to any 2D spaces and may therefore be transferred to aid prediction in entirely novel 2D environments . This is powerful , since unobserved relations between observed stimuli can be inferred in a zero-shot manner . However , these relational rules are not ‘ known ’ a priori and therefore must be learnt . We therefore train across multiple different spatial environments which share the same underlying 4-connected Euclidean structure ( Figure 1a ) - this means the model must learn and generalise the abstract structure of space to use for prediction in new environments . To perform on these tasks , the three modifications to the transformer are : 1 . Recall equation 1 ; yt = softmax ( qtK T √ dk ) V , where Q = HWq , K = HWk , V = HWv , and H is a matrix of inputs and position encodings ( i.e . its rows are ht = [ xt , et ] ) . We restrict these weight matrices such that queries ( Q ) and keys ( K ) are the same ; Q , K = EWe . We refer to this matrix as Ẽ . Thus the keys and queries only focus on position encodings . Meanwhile , values are exclusively dependent on the stimulus component of H i.e . V = XWx . We refer to this matrix as X̃ . yt = softmax ( qtK T √ dk ) V → yt = softmax ( ẽtẼ T √ dk ) X̃ ( 2 ) This is an extreme version of the realisation that , in transformers , best performance is when position encodings are used to compute keys and queries , but not values . 2 . We use causal transformers ; the key and value matrices contain the projected position encodings and sensory stimuli respectively at all previous time-steps ( i.e . e < t and x < t ) . This is equivalent to causal ‘ unmasking ’ as the agent wanders the environment accumulating new experiences ( not-yet-experienced stimulus-position pairs are inaccessible to the agent ) . Meanwhile the query at each time-point is the present positional encoding et . 3 . The position encodings are recurrently generated ( as in Wang et al . ( 2019 ) ; Liu et al . ( 2020 ) ) ; et+1 = σ ( etWa ) , where Wa is a learnable action-dependent weight matrix , and σ ( · · · ) is a non-linear activation function . This means that unlike traditional transformers , position encodings can be optimised and not the same for every sequence . It now becomes interesting to see what representations are learned . These modifications are sufficient to learn spatial representations , in the position encodings , that mimic representations observed in the brain ( Figure 1C ; see Appendix for model and training details ) . The rest of this paper now explains why this is not a surprising result ; namely we show that a transformer with recurrent positional encodings is closely related to current neuroscience models of the hippocampus and surrounding cortex ( Whittington et al. , 2020 ; Uria et al. , 2020 ) . Here we focus on the Tolman-Eichenbaum Machine ( TEM ) ( Whittington et al. , 2020 ) , though the same principles apply for Uria et al . ( 2020 ) . The critical points are : 1 ) the memory component of TEM can be viewed as a transformer selfattention , since the TEM memory network is analogous to a Hopfield network ( Hopfield , 1982 ) which have recently been shown to be closely related to transformers ( Ramsauer et al. , 2020 ) ; 2 ) TEM path integration ( see below ) can be viewed as a way to learn a position encoding . | The paper postulates a task that requires the ML model to capture the spatial nature of the task in order to perform well. The chosen ML method is a Transformer with positional encoder, where the encoding is not fixed, but is learnable, making it possible for the transformer to "choose" what kind of representation would it "prefer" for the task at hand. The Transformer is modified to act comparably to a neuroscientific model of hippocampal function (TEM). After the learning process the authors discover, that, when visualized, the representation that has emerged resembles the grid-cell pattern similar to the one that is being encoded by hippocampal place and grid cells. From there the works suggests the existence of a special relationship between the Transformer architecture and the mechanism by which hippocampus encodes spatial information. | SP:86e84595489ca0f10e58c0977135aa001db5166f |
Relating transformers to models and neural representations of the hippocampal formation | 1 INTRODUCTION . The last ten years have seen dramatic developments using deep neural networks , from computer vision ( Krizhevsky et al. , 2012 ) to natural language processing and beyond ( Vaswani et al. , 2017 ) . During the same time , neuroscientists have used these tools to build models of the brain that explain neural recordings at a precision not seen before ( Yamins et al. , 2014 ; Banino et al. , 2018 ; Whittington et al. , 2020 ) . For example , representations from convolutional neural networks ( Lecun et al. , 1998 ) predict neurons in visual and inferior temporal cortex ( Yamins et al. , 2014 ; Khaligh-Razavi & Kriegeskorte , 2014 ) , representations from transformer neural networks ( Vaswani et al. , 2017 ) predict brain representations in language areas ( Schrimpf et al. , 2020 ) , and lastly recurrent neural networks ( Cueva & Wei , 2018 ; Banino et al. , 2018 ; Sorscher et al. , 2019 ) have been shown to recapitulate grid cells ( Hafting et al. , 2005 ) from medial entorhinal cortex . Being able to use models from machine learning to predict brain representations provides a deeper understanding into the mechanistic computations of the respective brain areas , and offers deeper insight into the nature of the models . As well as using off-the-shelf machine learning models , neuroscience has developed bespoke deep learning models ( mixing together recurrent networks with memory networks ) that learn neural representations that mimic the exquisite spatial representations found in hippocampus and entorhinal cortex ( Whittington et al. , 2020 ; Uria et al. , 2020 ) , including grid cells ( Hafting et al. , 2005 ) , band cells ( Krupic et al. , 2012 ) , and place cells ( O ’ Keefe & Dostrovsky , 1971 ) . However , since these models are bespoke , it is not clear whether they , and by implication the hippocampal architecture , are capable of the general purpose computations of the kind studied in machine learning . In this work we 1 ) show that transformers ( with a little twist ) recapitulate spatial representations found in the brain ; 2 ) show a close mathematical relationship of this transformer to current hippocampal models from neuroscience ( with a focus on Whittington et al . ( 2020 ) though the same is true for Uria et al . ( 2020 ) ) ; 3 ) offer a novel take on the computational role of the hippocampus , and an instantiation of hippocampal indexing theory ( Teyler & Rudy , 2007 ) ; 4 ) offer novel insights on the role of positional encodings in transformers . 5 ) discuss whether similar computational principles might apply to broader cognitive domains , such as language , either in the hippocampal formation or in neocortical circuits . Note , we are not saying the brain is closely related to transformers because it learns the same neural representations , instead we are saying the relationship is close because we have shown a mathe- matical relationship between transformers and carefully formulated neuroscience models of the hippocampal formation . This relationship helps us get a better understanding of hippocampal models , it also suggests a new mechanism for place cells that would not be possible without this mathematical relationship , and finally it tells us something formal about position encodings in transformers . 2 TRANSFORMERS . Transformer Neural Networks ( Vaswani et al. , 2017 ) are highly successful machine learning algorithms . Originally developed for language , transformers perform well on other tasks that can be posed sequentially , such as mathematical understanding , logic problems ( Brown et al. , 2020 ) , and image processing ( Dosovitskiy et al. , 2020 ) . Transformers accept a set of observations ; X = { x1 , x2 , x3 , · · · , xT } ( xt could be a word embedding or image patch etc ) , and aim to predict missing elements of that set . The missing elements could be in the future , i.e . xt > T , or could be a missing part of a sentence or image , i.e . { x1 = the , x2 = cat , x3 = sat , x4 = ? , x5 = the , x6 = mat } . Self-attention . The core mechanism of transformers is self-attention . Self-attention allows each element to ‘ attend ’ to all other elements , and update itself accordingly . In the example data-set above , the 4th element ( ? ) could attend to the 2nd ( cat ) , 3rd ( sat ) , and 6th ( mat ) to understand it should be on . Formally , to attend to another element each element ( xt is a row vector ) emits a query ( qt = xtWq ) and compares it to other elements keys ( kτ = xtWk ) . Each element is then updated using yt = ∑ τ κ ( qt , kτ ) vτ , where κ ( qt , kτ ) is kernel describing the similarity of qt to kτ and vτ is the value computed by each element vτ = xtWv . Intuitively , the similarity measure κ ( qt , kτ ) places more emphasis on the elements that are relevant for prediction ; in this example , the keys may contain information about whether the word is a noun , verb or adjective , while the query may ‘ ask ’ for any elements that are nouns or verbs - elements that match this criteria ( large κ ( qt , kτ ) , i.e . cat , sat , mat ) are ‘ attended ’ to and therefore contribute more to the output yt . Typically , the similarity measure is a softmax i.e . κ ( qt , kτ ) = e βqt·kτ∑ τ′ e βqt·kτ′ . These equations can be succinctly expressed in matrix form , with all elements updated simultaneously : yt = softmax ( qtK T √ dk ) V → Y = softmax ( QK T √ dk ) V ( 1 ) Here Q , K , V are matrices with rows filled by qt , kt , vt respectively , and the softmax is taken independently for each row . After this update , each yt is then sent through a deep network ( fθ ( · · · ) ) typically consisting of residual ( He et al. , 2016 ) and layer-normalisation ( Ba et al. , 2016b ) layers to produce zt = fθ ( yt ) . Z is the output of the transformer which can then be used for prediction , or sent through subsequent transformer blocks . Position encodings . Self-attention is permutation invariant and so tells you nothing about order of the inputs . Should the data be sequential ( i.e . meaning depends on the order of elements , such as in language , or navigation as we will see later ! ) , it is necessary to additionally encode the position/ where x is in the sequence . This is typically done by adding a ‘ position encoding ’ that uniquely identifies each time-step ( et - typically sines and cosines ) to each input : xt ← xt+et . Alternatively the position embedding can be appended i.e . ht = [ xt , et ] , with self attention then performed using ht as input . 3 TRANSFORMERS LEARN ENTORHINAL REPRESENTATIONS . Here we show that transformers ( with a small modification ) recapitulate spatial representations - grid and band cells - when trained on tasks that require abstract spatial knowledge . Spatial understanding task . The task ( more detail in Appendix ) is to predict upcoming sensory observations xt+1 conditioned on taking an action at while moving around spatial environments ( Figure 1a ) . For example , after seeing { ( x1 = cat , a1 = North ) , ( x2 = dog , a2 = East ) , ( x3 = frog , a3 = South ) , ( x4 = pig , a4 = West ) , ( x5 = ? , a5 = · · · ) } , the aim is to predict x5 = cat . For simplicity , we treat sensory observations as one-hot vectors , thus the prediction problem is a classification problem . e0 x0 e1 x1 e2 x2 e3 ? y3 q0 k0 v0 q1 k1 v1 q2 k2 v2 q3 iWv iWq iWk Transformer ( with causal attention mask ) Input with position encoding Query Key , Value A B D B C A A D B B C A D A D C D C Environment 1 Environment 2 Observations ( x ) Actions ( a ) A B A ? C AAD ? D Recurrent connections W ( a0 ) W ( a1 ) W ( a2 ) With linear activations With ReLu activationsReal grid cell rate maps Cell rate maps Autocorrelations Cell rate maps Autocorrelations a b dc e f Band cells Figure 1 : ( a ) Sequence prediction in spatial navigation tasks test abstract spatial understanding since some sensory predictions can only be done by knowing ( generalising ) certain rules e.g . North + East + South + West = 0 or Parent + Sibling + Niece = 0 . Note , we use sequences drawn from much larger graphs . ( b ) Transformer with recurrent position encodings . ( c ) Real grid cell rate-maps ( Hafting et al. , 2005 ) . ( d-f ) Learned position embedding rate-maps ( i.e . average activity at each spatial location ; plots are spatially smoothed ) . ( d-e ) Resembling grid cells with ( e ) linear activation or ( e ) ReLu activation post transition . ( f ) Resembling band cells ( Krupic et al. , 2012 ) . When faced with an unseen stimulus-action pair ( e.g . x4 = pig , a4 = West above ; an action you have never taken at that stimulus before ) , successful prediction requires more than just remembering specific sequences of stimulus-action pairs ; knowledge of the rules of space must be known ; i.e . North + East + South + West = 0 allows prediction of x5 = cat . Crucially , such rules generalise to any 2D spaces and may therefore be transferred to aid prediction in entirely novel 2D environments . This is powerful , since unobserved relations between observed stimuli can be inferred in a zero-shot manner . However , these relational rules are not ‘ known ’ a priori and therefore must be learnt . We therefore train across multiple different spatial environments which share the same underlying 4-connected Euclidean structure ( Figure 1a ) - this means the model must learn and generalise the abstract structure of space to use for prediction in new environments . To perform on these tasks , the three modifications to the transformer are : 1 . Recall equation 1 ; yt = softmax ( qtK T √ dk ) V , where Q = HWq , K = HWk , V = HWv , and H is a matrix of inputs and position encodings ( i.e . its rows are ht = [ xt , et ] ) . We restrict these weight matrices such that queries ( Q ) and keys ( K ) are the same ; Q , K = EWe . We refer to this matrix as Ẽ . Thus the keys and queries only focus on position encodings . Meanwhile , values are exclusively dependent on the stimulus component of H i.e . V = XWx . We refer to this matrix as X̃ . yt = softmax ( qtK T √ dk ) V → yt = softmax ( ẽtẼ T √ dk ) X̃ ( 2 ) This is an extreme version of the realisation that , in transformers , best performance is when position encodings are used to compute keys and queries , but not values . 2 . We use causal transformers ; the key and value matrices contain the projected position encodings and sensory stimuli respectively at all previous time-steps ( i.e . e < t and x < t ) . This is equivalent to causal ‘ unmasking ’ as the agent wanders the environment accumulating new experiences ( not-yet-experienced stimulus-position pairs are inaccessible to the agent ) . Meanwhile the query at each time-point is the present positional encoding et . 3 . The position encodings are recurrently generated ( as in Wang et al . ( 2019 ) ; Liu et al . ( 2020 ) ) ; et+1 = σ ( etWa ) , where Wa is a learnable action-dependent weight matrix , and σ ( · · · ) is a non-linear activation function . This means that unlike traditional transformers , position encodings can be optimised and not the same for every sequence . It now becomes interesting to see what representations are learned . These modifications are sufficient to learn spatial representations , in the position encodings , that mimic representations observed in the brain ( Figure 1C ; see Appendix for model and training details ) . The rest of this paper now explains why this is not a surprising result ; namely we show that a transformer with recurrent positional encodings is closely related to current neuroscience models of the hippocampus and surrounding cortex ( Whittington et al. , 2020 ; Uria et al. , 2020 ) . Here we focus on the Tolman-Eichenbaum Machine ( TEM ) ( Whittington et al. , 2020 ) , though the same principles apply for Uria et al . ( 2020 ) . The critical points are : 1 ) the memory component of TEM can be viewed as a transformer selfattention , since the TEM memory network is analogous to a Hopfield network ( Hopfield , 1982 ) which have recently been shown to be closely related to transformers ( Ramsauer et al. , 2020 ) ; 2 ) TEM path integration ( see below ) can be viewed as a way to learn a position encoding . | This paper demonstrates emergence of grid and place cells in transformer architectures with positional encodings. Transformers are related to TEM models and Hopfield networks in computational neuroscience. A neurobiological model of TEM-transformers is presented, which contains feature neurons and memory neurons. It is shown that memory neurons resemble place cells in the hippocampus. Remapping phenomenon is also discussed. | SP:86e84595489ca0f10e58c0977135aa001db5166f |
Relating transformers to models and neural representations of the hippocampal formation | 1 INTRODUCTION . The last ten years have seen dramatic developments using deep neural networks , from computer vision ( Krizhevsky et al. , 2012 ) to natural language processing and beyond ( Vaswani et al. , 2017 ) . During the same time , neuroscientists have used these tools to build models of the brain that explain neural recordings at a precision not seen before ( Yamins et al. , 2014 ; Banino et al. , 2018 ; Whittington et al. , 2020 ) . For example , representations from convolutional neural networks ( Lecun et al. , 1998 ) predict neurons in visual and inferior temporal cortex ( Yamins et al. , 2014 ; Khaligh-Razavi & Kriegeskorte , 2014 ) , representations from transformer neural networks ( Vaswani et al. , 2017 ) predict brain representations in language areas ( Schrimpf et al. , 2020 ) , and lastly recurrent neural networks ( Cueva & Wei , 2018 ; Banino et al. , 2018 ; Sorscher et al. , 2019 ) have been shown to recapitulate grid cells ( Hafting et al. , 2005 ) from medial entorhinal cortex . Being able to use models from machine learning to predict brain representations provides a deeper understanding into the mechanistic computations of the respective brain areas , and offers deeper insight into the nature of the models . As well as using off-the-shelf machine learning models , neuroscience has developed bespoke deep learning models ( mixing together recurrent networks with memory networks ) that learn neural representations that mimic the exquisite spatial representations found in hippocampus and entorhinal cortex ( Whittington et al. , 2020 ; Uria et al. , 2020 ) , including grid cells ( Hafting et al. , 2005 ) , band cells ( Krupic et al. , 2012 ) , and place cells ( O ’ Keefe & Dostrovsky , 1971 ) . However , since these models are bespoke , it is not clear whether they , and by implication the hippocampal architecture , are capable of the general purpose computations of the kind studied in machine learning . In this work we 1 ) show that transformers ( with a little twist ) recapitulate spatial representations found in the brain ; 2 ) show a close mathematical relationship of this transformer to current hippocampal models from neuroscience ( with a focus on Whittington et al . ( 2020 ) though the same is true for Uria et al . ( 2020 ) ) ; 3 ) offer a novel take on the computational role of the hippocampus , and an instantiation of hippocampal indexing theory ( Teyler & Rudy , 2007 ) ; 4 ) offer novel insights on the role of positional encodings in transformers . 5 ) discuss whether similar computational principles might apply to broader cognitive domains , such as language , either in the hippocampal formation or in neocortical circuits . Note , we are not saying the brain is closely related to transformers because it learns the same neural representations , instead we are saying the relationship is close because we have shown a mathe- matical relationship between transformers and carefully formulated neuroscience models of the hippocampal formation . This relationship helps us get a better understanding of hippocampal models , it also suggests a new mechanism for place cells that would not be possible without this mathematical relationship , and finally it tells us something formal about position encodings in transformers . 2 TRANSFORMERS . Transformer Neural Networks ( Vaswani et al. , 2017 ) are highly successful machine learning algorithms . Originally developed for language , transformers perform well on other tasks that can be posed sequentially , such as mathematical understanding , logic problems ( Brown et al. , 2020 ) , and image processing ( Dosovitskiy et al. , 2020 ) . Transformers accept a set of observations ; X = { x1 , x2 , x3 , · · · , xT } ( xt could be a word embedding or image patch etc ) , and aim to predict missing elements of that set . The missing elements could be in the future , i.e . xt > T , or could be a missing part of a sentence or image , i.e . { x1 = the , x2 = cat , x3 = sat , x4 = ? , x5 = the , x6 = mat } . Self-attention . The core mechanism of transformers is self-attention . Self-attention allows each element to ‘ attend ’ to all other elements , and update itself accordingly . In the example data-set above , the 4th element ( ? ) could attend to the 2nd ( cat ) , 3rd ( sat ) , and 6th ( mat ) to understand it should be on . Formally , to attend to another element each element ( xt is a row vector ) emits a query ( qt = xtWq ) and compares it to other elements keys ( kτ = xtWk ) . Each element is then updated using yt = ∑ τ κ ( qt , kτ ) vτ , where κ ( qt , kτ ) is kernel describing the similarity of qt to kτ and vτ is the value computed by each element vτ = xtWv . Intuitively , the similarity measure κ ( qt , kτ ) places more emphasis on the elements that are relevant for prediction ; in this example , the keys may contain information about whether the word is a noun , verb or adjective , while the query may ‘ ask ’ for any elements that are nouns or verbs - elements that match this criteria ( large κ ( qt , kτ ) , i.e . cat , sat , mat ) are ‘ attended ’ to and therefore contribute more to the output yt . Typically , the similarity measure is a softmax i.e . κ ( qt , kτ ) = e βqt·kτ∑ τ′ e βqt·kτ′ . These equations can be succinctly expressed in matrix form , with all elements updated simultaneously : yt = softmax ( qtK T √ dk ) V → Y = softmax ( QK T √ dk ) V ( 1 ) Here Q , K , V are matrices with rows filled by qt , kt , vt respectively , and the softmax is taken independently for each row . After this update , each yt is then sent through a deep network ( fθ ( · · · ) ) typically consisting of residual ( He et al. , 2016 ) and layer-normalisation ( Ba et al. , 2016b ) layers to produce zt = fθ ( yt ) . Z is the output of the transformer which can then be used for prediction , or sent through subsequent transformer blocks . Position encodings . Self-attention is permutation invariant and so tells you nothing about order of the inputs . Should the data be sequential ( i.e . meaning depends on the order of elements , such as in language , or navigation as we will see later ! ) , it is necessary to additionally encode the position/ where x is in the sequence . This is typically done by adding a ‘ position encoding ’ that uniquely identifies each time-step ( et - typically sines and cosines ) to each input : xt ← xt+et . Alternatively the position embedding can be appended i.e . ht = [ xt , et ] , with self attention then performed using ht as input . 3 TRANSFORMERS LEARN ENTORHINAL REPRESENTATIONS . Here we show that transformers ( with a small modification ) recapitulate spatial representations - grid and band cells - when trained on tasks that require abstract spatial knowledge . Spatial understanding task . The task ( more detail in Appendix ) is to predict upcoming sensory observations xt+1 conditioned on taking an action at while moving around spatial environments ( Figure 1a ) . For example , after seeing { ( x1 = cat , a1 = North ) , ( x2 = dog , a2 = East ) , ( x3 = frog , a3 = South ) , ( x4 = pig , a4 = West ) , ( x5 = ? , a5 = · · · ) } , the aim is to predict x5 = cat . For simplicity , we treat sensory observations as one-hot vectors , thus the prediction problem is a classification problem . e0 x0 e1 x1 e2 x2 e3 ? y3 q0 k0 v0 q1 k1 v1 q2 k2 v2 q3 iWv iWq iWk Transformer ( with causal attention mask ) Input with position encoding Query Key , Value A B D B C A A D B B C A D A D C D C Environment 1 Environment 2 Observations ( x ) Actions ( a ) A B A ? C AAD ? D Recurrent connections W ( a0 ) W ( a1 ) W ( a2 ) With linear activations With ReLu activationsReal grid cell rate maps Cell rate maps Autocorrelations Cell rate maps Autocorrelations a b dc e f Band cells Figure 1 : ( a ) Sequence prediction in spatial navigation tasks test abstract spatial understanding since some sensory predictions can only be done by knowing ( generalising ) certain rules e.g . North + East + South + West = 0 or Parent + Sibling + Niece = 0 . Note , we use sequences drawn from much larger graphs . ( b ) Transformer with recurrent position encodings . ( c ) Real grid cell rate-maps ( Hafting et al. , 2005 ) . ( d-f ) Learned position embedding rate-maps ( i.e . average activity at each spatial location ; plots are spatially smoothed ) . ( d-e ) Resembling grid cells with ( e ) linear activation or ( e ) ReLu activation post transition . ( f ) Resembling band cells ( Krupic et al. , 2012 ) . When faced with an unseen stimulus-action pair ( e.g . x4 = pig , a4 = West above ; an action you have never taken at that stimulus before ) , successful prediction requires more than just remembering specific sequences of stimulus-action pairs ; knowledge of the rules of space must be known ; i.e . North + East + South + West = 0 allows prediction of x5 = cat . Crucially , such rules generalise to any 2D spaces and may therefore be transferred to aid prediction in entirely novel 2D environments . This is powerful , since unobserved relations between observed stimuli can be inferred in a zero-shot manner . However , these relational rules are not ‘ known ’ a priori and therefore must be learnt . We therefore train across multiple different spatial environments which share the same underlying 4-connected Euclidean structure ( Figure 1a ) - this means the model must learn and generalise the abstract structure of space to use for prediction in new environments . To perform on these tasks , the three modifications to the transformer are : 1 . Recall equation 1 ; yt = softmax ( qtK T √ dk ) V , where Q = HWq , K = HWk , V = HWv , and H is a matrix of inputs and position encodings ( i.e . its rows are ht = [ xt , et ] ) . We restrict these weight matrices such that queries ( Q ) and keys ( K ) are the same ; Q , K = EWe . We refer to this matrix as Ẽ . Thus the keys and queries only focus on position encodings . Meanwhile , values are exclusively dependent on the stimulus component of H i.e . V = XWx . We refer to this matrix as X̃ . yt = softmax ( qtK T √ dk ) V → yt = softmax ( ẽtẼ T √ dk ) X̃ ( 2 ) This is an extreme version of the realisation that , in transformers , best performance is when position encodings are used to compute keys and queries , but not values . 2 . We use causal transformers ; the key and value matrices contain the projected position encodings and sensory stimuli respectively at all previous time-steps ( i.e . e < t and x < t ) . This is equivalent to causal ‘ unmasking ’ as the agent wanders the environment accumulating new experiences ( not-yet-experienced stimulus-position pairs are inaccessible to the agent ) . Meanwhile the query at each time-point is the present positional encoding et . 3 . The position encodings are recurrently generated ( as in Wang et al . ( 2019 ) ; Liu et al . ( 2020 ) ) ; et+1 = σ ( etWa ) , where Wa is a learnable action-dependent weight matrix , and σ ( · · · ) is a non-linear activation function . This means that unlike traditional transformers , position encodings can be optimised and not the same for every sequence . It now becomes interesting to see what representations are learned . These modifications are sufficient to learn spatial representations , in the position encodings , that mimic representations observed in the brain ( Figure 1C ; see Appendix for model and training details ) . The rest of this paper now explains why this is not a surprising result ; namely we show that a transformer with recurrent positional encodings is closely related to current neuroscience models of the hippocampus and surrounding cortex ( Whittington et al. , 2020 ; Uria et al. , 2020 ) . Here we focus on the Tolman-Eichenbaum Machine ( TEM ) ( Whittington et al. , 2020 ) , though the same principles apply for Uria et al . ( 2020 ) . The critical points are : 1 ) the memory component of TEM can be viewed as a transformer selfattention , since the TEM memory network is analogous to a Hopfield network ( Hopfield , 1982 ) which have recently been shown to be closely related to transformers ( Ramsauer et al. , 2020 ) ; 2 ) TEM path integration ( see below ) can be viewed as a way to learn a position encoding . | This well-written and clear paper clarifies the relationship between transformers and a recent exciting model of the medial temporal lobe in neuroscience. The authors find that a transformer with recurrent positional encodings trained on a spatial navigation task ends up learning representations that resemble some classic findings from the medial temporal lobe in the neuroscience (e.g., grid and place cells). This finding alone is not that surprising -- as the authors note, a variety of groups have previously found such representations are learned in navigation tasks with other models (particularly place cells). The major contribution of this paper is that the authors then demonstrate the connection between transformers and a contemporary model of the hippocampal formation, the Tolman-Eichenbaum Machine (TEM). Specifically, the memory retrieval process in the TEM resembles self-attention and the path integration representations are comparable to a transformer's learned position encodings. I believe that by offering a novel perspective on transformers and models of biological memory systems, this work may lead to fruitful future work both in machine learning and neuroscience. | SP:86e84595489ca0f10e58c0977135aa001db5166f |
A Fair Generative Model Using Total Variation Distance | 1 INTRODUCTION . High-quality realistic samples synthesized thanks to recent advances in generative models ( Brock et al. , 2019 ; Goodfellow et al. , 2014 ; Karras et al. , 2019 ) have played a crucial role to enrich training data for a widening array of applications such as face recognition , natural language processing , and medical imaging ( Wang et al. , 2019 ; Chang et al. , 2018 ; Yi et al. , 2019 ) . One challenge concerning fairness arises when generative models are built upon biased training data that preserve unbalanced representations of demographic groups . Any existing bias in the dataset can readily be propagated to the learned model , thus producing generations that are biased towards certain demographics . The unbalanced generated samples may often yield undesirable performances against underrepresented groups for downstream applications . One natural way to ensure fair sample generation is to exploit demographic labels ( if available ) to build a fair generative model , e.g. , via conditional GAN ( Mirza & Osindero , 2014 ; Odena et al. , 2017 ; Miyato & Koyama , 2018 ) which employs such labels to easily generate an arbitrary number of samples for minority groups . In many practically-relevant scenarios , however , such labels are not often available . To address the challenge , one pioneering work ( Choi et al. , 2020 ) develops a novel debiasing technique that employs the reweighting idea ( Ren et al. , 2018 ; Kamiran & Calders , 2012 ; Byrd & Lipton , 2019 ) to put more weights to underrepresented samples , thereby promoting fair sample generation across demographic groups . One key feature of the technique is to identify the bias ( reflected in the weights ) via a small and unlabelled reference dataset . While it enjoys significant fairness performance for moderate sizes of the reference dataset , it may provide a marginal gain for a more practically-relevant case of a small set size where the weight estimation is often inaccurate , as hinted by the meta-learning literature ( Ren et al. , 2018 ; Shu et al. , 2020 ) . We also find such phenomenon in our experiments ; see Table 2 for details . On the other hand , one recent study ( Roh et al. , 2020 ) sheds lights on addressing the small set size issue . Roh et al . ( 2020 ) propose a robust training approach by employing the Jensen-Shannon divergence ( or equivalently mutual information ( Majtey et al. , 2005 ) ) between poisoned training and clean reference samples . It then takes the divergence as a regularization term in order to promote robust training . One key benefit of Roh et al . ( 2020 ) in light of the reweighting-based approaches ( Ren et al. , 2018 ; Choi et al. , 2020 ) is that the robustness performance is guaranteed even for a small size of the clean reference set down to 5 % relative to the training set size . This implies that the divergence-based regularization approach makes a more efficient use of reference data for robustness , as compared to the reweighting technique . Contribution : Inspired by the key benefit featured in Roh et al . ( 2020 ) , we address the small set issue by another well-known statistical measure , in particular , total variation distance ( TVD ) . See Section 3.2 for details on the rationale behind the use of TVD . Similarly we introduce a reference dataset which is balanced and unlabelled . We then employ it to formulate the TVD between the generated and reference sample distributions , which can serve as quantifying the degree of unfairness . We then promote fair sample generation by adding the TVD into a conventional optimization ( e.g. , GAN-based optimization ( Goodfellow et al. , 2014 ; Nowozin et al. , 2016 ; Arjovsky et al. , 2017 ) ) . Motivated by the variational optimization technique w.r.t . TVD ( Villani , 2009 ; Nowozin et al. , 2016 ; Arjovsky et al. , 2017 ) , we translate the TVD-regularization term into a function optimization . We also conduct extensive experiments on three benchmark real datasets : CelebA ( Liu et al. , 2015 ) , UTKFace ( Zhang et al. , 2017 ) , and FairFace ( Karkkainen & Joo , 2021 ) . We demonstrate via simulation that the proposed framework can significantly boost up the fairness performance while offering high-quality realistic samples reflected in low FID . We also find that our approach outperforms the state of the art ( Choi et al. , 2020 ) , particularly being robust to the balanced reference set size : the significant improvements preserve for a wide range of the reference set size down to 1 % relative to training data ( more preferable in reality ) . Related works : In addition to Choi et al . ( 2020 ) , Tan et al . ( 2020 ) propose a different way that promotes fair sample generation by smartly perturbing the input distribution of a pre-trained generative model with the help of a classifier for sensitive attributes . The key distinction w.r.t . ours is that it relies upon the additional classifier which requires the use of demographic labels to obtain . Yu et al . ( 2020 ) employ demographic labels for minority groups to generate a wide variety of samples with improved data converge by harmonizing GAN and MLE ideas . A distinction w.r.t . ours is that it requires the knowledge on demographic labels . Jalal et al . ( 2021 ) consider a fair generative model yet in a different context , image reconstruction . The goal of the task is to ensure fair sample generation of restored images from degraded versions . Since it relies upon the degraded images , it is not directly comparable to ours . Another line of fair generative modeling focuses on label bias , instead of representation bias ( Xu et al. , 2018 ; 2019a ; b ; Sattigeri et al. , 2019 ; Jang et al. , 2021 ; Kyono et al. , 2021 ) . The goal therein is to develop a generative model such that the generated decision labels are statistically independent of the given demographic labels . Again , these are not directly comparable to ours , as they require the use of demographic labels . The variational optimization technique w.r.t . TVD that gives an inspiration to our work has originated from Villani ( 2003 ; 2009 ) , wherein the author shows that the TVD can be expressed as a function optimization in the context of transport theory . The technique was recently applied to the GAN context ( Nowozin et al. , 2016 ) . The TVD has also served as a useful tool for quantifying various fairness measures in fair classifiers that pursue individual fairness ( Dwork et al. , 2012 ; Dwork & Ilvento , 2018 ) and group fairness ( Gordaliza et al. , 2019 ; Wang et al. , 2020a ; Farokhi , 2021 ) . 2 PROBLEM FORMULATION . Setup : Fig . 1 illustrates the problem setting for a fair generative model that we focus on herein . We consider a challenging yet practically-relevant scenario wherein demographic information ( or that we call sensitive attribute ) , say z ∈ Z , is not available . Under this blind setting , the goal of the problem is to construct a fair generative model that ensures the produced samples to have the same size across distinct demographics . We assume that there are two types of data given in the problem : ( i ) training data Dbias : = { x ( i ) bias } mbias i=1 ; ( ii ) reference data Dref : = { x ( i ) ref } mref i=1 . Since we consider the setting where training data is potentially biased , we use the word “ bias ” in the associated notations . Here mbias denotes the number of training examples . Let Pbias be data distribution which each training data x ( i ) bias ∈ X is generated upon . In a biased scenario having female-vs-male sensitive attribute , e.g. , z = 0 ( female ) and z = 1 ( male ) , we may have Pbias ( Z = 0 ) > Pbias ( Z = 1 ) . As in Choi et al . ( 2020 ) , we also employ a balanced yet small reference dataset for the purpose of promoting fair sample generation , which can be obtained via a set of carefully-designed data collection protocols employed in organizations like World Bank and biotech companies ( Choi et al. , 2020 ; 23 & me , 2016 ; Hong , 2016 ) . Let Pref be the corresponding data distribution defined on X : i=1 ( potentially biased ) and balanced reference data { x ( i ) ref } mref i=1 . The entire structure of the proposed model will be illustrated in detail in Fig . 2 . Here mbias ( or mref ) denotes the number of training samples ( or reference samples ) . Pref ( Z = 0 ) ≈ Pref ( Z = 1 ) . Since the balanced reference set is often challenging to collect , typically the number of the reference samples is much smaller than that of training examples : mref mbias . Denote by x̂ : = G ( w ) ∈ X the generated sample fed by a random noise input w ∈ W . We assume that the generated samples have the same support X as training and reference samples . Let PG and PW be distributions w.r.t . generated samples and the random noise input respectively . As a fairness measure that will be employed for the purpose of evaluating our framework to be presented in Section 3 , we consider fairness discrepancy proposed by Choi et al . ( 2020 ) . It quantifies how PG differs from Pref w.r.t . a certain sensitive attribute , formally defined below . Definition 1 ( Fairness Discrepancy ( Choi et al. , 2020 ) ) . Fairness discrepancy between Pref and PG w.r.t . a sensitive attribute z ∈ { z1 , . . . , z|Z| } is defined as : F ( Pref , PG ) : = ‖pref ( z ) − pG ( z ) ‖2 ( 1 ) where pref ( z ) : = Pref ( Z = z1 ) Pref ( Z = z2 ) ... Pref ( Z = z|Z| ) and pG ( z ) : = PG ( Ẑ = z1 ) PG ( Ẑ = z2 ) ... PG ( Ẑ = z|Z| ) . Here Ẑ denotes the prediction of the sensitive attribute w.r.t . a generated sample , yielded by a pre-trained classification model which we call attribute classifier as in Choi et al . ( 2020 ) . The attribute classifier is employed only for the purpose of evaluation , and is trained based on another real dataset , e.g. , like the one mentioned in Choi et al . ( 2020 ) : the standard train and validation splits of CelebA ( Liu et al. , 2015 ) . For faithful evaluation , we consider a vast number of generated samples ( i.e. , more than 10,000 ) as well as employ highly-accurate attribute classifiers , around 98 % accuracy of gender classifier for instance . As a measure for the quality of generated samples that may compete with the fairness measure , we employ a well-known measure : Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) . It is defined as the Fréchet distance ( Fréchet , 1957 ) ( also known as the second-order Wasserstein distance ( Wasserstein , 1969 ) ) between real and generated samples approximated via the Gaussian distribution . The lower FID , the more realistic and diverse the generated samples are . For a more precise measure that reveals sample quality of each sensitive group , we consider FID computed within each demographic , called intra FID ( Miyato & Koyama , 2018 ; Zhang et al. , 2019 ; Wang et al. , 2020b ) . Computing intra FID requires the knowledge on group identities of generated samples . Since demographic labels are not available in our context , we rely upon the attribute classifier ( that we introduced above ) for predicting demographic information of the generated samples . GAN-based generative model : Our framework ( to be presented soon ) builds upon one powerful generative model : Generative Adversarial Network ( GAN ) ( Goodfellow et al. , 2014 ) . The GAN comprises two competing players : ( i ) discriminator D ( · ) that wishes to discriminate real samples against generated samples ; and ( ii ) generator G ( · ) that intends to fool the discriminator by producing realistic generated samples . In particular , we consider a general optimization framework ( Tseng et al. , 2021 ) which subsumes many GAN variants as special cases : ( Discriminator ) max D EPbias [ fD ( D ( X ) ) ] + EPG [ fG ( D ( X ) ) ] ; ( Generator ) min G −EPG [ gG ( D ( X ) ) ] ( 2 ) where fD , fG , and gG indicate certain functions that vary depending on an employed GAN approach . For instance , the choice of ( fD ( t ) , fG ( t ) , gG ( t ) ) = ( t , −t , t ) together with Lipschitz-1 condition on D leads to the prominent WGAN optimization ( Arjovsky et al. , 2017 ) . Another choice of ( fD ( t ) , fG ( t ) , gG ( t ) ) = ( min { 0 , −1 + t } , min { 0 , −1− t } , t ) yields a hinge-loss-based GAN ( Lim & Ye , 2017 ; Tran et al. , 2017 ) . We adopt this as a base framework that we will add a fairness aspect into in the next section . | In their paper, the authors propose a novel generative adversarial network (GAN) which, more than realistic outputs, promotes fairness in the generative distribution. Specifically, when the training dataset is trademarked by some _unfair_ property (the authors' example being unevenly distributed gender or ethnicity), their proposed method-- a "TVD based" approach, which I will call TVD-GAN going forward --will rectify this in their generative distribution. The generative distribution is evaluated using an $L_2$ norm between the likelihoods of some samples given a reference distribution and the generative distribution ($\mathbb{P}_{\text{ref}}$ and $\mathbb{P}_G$, respectively in eq.(1)). TVD-GAN minimises the "total variational distance" (TVD), through regularisation in eq.(5). As claimed on p.4, TVD "offers the best trade-off performances in fairness and sample quality compared to other divergence measures", as the results indeed seem to suggest. The main goal-- I believe --is to provide a more realistic sample using the larger training dataset, while still maintaining a fair distribution using the smaller reference dataset. | SP:82d647bf36fa365bb123bbd328f2a4253c565cb8 |
A Fair Generative Model Using Total Variation Distance | 1 INTRODUCTION . High-quality realistic samples synthesized thanks to recent advances in generative models ( Brock et al. , 2019 ; Goodfellow et al. , 2014 ; Karras et al. , 2019 ) have played a crucial role to enrich training data for a widening array of applications such as face recognition , natural language processing , and medical imaging ( Wang et al. , 2019 ; Chang et al. , 2018 ; Yi et al. , 2019 ) . One challenge concerning fairness arises when generative models are built upon biased training data that preserve unbalanced representations of demographic groups . Any existing bias in the dataset can readily be propagated to the learned model , thus producing generations that are biased towards certain demographics . The unbalanced generated samples may often yield undesirable performances against underrepresented groups for downstream applications . One natural way to ensure fair sample generation is to exploit demographic labels ( if available ) to build a fair generative model , e.g. , via conditional GAN ( Mirza & Osindero , 2014 ; Odena et al. , 2017 ; Miyato & Koyama , 2018 ) which employs such labels to easily generate an arbitrary number of samples for minority groups . In many practically-relevant scenarios , however , such labels are not often available . To address the challenge , one pioneering work ( Choi et al. , 2020 ) develops a novel debiasing technique that employs the reweighting idea ( Ren et al. , 2018 ; Kamiran & Calders , 2012 ; Byrd & Lipton , 2019 ) to put more weights to underrepresented samples , thereby promoting fair sample generation across demographic groups . One key feature of the technique is to identify the bias ( reflected in the weights ) via a small and unlabelled reference dataset . While it enjoys significant fairness performance for moderate sizes of the reference dataset , it may provide a marginal gain for a more practically-relevant case of a small set size where the weight estimation is often inaccurate , as hinted by the meta-learning literature ( Ren et al. , 2018 ; Shu et al. , 2020 ) . We also find such phenomenon in our experiments ; see Table 2 for details . On the other hand , one recent study ( Roh et al. , 2020 ) sheds lights on addressing the small set size issue . Roh et al . ( 2020 ) propose a robust training approach by employing the Jensen-Shannon divergence ( or equivalently mutual information ( Majtey et al. , 2005 ) ) between poisoned training and clean reference samples . It then takes the divergence as a regularization term in order to promote robust training . One key benefit of Roh et al . ( 2020 ) in light of the reweighting-based approaches ( Ren et al. , 2018 ; Choi et al. , 2020 ) is that the robustness performance is guaranteed even for a small size of the clean reference set down to 5 % relative to the training set size . This implies that the divergence-based regularization approach makes a more efficient use of reference data for robustness , as compared to the reweighting technique . Contribution : Inspired by the key benefit featured in Roh et al . ( 2020 ) , we address the small set issue by another well-known statistical measure , in particular , total variation distance ( TVD ) . See Section 3.2 for details on the rationale behind the use of TVD . Similarly we introduce a reference dataset which is balanced and unlabelled . We then employ it to formulate the TVD between the generated and reference sample distributions , which can serve as quantifying the degree of unfairness . We then promote fair sample generation by adding the TVD into a conventional optimization ( e.g. , GAN-based optimization ( Goodfellow et al. , 2014 ; Nowozin et al. , 2016 ; Arjovsky et al. , 2017 ) ) . Motivated by the variational optimization technique w.r.t . TVD ( Villani , 2009 ; Nowozin et al. , 2016 ; Arjovsky et al. , 2017 ) , we translate the TVD-regularization term into a function optimization . We also conduct extensive experiments on three benchmark real datasets : CelebA ( Liu et al. , 2015 ) , UTKFace ( Zhang et al. , 2017 ) , and FairFace ( Karkkainen & Joo , 2021 ) . We demonstrate via simulation that the proposed framework can significantly boost up the fairness performance while offering high-quality realistic samples reflected in low FID . We also find that our approach outperforms the state of the art ( Choi et al. , 2020 ) , particularly being robust to the balanced reference set size : the significant improvements preserve for a wide range of the reference set size down to 1 % relative to training data ( more preferable in reality ) . Related works : In addition to Choi et al . ( 2020 ) , Tan et al . ( 2020 ) propose a different way that promotes fair sample generation by smartly perturbing the input distribution of a pre-trained generative model with the help of a classifier for sensitive attributes . The key distinction w.r.t . ours is that it relies upon the additional classifier which requires the use of demographic labels to obtain . Yu et al . ( 2020 ) employ demographic labels for minority groups to generate a wide variety of samples with improved data converge by harmonizing GAN and MLE ideas . A distinction w.r.t . ours is that it requires the knowledge on demographic labels . Jalal et al . ( 2021 ) consider a fair generative model yet in a different context , image reconstruction . The goal of the task is to ensure fair sample generation of restored images from degraded versions . Since it relies upon the degraded images , it is not directly comparable to ours . Another line of fair generative modeling focuses on label bias , instead of representation bias ( Xu et al. , 2018 ; 2019a ; b ; Sattigeri et al. , 2019 ; Jang et al. , 2021 ; Kyono et al. , 2021 ) . The goal therein is to develop a generative model such that the generated decision labels are statistically independent of the given demographic labels . Again , these are not directly comparable to ours , as they require the use of demographic labels . The variational optimization technique w.r.t . TVD that gives an inspiration to our work has originated from Villani ( 2003 ; 2009 ) , wherein the author shows that the TVD can be expressed as a function optimization in the context of transport theory . The technique was recently applied to the GAN context ( Nowozin et al. , 2016 ) . The TVD has also served as a useful tool for quantifying various fairness measures in fair classifiers that pursue individual fairness ( Dwork et al. , 2012 ; Dwork & Ilvento , 2018 ) and group fairness ( Gordaliza et al. , 2019 ; Wang et al. , 2020a ; Farokhi , 2021 ) . 2 PROBLEM FORMULATION . Setup : Fig . 1 illustrates the problem setting for a fair generative model that we focus on herein . We consider a challenging yet practically-relevant scenario wherein demographic information ( or that we call sensitive attribute ) , say z ∈ Z , is not available . Under this blind setting , the goal of the problem is to construct a fair generative model that ensures the produced samples to have the same size across distinct demographics . We assume that there are two types of data given in the problem : ( i ) training data Dbias : = { x ( i ) bias } mbias i=1 ; ( ii ) reference data Dref : = { x ( i ) ref } mref i=1 . Since we consider the setting where training data is potentially biased , we use the word “ bias ” in the associated notations . Here mbias denotes the number of training examples . Let Pbias be data distribution which each training data x ( i ) bias ∈ X is generated upon . In a biased scenario having female-vs-male sensitive attribute , e.g. , z = 0 ( female ) and z = 1 ( male ) , we may have Pbias ( Z = 0 ) > Pbias ( Z = 1 ) . As in Choi et al . ( 2020 ) , we also employ a balanced yet small reference dataset for the purpose of promoting fair sample generation , which can be obtained via a set of carefully-designed data collection protocols employed in organizations like World Bank and biotech companies ( Choi et al. , 2020 ; 23 & me , 2016 ; Hong , 2016 ) . Let Pref be the corresponding data distribution defined on X : i=1 ( potentially biased ) and balanced reference data { x ( i ) ref } mref i=1 . The entire structure of the proposed model will be illustrated in detail in Fig . 2 . Here mbias ( or mref ) denotes the number of training samples ( or reference samples ) . Pref ( Z = 0 ) ≈ Pref ( Z = 1 ) . Since the balanced reference set is often challenging to collect , typically the number of the reference samples is much smaller than that of training examples : mref mbias . Denote by x̂ : = G ( w ) ∈ X the generated sample fed by a random noise input w ∈ W . We assume that the generated samples have the same support X as training and reference samples . Let PG and PW be distributions w.r.t . generated samples and the random noise input respectively . As a fairness measure that will be employed for the purpose of evaluating our framework to be presented in Section 3 , we consider fairness discrepancy proposed by Choi et al . ( 2020 ) . It quantifies how PG differs from Pref w.r.t . a certain sensitive attribute , formally defined below . Definition 1 ( Fairness Discrepancy ( Choi et al. , 2020 ) ) . Fairness discrepancy between Pref and PG w.r.t . a sensitive attribute z ∈ { z1 , . . . , z|Z| } is defined as : F ( Pref , PG ) : = ‖pref ( z ) − pG ( z ) ‖2 ( 1 ) where pref ( z ) : = Pref ( Z = z1 ) Pref ( Z = z2 ) ... Pref ( Z = z|Z| ) and pG ( z ) : = PG ( Ẑ = z1 ) PG ( Ẑ = z2 ) ... PG ( Ẑ = z|Z| ) . Here Ẑ denotes the prediction of the sensitive attribute w.r.t . a generated sample , yielded by a pre-trained classification model which we call attribute classifier as in Choi et al . ( 2020 ) . The attribute classifier is employed only for the purpose of evaluation , and is trained based on another real dataset , e.g. , like the one mentioned in Choi et al . ( 2020 ) : the standard train and validation splits of CelebA ( Liu et al. , 2015 ) . For faithful evaluation , we consider a vast number of generated samples ( i.e. , more than 10,000 ) as well as employ highly-accurate attribute classifiers , around 98 % accuracy of gender classifier for instance . As a measure for the quality of generated samples that may compete with the fairness measure , we employ a well-known measure : Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) . It is defined as the Fréchet distance ( Fréchet , 1957 ) ( also known as the second-order Wasserstein distance ( Wasserstein , 1969 ) ) between real and generated samples approximated via the Gaussian distribution . The lower FID , the more realistic and diverse the generated samples are . For a more precise measure that reveals sample quality of each sensitive group , we consider FID computed within each demographic , called intra FID ( Miyato & Koyama , 2018 ; Zhang et al. , 2019 ; Wang et al. , 2020b ) . Computing intra FID requires the knowledge on group identities of generated samples . Since demographic labels are not available in our context , we rely upon the attribute classifier ( that we introduced above ) for predicting demographic information of the generated samples . GAN-based generative model : Our framework ( to be presented soon ) builds upon one powerful generative model : Generative Adversarial Network ( GAN ) ( Goodfellow et al. , 2014 ) . The GAN comprises two competing players : ( i ) discriminator D ( · ) that wishes to discriminate real samples against generated samples ; and ( ii ) generator G ( · ) that intends to fool the discriminator by producing realistic generated samples . In particular , we consider a general optimization framework ( Tseng et al. , 2021 ) which subsumes many GAN variants as special cases : ( Discriminator ) max D EPbias [ fD ( D ( X ) ) ] + EPG [ fG ( D ( X ) ) ] ; ( Generator ) min G −EPG [ gG ( D ( X ) ) ] ( 2 ) where fD , fG , and gG indicate certain functions that vary depending on an employed GAN approach . For instance , the choice of ( fD ( t ) , fG ( t ) , gG ( t ) ) = ( t , −t , t ) together with Lipschitz-1 condition on D leads to the prominent WGAN optimization ( Arjovsky et al. , 2017 ) . Another choice of ( fD ( t ) , fG ( t ) , gG ( t ) ) = ( min { 0 , −1 + t } , min { 0 , −1− t } , t ) yields a hinge-loss-based GAN ( Lim & Ye , 2017 ; Tran et al. , 2017 ) . We adopt this as a base framework that we will add a fairness aspect into in the next section . | This paper focuses on the fair generative model problem. In particular, "fairness" lies in the balance of representation among groups in the data. The paper proposes to use TV as the distance metric in order to capture the difference between the generated distribution and the reference distribution. Empirical results are also provided. | SP:82d647bf36fa365bb123bbd328f2a4253c565cb8 |
A Fair Generative Model Using Total Variation Distance | 1 INTRODUCTION . High-quality realistic samples synthesized thanks to recent advances in generative models ( Brock et al. , 2019 ; Goodfellow et al. , 2014 ; Karras et al. , 2019 ) have played a crucial role to enrich training data for a widening array of applications such as face recognition , natural language processing , and medical imaging ( Wang et al. , 2019 ; Chang et al. , 2018 ; Yi et al. , 2019 ) . One challenge concerning fairness arises when generative models are built upon biased training data that preserve unbalanced representations of demographic groups . Any existing bias in the dataset can readily be propagated to the learned model , thus producing generations that are biased towards certain demographics . The unbalanced generated samples may often yield undesirable performances against underrepresented groups for downstream applications . One natural way to ensure fair sample generation is to exploit demographic labels ( if available ) to build a fair generative model , e.g. , via conditional GAN ( Mirza & Osindero , 2014 ; Odena et al. , 2017 ; Miyato & Koyama , 2018 ) which employs such labels to easily generate an arbitrary number of samples for minority groups . In many practically-relevant scenarios , however , such labels are not often available . To address the challenge , one pioneering work ( Choi et al. , 2020 ) develops a novel debiasing technique that employs the reweighting idea ( Ren et al. , 2018 ; Kamiran & Calders , 2012 ; Byrd & Lipton , 2019 ) to put more weights to underrepresented samples , thereby promoting fair sample generation across demographic groups . One key feature of the technique is to identify the bias ( reflected in the weights ) via a small and unlabelled reference dataset . While it enjoys significant fairness performance for moderate sizes of the reference dataset , it may provide a marginal gain for a more practically-relevant case of a small set size where the weight estimation is often inaccurate , as hinted by the meta-learning literature ( Ren et al. , 2018 ; Shu et al. , 2020 ) . We also find such phenomenon in our experiments ; see Table 2 for details . On the other hand , one recent study ( Roh et al. , 2020 ) sheds lights on addressing the small set size issue . Roh et al . ( 2020 ) propose a robust training approach by employing the Jensen-Shannon divergence ( or equivalently mutual information ( Majtey et al. , 2005 ) ) between poisoned training and clean reference samples . It then takes the divergence as a regularization term in order to promote robust training . One key benefit of Roh et al . ( 2020 ) in light of the reweighting-based approaches ( Ren et al. , 2018 ; Choi et al. , 2020 ) is that the robustness performance is guaranteed even for a small size of the clean reference set down to 5 % relative to the training set size . This implies that the divergence-based regularization approach makes a more efficient use of reference data for robustness , as compared to the reweighting technique . Contribution : Inspired by the key benefit featured in Roh et al . ( 2020 ) , we address the small set issue by another well-known statistical measure , in particular , total variation distance ( TVD ) . See Section 3.2 for details on the rationale behind the use of TVD . Similarly we introduce a reference dataset which is balanced and unlabelled . We then employ it to formulate the TVD between the generated and reference sample distributions , which can serve as quantifying the degree of unfairness . We then promote fair sample generation by adding the TVD into a conventional optimization ( e.g. , GAN-based optimization ( Goodfellow et al. , 2014 ; Nowozin et al. , 2016 ; Arjovsky et al. , 2017 ) ) . Motivated by the variational optimization technique w.r.t . TVD ( Villani , 2009 ; Nowozin et al. , 2016 ; Arjovsky et al. , 2017 ) , we translate the TVD-regularization term into a function optimization . We also conduct extensive experiments on three benchmark real datasets : CelebA ( Liu et al. , 2015 ) , UTKFace ( Zhang et al. , 2017 ) , and FairFace ( Karkkainen & Joo , 2021 ) . We demonstrate via simulation that the proposed framework can significantly boost up the fairness performance while offering high-quality realistic samples reflected in low FID . We also find that our approach outperforms the state of the art ( Choi et al. , 2020 ) , particularly being robust to the balanced reference set size : the significant improvements preserve for a wide range of the reference set size down to 1 % relative to training data ( more preferable in reality ) . Related works : In addition to Choi et al . ( 2020 ) , Tan et al . ( 2020 ) propose a different way that promotes fair sample generation by smartly perturbing the input distribution of a pre-trained generative model with the help of a classifier for sensitive attributes . The key distinction w.r.t . ours is that it relies upon the additional classifier which requires the use of demographic labels to obtain . Yu et al . ( 2020 ) employ demographic labels for minority groups to generate a wide variety of samples with improved data converge by harmonizing GAN and MLE ideas . A distinction w.r.t . ours is that it requires the knowledge on demographic labels . Jalal et al . ( 2021 ) consider a fair generative model yet in a different context , image reconstruction . The goal of the task is to ensure fair sample generation of restored images from degraded versions . Since it relies upon the degraded images , it is not directly comparable to ours . Another line of fair generative modeling focuses on label bias , instead of representation bias ( Xu et al. , 2018 ; 2019a ; b ; Sattigeri et al. , 2019 ; Jang et al. , 2021 ; Kyono et al. , 2021 ) . The goal therein is to develop a generative model such that the generated decision labels are statistically independent of the given demographic labels . Again , these are not directly comparable to ours , as they require the use of demographic labels . The variational optimization technique w.r.t . TVD that gives an inspiration to our work has originated from Villani ( 2003 ; 2009 ) , wherein the author shows that the TVD can be expressed as a function optimization in the context of transport theory . The technique was recently applied to the GAN context ( Nowozin et al. , 2016 ) . The TVD has also served as a useful tool for quantifying various fairness measures in fair classifiers that pursue individual fairness ( Dwork et al. , 2012 ; Dwork & Ilvento , 2018 ) and group fairness ( Gordaliza et al. , 2019 ; Wang et al. , 2020a ; Farokhi , 2021 ) . 2 PROBLEM FORMULATION . Setup : Fig . 1 illustrates the problem setting for a fair generative model that we focus on herein . We consider a challenging yet practically-relevant scenario wherein demographic information ( or that we call sensitive attribute ) , say z ∈ Z , is not available . Under this blind setting , the goal of the problem is to construct a fair generative model that ensures the produced samples to have the same size across distinct demographics . We assume that there are two types of data given in the problem : ( i ) training data Dbias : = { x ( i ) bias } mbias i=1 ; ( ii ) reference data Dref : = { x ( i ) ref } mref i=1 . Since we consider the setting where training data is potentially biased , we use the word “ bias ” in the associated notations . Here mbias denotes the number of training examples . Let Pbias be data distribution which each training data x ( i ) bias ∈ X is generated upon . In a biased scenario having female-vs-male sensitive attribute , e.g. , z = 0 ( female ) and z = 1 ( male ) , we may have Pbias ( Z = 0 ) > Pbias ( Z = 1 ) . As in Choi et al . ( 2020 ) , we also employ a balanced yet small reference dataset for the purpose of promoting fair sample generation , which can be obtained via a set of carefully-designed data collection protocols employed in organizations like World Bank and biotech companies ( Choi et al. , 2020 ; 23 & me , 2016 ; Hong , 2016 ) . Let Pref be the corresponding data distribution defined on X : i=1 ( potentially biased ) and balanced reference data { x ( i ) ref } mref i=1 . The entire structure of the proposed model will be illustrated in detail in Fig . 2 . Here mbias ( or mref ) denotes the number of training samples ( or reference samples ) . Pref ( Z = 0 ) ≈ Pref ( Z = 1 ) . Since the balanced reference set is often challenging to collect , typically the number of the reference samples is much smaller than that of training examples : mref mbias . Denote by x̂ : = G ( w ) ∈ X the generated sample fed by a random noise input w ∈ W . We assume that the generated samples have the same support X as training and reference samples . Let PG and PW be distributions w.r.t . generated samples and the random noise input respectively . As a fairness measure that will be employed for the purpose of evaluating our framework to be presented in Section 3 , we consider fairness discrepancy proposed by Choi et al . ( 2020 ) . It quantifies how PG differs from Pref w.r.t . a certain sensitive attribute , formally defined below . Definition 1 ( Fairness Discrepancy ( Choi et al. , 2020 ) ) . Fairness discrepancy between Pref and PG w.r.t . a sensitive attribute z ∈ { z1 , . . . , z|Z| } is defined as : F ( Pref , PG ) : = ‖pref ( z ) − pG ( z ) ‖2 ( 1 ) where pref ( z ) : = Pref ( Z = z1 ) Pref ( Z = z2 ) ... Pref ( Z = z|Z| ) and pG ( z ) : = PG ( Ẑ = z1 ) PG ( Ẑ = z2 ) ... PG ( Ẑ = z|Z| ) . Here Ẑ denotes the prediction of the sensitive attribute w.r.t . a generated sample , yielded by a pre-trained classification model which we call attribute classifier as in Choi et al . ( 2020 ) . The attribute classifier is employed only for the purpose of evaluation , and is trained based on another real dataset , e.g. , like the one mentioned in Choi et al . ( 2020 ) : the standard train and validation splits of CelebA ( Liu et al. , 2015 ) . For faithful evaluation , we consider a vast number of generated samples ( i.e. , more than 10,000 ) as well as employ highly-accurate attribute classifiers , around 98 % accuracy of gender classifier for instance . As a measure for the quality of generated samples that may compete with the fairness measure , we employ a well-known measure : Fréchet Inception Distance ( FID ) ( Heusel et al. , 2017 ) . It is defined as the Fréchet distance ( Fréchet , 1957 ) ( also known as the second-order Wasserstein distance ( Wasserstein , 1969 ) ) between real and generated samples approximated via the Gaussian distribution . The lower FID , the more realistic and diverse the generated samples are . For a more precise measure that reveals sample quality of each sensitive group , we consider FID computed within each demographic , called intra FID ( Miyato & Koyama , 2018 ; Zhang et al. , 2019 ; Wang et al. , 2020b ) . Computing intra FID requires the knowledge on group identities of generated samples . Since demographic labels are not available in our context , we rely upon the attribute classifier ( that we introduced above ) for predicting demographic information of the generated samples . GAN-based generative model : Our framework ( to be presented soon ) builds upon one powerful generative model : Generative Adversarial Network ( GAN ) ( Goodfellow et al. , 2014 ) . The GAN comprises two competing players : ( i ) discriminator D ( · ) that wishes to discriminate real samples against generated samples ; and ( ii ) generator G ( · ) that intends to fool the discriminator by producing realistic generated samples . In particular , we consider a general optimization framework ( Tseng et al. , 2021 ) which subsumes many GAN variants as special cases : ( Discriminator ) max D EPbias [ fD ( D ( X ) ) ] + EPG [ fG ( D ( X ) ) ] ; ( Generator ) min G −EPG [ gG ( D ( X ) ) ] ( 2 ) where fD , fG , and gG indicate certain functions that vary depending on an employed GAN approach . For instance , the choice of ( fD ( t ) , fG ( t ) , gG ( t ) ) = ( t , −t , t ) together with Lipschitz-1 condition on D leads to the prominent WGAN optimization ( Arjovsky et al. , 2017 ) . Another choice of ( fD ( t ) , fG ( t ) , gG ( t ) ) = ( min { 0 , −1 + t } , min { 0 , −1− t } , t ) yields a hinge-loss-based GAN ( Lim & Ye , 2017 ; Tran et al. , 2017 ) . We adopt this as a base framework that we will add a fairness aspect into in the next section . | This paper studies fairness in generative models. When the training data is biased, it may result in imbalanced generated samples. They assume that they have access to scarce reference samples that are balanced. They measure the unfairness of the model through total variation and propose a fairness-aware model that is regulated by total variation distance between the generated samples (biased) and reference samples (unbiased). The authors further assume that the sensitive attributes are not available. Finally, they provide numerical experiments to demonstrate the performance of their proposed framework. | SP:82d647bf36fa365bb123bbd328f2a4253c565cb8 |
Synthetic Reduced Nearest Neighbor Model for Regression | 1 INTRODUCTION . One of the main topics of research in Machine Learning is the relation between the features and output responses Hastie et al . ( 2009 ) ; Santosa & Symes ( 1986 ) ; Tibshirani ( 1996 ) ; Criminisi & Shotton ( 2013 ) . Synthetic Reduced Nearest Neighbor ( SRNN ) models are shown to be an effective tool in determining the relationships between features of the inputs and the sub-clusters of each class in supervised learning tasks Tavallali et al . ( 2020b ) . However , existing prototype nearest neighbor models such as SRNN are constrained to classification problems , and to the best of our knowledge , there remains a gap in extending these algorithms towards regression tasks . Such regression reduced nearest neighbor models may find extensive applications in epidemiological studies Tavallali et al . ( 2020a ) ; Cisneros et al . ( 2021 ) , medical studies Criminisi & Shotton ( 2013 ) ; Graf et al . ( 2011b ; a ) , and other applied regression tasks in general Tibshirani ( 1996 ) . To address this gap in the state of the art , we propose a novel algorithm for the optimization and construction of Regression Synthetic Reduced Nearest Neighbor ( Reg-SRNN ) models . The proposed Reg-SRNN is capable of discovering various modalities of the input data , and relates those to the modalities of the output responses . The Reg-SRNN algorithm is designed to handle both single-response and multi-response regression . The multi-response regression consists of learning the relation between input samples and several ground-truth output responses . Reg-SRNN partitions the input space into piecewise constant regions , where each region is represented by a centroid and its output response . From this perspective , Reg-SRNN is similar to other piecewise constant models , such as Li & Martin ( 2017 ) ; Begon et al . ( 2017 ) ; Bertsimas & Dunn ( 2017 ) ; Tavallali et al . ( 2019 ; 2020c ) . Reg-SRNN is capable of learning an accurate relation between each cluster of the data and its corresponding output responses . Therefore , Reg-SRNN can also provide enhanced interpretablity by reducing the information content of clusters into a compressed representation manifested in their centroids . The technical contributions of this paper include the proposal of a novel initialization that by itself is competitive to other existing regression models . This is proceeded by an expectation maximization algorithm for directly minimizing the least squares error of the mode . The proposed optimization algorithm is provably convergent , and it is shown that it monotonically decreases the loss function . Therefore , the algorithm has a convergence guarantee on minimizing the loss function and achieving a local optimum.It is also worth mentioning that the algorithm does not cycle . The proposed optimization algorithm consists of two steps and is inspired by K-means algorithm Lloyd ( 1982 ) . One step is the assignment step and is composed of finding samples assignments and proper output response of the centroid . Second step is the update step where the centroid is optimized such that the loss function is decreased . The centroid step is affected by all the samples and we will show that this update step is a kind of NP-hard weighted binary classification problem . The update step is computed through a surrogate objective function that is similar to SVM . We establish that the algorithm is efficient because of its linear computational complexity . Finally , the model is evaluated on various datasets with various sizes and dimensionalities , the results of which demonstrate that Reg-SRNN is capable of competing and even over-matching similar regression models . Accordingly , the main contributions of this paper are as follows : • We propose a novel algorithm for initialization of SRNN models to extend their application to regression tasks . • We develop an optimization algorithm for regression SRNN models with guarantees on convergence . • Through experimental evaluation , we demonstrated the feasibility of our proposed regression SRNN model in filling the gap between more complex models ( such as random forests ) and basic and interpretable models such as linear regression and decision trees . 2 RELATED WORK . A regression task consists of learning the relation between samples of the input space and a numerical output space . More specifically , regression is a supervised learning task of mapping inputs ( independent variable X ) to the output Y , which is a continuous vector ( Y ∈ Rd ) . If the dimensionality of the output d ≥ 2 , the task is known as multi-response regression . Regression has been the workhorse of numerous fields Tai ( 2021 ) , and various regression models have been developed and expanded fundamentally over the recent decades Hastie et al . ( 2009 ) . This expansion has been so rampant such that listing all such models and their relationships is a difficult task and is out of the scope of this work . However , a brief review of the recent models is presented in this paper . A common objective function for regression is to minimize the least squares error : ||Ŷ − Y ||2 ( 1 ) Where Ŷ is the prediction . According to the Gauss-Markov theorem Gauss ( 1823 ) , the least squares error can be an unbiased linear model of minimum variance of the data under certain assumptions . Ordinary least squares may fail to properly predict outcomes if it is applied to settings where the Gauss-Markov assumptions are not held . Therefore , it is important to understand the assumptions and occasionally apply the proper changes to the objective function of equation 1 to modify the model Tai ( 2021 ) . Manifestations of such changes include imposing regulations or constraints over the objective function . The literature on ordinary least squares estimation has extensively dealt with some of the well-known concerns that might violate the assumptions , such as Ridge Hoerl & Kennard ( 1970a ; b ) , Lasso Tibshirani ( 1996 ) , Elastic Net Zou & Hastie ( 2005 ) , trees Quinlan ( 2014 ) , forest Breiman ( 2001 ) , boosting Bühlmann & Yu ( 2003 ) and others . Common regression models include bagging , boosting , random forest Criminisi & Shotton ( 2013 ) , oblique trees Murthy et al . ( 1994 ) ; Norouzi et al . ( 2015 ) ; Heath et al . ( 1993 ) , and regression SVM Drucker et al . ( 1997 ) . In the context of regression trees , various approaches of inducing a tree are presented in the literature . Most decision tree induction methods are concentrated on the splitting criterion used at the growing phase of the tree Ikonomovska et al . ( 2011 ) ; Levatić et al . ( 2014 ) . Application of decision tree algorithms to multi-response regression has been previously considered in the literature Breiman et al . ( 1984 ) ; De ’ Ath ( 2002 ) . In Breiman et al . ( 1984 ) ; Quinlan ( 1986 ) , authors consider training a decision tree for each individual output response . However , such approach constructs a large model specially if the number of output responses are high . Another approach proposed in De ’ Ath ( 2002 ) consists of constructing a single decision tree for all the output responses . In other words , the model predicts all the output values simultaneously through a single decision tree . However , a model for all the outputs might not be sufficient Kocev et al . ( 2009 ) because they train model for single response rather than the true problem which is a multi-response regression . Authors in Kocev et al . ( 2009 ) have explored two approaches to the multi-response regression problem by comparing learning a model for each output separately ( i.e. , multiple regression trees ) , and learning one model for all outputs simultaneously ( i.e. , a single multi-target regression tree ) . In order to improve predictive performance , Kocev et al . ( 2013 ) has also considered two ensemble learning techniques , namely , bagging Breiman ( 1996 ) ; Liang et al . ( 2011 ) and random forests Breiman ( 2001 ) for regression trees and multi-target regression trees . The results showed that multi-target regression trees created more accurate and compact models . A related topic to the problem of this paper is nearest neighbor regression . Nearest neighbor regression and local estimators are well-established methods in the literature of ordinary univariate location estimators ( Benedetti ( 1977 ) ; Stone ( 1974 ) ; Tukey et al . ( 1977 ) . However , as per our extensive search , there remains a gap in prototype nearest neighbor approaches to regression . The only work that considered a similar model and optimization to SRNN was Huang ( 2017 ) . However , the proposed algorithm does not have guarantee of convergence or achieving some sort of optimum solution . 3 PROPOSED METHOD . 3.1 PRELIMINARIES . Assume a dataset consisting of tuples ( xi , yi ) where x , y and i represent input features , output responses and index number . Each tuple represent a data xi and its corresponding output response yi . Here , xi ∈ R D and yi ∈ R d. The Regression Synthetic Reduced Nearest Neighbor ( Reg . SRNN ) consists of K tuples of synthetically produced centroids/prototypes ( cj , ŷj ) where c , ŷj and j represent the centroid ’ s point in the input space , output prediction and index . At the inference time , the Reg . SRNN operates like a nearest neighbor model where the centroids are used as the samples . The problem of training Reg . SRNN is as follows : min { ( cj , ŷj ) } K1 N ∑ i=1 ||yi − ŷj∗ i ||2 s.t j∗i = argmin { j } K 1 d ( xi − cj ) ( 2 ) where d ( . ) is a distance metric . Through this paper we use the l-2 norm as the distance metric : d ( xi − cj ) = √ ||xi − cj ||2 ( 3 ) Essentially the prediction of the model consists of the output prediction of closest centroid to the input sample . Officially , we define Reg . SRNN as follows : NN ( x ) = K ∑ j=1 yjI ( x ∈ Rj ) ( 4 ) Where , NN ( . ) represents a nearest neighbor function of the K centroids . I ( . ) is an indicator function that produces 1 if the input x is in the region of Rj . Rj represents the region where the closest centroid to the points in that region is cj . 3.2 INITIALIZATION . Numerical optimization algorithms require initialization ( cite num opt ) . In this paper , we propose a novel initialization for the regression SRNN . Previous work on the initialization of SRNN models consisted of learning a K-means model for each class of the data . For example , in case of M classes and K centroids , K M centroids are learned for each class as initialization of the SRNN Kusner et al . ( 2014 ) ; Wang et al . ( 2016 ) ; Zhong et al . ( 2017 ) ; Tavallali et al . ( 2020b ) . However , such approach is not applicable to the regression . This initialization also have close ties with naive Bayes and density estimation Silverman ( 2018 ) . Here , we expand this initialization to the case of Reg . SRNN . Intuitively , the output responses can consist of several modalities . In other words , it is possible that the output responses are generated from several distributions . The clusters of such distributions can be approximated by running a K-means over the output space ( M centroids ) . Assume that Sm represents the set of samples assigned to each output cluster . Next step consists of learning KN|Sm| centroids over the input features of the Sm for all M clusters . In other words , we learn centroids over the input features of each output cluster relative to the population of that cluster . The found centroids at the second step are used as initialization for the Reg . SRNN . At this step , ŷj is found using the following formula : ŷj = mean ( yi ∈ Sj ) ( 5 ) where Sj represents the set of samples that are assigned to j th centroid . Sj essentially consists of samples where jth centroid is the closest centroid to them . mean ( . ) represents the average of its input set . Note that Sj ∈ Rj . | The paper presents a novel regression model based on Nearest Neighbor. In particular authors adapt the synthetic reduced NN model used for classification to the regression case. The main intuition is to simplify the regression problem by finding clusters of inputs and targets. Authors presents some theoretical properties of the method as well as few experimental results. | SP:4e742c63860ef00798f23d647b46af94de0abad4 |
Synthetic Reduced Nearest Neighbor Model for Regression | 1 INTRODUCTION . One of the main topics of research in Machine Learning is the relation between the features and output responses Hastie et al . ( 2009 ) ; Santosa & Symes ( 1986 ) ; Tibshirani ( 1996 ) ; Criminisi & Shotton ( 2013 ) . Synthetic Reduced Nearest Neighbor ( SRNN ) models are shown to be an effective tool in determining the relationships between features of the inputs and the sub-clusters of each class in supervised learning tasks Tavallali et al . ( 2020b ) . However , existing prototype nearest neighbor models such as SRNN are constrained to classification problems , and to the best of our knowledge , there remains a gap in extending these algorithms towards regression tasks . Such regression reduced nearest neighbor models may find extensive applications in epidemiological studies Tavallali et al . ( 2020a ) ; Cisneros et al . ( 2021 ) , medical studies Criminisi & Shotton ( 2013 ) ; Graf et al . ( 2011b ; a ) , and other applied regression tasks in general Tibshirani ( 1996 ) . To address this gap in the state of the art , we propose a novel algorithm for the optimization and construction of Regression Synthetic Reduced Nearest Neighbor ( Reg-SRNN ) models . The proposed Reg-SRNN is capable of discovering various modalities of the input data , and relates those to the modalities of the output responses . The Reg-SRNN algorithm is designed to handle both single-response and multi-response regression . The multi-response regression consists of learning the relation between input samples and several ground-truth output responses . Reg-SRNN partitions the input space into piecewise constant regions , where each region is represented by a centroid and its output response . From this perspective , Reg-SRNN is similar to other piecewise constant models , such as Li & Martin ( 2017 ) ; Begon et al . ( 2017 ) ; Bertsimas & Dunn ( 2017 ) ; Tavallali et al . ( 2019 ; 2020c ) . Reg-SRNN is capable of learning an accurate relation between each cluster of the data and its corresponding output responses . Therefore , Reg-SRNN can also provide enhanced interpretablity by reducing the information content of clusters into a compressed representation manifested in their centroids . The technical contributions of this paper include the proposal of a novel initialization that by itself is competitive to other existing regression models . This is proceeded by an expectation maximization algorithm for directly minimizing the least squares error of the mode . The proposed optimization algorithm is provably convergent , and it is shown that it monotonically decreases the loss function . Therefore , the algorithm has a convergence guarantee on minimizing the loss function and achieving a local optimum.It is also worth mentioning that the algorithm does not cycle . The proposed optimization algorithm consists of two steps and is inspired by K-means algorithm Lloyd ( 1982 ) . One step is the assignment step and is composed of finding samples assignments and proper output response of the centroid . Second step is the update step where the centroid is optimized such that the loss function is decreased . The centroid step is affected by all the samples and we will show that this update step is a kind of NP-hard weighted binary classification problem . The update step is computed through a surrogate objective function that is similar to SVM . We establish that the algorithm is efficient because of its linear computational complexity . Finally , the model is evaluated on various datasets with various sizes and dimensionalities , the results of which demonstrate that Reg-SRNN is capable of competing and even over-matching similar regression models . Accordingly , the main contributions of this paper are as follows : • We propose a novel algorithm for initialization of SRNN models to extend their application to regression tasks . • We develop an optimization algorithm for regression SRNN models with guarantees on convergence . • Through experimental evaluation , we demonstrated the feasibility of our proposed regression SRNN model in filling the gap between more complex models ( such as random forests ) and basic and interpretable models such as linear regression and decision trees . 2 RELATED WORK . A regression task consists of learning the relation between samples of the input space and a numerical output space . More specifically , regression is a supervised learning task of mapping inputs ( independent variable X ) to the output Y , which is a continuous vector ( Y ∈ Rd ) . If the dimensionality of the output d ≥ 2 , the task is known as multi-response regression . Regression has been the workhorse of numerous fields Tai ( 2021 ) , and various regression models have been developed and expanded fundamentally over the recent decades Hastie et al . ( 2009 ) . This expansion has been so rampant such that listing all such models and their relationships is a difficult task and is out of the scope of this work . However , a brief review of the recent models is presented in this paper . A common objective function for regression is to minimize the least squares error : ||Ŷ − Y ||2 ( 1 ) Where Ŷ is the prediction . According to the Gauss-Markov theorem Gauss ( 1823 ) , the least squares error can be an unbiased linear model of minimum variance of the data under certain assumptions . Ordinary least squares may fail to properly predict outcomes if it is applied to settings where the Gauss-Markov assumptions are not held . Therefore , it is important to understand the assumptions and occasionally apply the proper changes to the objective function of equation 1 to modify the model Tai ( 2021 ) . Manifestations of such changes include imposing regulations or constraints over the objective function . The literature on ordinary least squares estimation has extensively dealt with some of the well-known concerns that might violate the assumptions , such as Ridge Hoerl & Kennard ( 1970a ; b ) , Lasso Tibshirani ( 1996 ) , Elastic Net Zou & Hastie ( 2005 ) , trees Quinlan ( 2014 ) , forest Breiman ( 2001 ) , boosting Bühlmann & Yu ( 2003 ) and others . Common regression models include bagging , boosting , random forest Criminisi & Shotton ( 2013 ) , oblique trees Murthy et al . ( 1994 ) ; Norouzi et al . ( 2015 ) ; Heath et al . ( 1993 ) , and regression SVM Drucker et al . ( 1997 ) . In the context of regression trees , various approaches of inducing a tree are presented in the literature . Most decision tree induction methods are concentrated on the splitting criterion used at the growing phase of the tree Ikonomovska et al . ( 2011 ) ; Levatić et al . ( 2014 ) . Application of decision tree algorithms to multi-response regression has been previously considered in the literature Breiman et al . ( 1984 ) ; De ’ Ath ( 2002 ) . In Breiman et al . ( 1984 ) ; Quinlan ( 1986 ) , authors consider training a decision tree for each individual output response . However , such approach constructs a large model specially if the number of output responses are high . Another approach proposed in De ’ Ath ( 2002 ) consists of constructing a single decision tree for all the output responses . In other words , the model predicts all the output values simultaneously through a single decision tree . However , a model for all the outputs might not be sufficient Kocev et al . ( 2009 ) because they train model for single response rather than the true problem which is a multi-response regression . Authors in Kocev et al . ( 2009 ) have explored two approaches to the multi-response regression problem by comparing learning a model for each output separately ( i.e. , multiple regression trees ) , and learning one model for all outputs simultaneously ( i.e. , a single multi-target regression tree ) . In order to improve predictive performance , Kocev et al . ( 2013 ) has also considered two ensemble learning techniques , namely , bagging Breiman ( 1996 ) ; Liang et al . ( 2011 ) and random forests Breiman ( 2001 ) for regression trees and multi-target regression trees . The results showed that multi-target regression trees created more accurate and compact models . A related topic to the problem of this paper is nearest neighbor regression . Nearest neighbor regression and local estimators are well-established methods in the literature of ordinary univariate location estimators ( Benedetti ( 1977 ) ; Stone ( 1974 ) ; Tukey et al . ( 1977 ) . However , as per our extensive search , there remains a gap in prototype nearest neighbor approaches to regression . The only work that considered a similar model and optimization to SRNN was Huang ( 2017 ) . However , the proposed algorithm does not have guarantee of convergence or achieving some sort of optimum solution . 3 PROPOSED METHOD . 3.1 PRELIMINARIES . Assume a dataset consisting of tuples ( xi , yi ) where x , y and i represent input features , output responses and index number . Each tuple represent a data xi and its corresponding output response yi . Here , xi ∈ R D and yi ∈ R d. The Regression Synthetic Reduced Nearest Neighbor ( Reg . SRNN ) consists of K tuples of synthetically produced centroids/prototypes ( cj , ŷj ) where c , ŷj and j represent the centroid ’ s point in the input space , output prediction and index . At the inference time , the Reg . SRNN operates like a nearest neighbor model where the centroids are used as the samples . The problem of training Reg . SRNN is as follows : min { ( cj , ŷj ) } K1 N ∑ i=1 ||yi − ŷj∗ i ||2 s.t j∗i = argmin { j } K 1 d ( xi − cj ) ( 2 ) where d ( . ) is a distance metric . Through this paper we use the l-2 norm as the distance metric : d ( xi − cj ) = √ ||xi − cj ||2 ( 3 ) Essentially the prediction of the model consists of the output prediction of closest centroid to the input sample . Officially , we define Reg . SRNN as follows : NN ( x ) = K ∑ j=1 yjI ( x ∈ Rj ) ( 4 ) Where , NN ( . ) represents a nearest neighbor function of the K centroids . I ( . ) is an indicator function that produces 1 if the input x is in the region of Rj . Rj represents the region where the closest centroid to the points in that region is cj . 3.2 INITIALIZATION . Numerical optimization algorithms require initialization ( cite num opt ) . In this paper , we propose a novel initialization for the regression SRNN . Previous work on the initialization of SRNN models consisted of learning a K-means model for each class of the data . For example , in case of M classes and K centroids , K M centroids are learned for each class as initialization of the SRNN Kusner et al . ( 2014 ) ; Wang et al . ( 2016 ) ; Zhong et al . ( 2017 ) ; Tavallali et al . ( 2020b ) . However , such approach is not applicable to the regression . This initialization also have close ties with naive Bayes and density estimation Silverman ( 2018 ) . Here , we expand this initialization to the case of Reg . SRNN . Intuitively , the output responses can consist of several modalities . In other words , it is possible that the output responses are generated from several distributions . The clusters of such distributions can be approximated by running a K-means over the output space ( M centroids ) . Assume that Sm represents the set of samples assigned to each output cluster . Next step consists of learning KN|Sm| centroids over the input features of the Sm for all M clusters . In other words , we learn centroids over the input features of each output cluster relative to the population of that cluster . The found centroids at the second step are used as initialization for the Reg . SRNN . At this step , ŷj is found using the following formula : ŷj = mean ( yi ∈ Sj ) ( 5 ) where Sj represents the set of samples that are assigned to j th centroid . Sj essentially consists of samples where jth centroid is the closest centroid to them . mean ( . ) represents the average of its input set . Note that Sj ∈ Rj . | The paper proposes a novel algorithm for learning prototype based nearest neighbor regression model. The algorithm minimizes an average of the quadratic loss on training examples w.r.t. the prototype centers and the prototype outputs by a block coordinate descent. The main contribution is the optimization algorithm finding the prototypes. | SP:4e742c63860ef00798f23d647b46af94de0abad4 |
Synthetic Reduced Nearest Neighbor Model for Regression | 1 INTRODUCTION . One of the main topics of research in Machine Learning is the relation between the features and output responses Hastie et al . ( 2009 ) ; Santosa & Symes ( 1986 ) ; Tibshirani ( 1996 ) ; Criminisi & Shotton ( 2013 ) . Synthetic Reduced Nearest Neighbor ( SRNN ) models are shown to be an effective tool in determining the relationships between features of the inputs and the sub-clusters of each class in supervised learning tasks Tavallali et al . ( 2020b ) . However , existing prototype nearest neighbor models such as SRNN are constrained to classification problems , and to the best of our knowledge , there remains a gap in extending these algorithms towards regression tasks . Such regression reduced nearest neighbor models may find extensive applications in epidemiological studies Tavallali et al . ( 2020a ) ; Cisneros et al . ( 2021 ) , medical studies Criminisi & Shotton ( 2013 ) ; Graf et al . ( 2011b ; a ) , and other applied regression tasks in general Tibshirani ( 1996 ) . To address this gap in the state of the art , we propose a novel algorithm for the optimization and construction of Regression Synthetic Reduced Nearest Neighbor ( Reg-SRNN ) models . The proposed Reg-SRNN is capable of discovering various modalities of the input data , and relates those to the modalities of the output responses . The Reg-SRNN algorithm is designed to handle both single-response and multi-response regression . The multi-response regression consists of learning the relation between input samples and several ground-truth output responses . Reg-SRNN partitions the input space into piecewise constant regions , where each region is represented by a centroid and its output response . From this perspective , Reg-SRNN is similar to other piecewise constant models , such as Li & Martin ( 2017 ) ; Begon et al . ( 2017 ) ; Bertsimas & Dunn ( 2017 ) ; Tavallali et al . ( 2019 ; 2020c ) . Reg-SRNN is capable of learning an accurate relation between each cluster of the data and its corresponding output responses . Therefore , Reg-SRNN can also provide enhanced interpretablity by reducing the information content of clusters into a compressed representation manifested in their centroids . The technical contributions of this paper include the proposal of a novel initialization that by itself is competitive to other existing regression models . This is proceeded by an expectation maximization algorithm for directly minimizing the least squares error of the mode . The proposed optimization algorithm is provably convergent , and it is shown that it monotonically decreases the loss function . Therefore , the algorithm has a convergence guarantee on minimizing the loss function and achieving a local optimum.It is also worth mentioning that the algorithm does not cycle . The proposed optimization algorithm consists of two steps and is inspired by K-means algorithm Lloyd ( 1982 ) . One step is the assignment step and is composed of finding samples assignments and proper output response of the centroid . Second step is the update step where the centroid is optimized such that the loss function is decreased . The centroid step is affected by all the samples and we will show that this update step is a kind of NP-hard weighted binary classification problem . The update step is computed through a surrogate objective function that is similar to SVM . We establish that the algorithm is efficient because of its linear computational complexity . Finally , the model is evaluated on various datasets with various sizes and dimensionalities , the results of which demonstrate that Reg-SRNN is capable of competing and even over-matching similar regression models . Accordingly , the main contributions of this paper are as follows : • We propose a novel algorithm for initialization of SRNN models to extend their application to regression tasks . • We develop an optimization algorithm for regression SRNN models with guarantees on convergence . • Through experimental evaluation , we demonstrated the feasibility of our proposed regression SRNN model in filling the gap between more complex models ( such as random forests ) and basic and interpretable models such as linear regression and decision trees . 2 RELATED WORK . A regression task consists of learning the relation between samples of the input space and a numerical output space . More specifically , regression is a supervised learning task of mapping inputs ( independent variable X ) to the output Y , which is a continuous vector ( Y ∈ Rd ) . If the dimensionality of the output d ≥ 2 , the task is known as multi-response regression . Regression has been the workhorse of numerous fields Tai ( 2021 ) , and various regression models have been developed and expanded fundamentally over the recent decades Hastie et al . ( 2009 ) . This expansion has been so rampant such that listing all such models and their relationships is a difficult task and is out of the scope of this work . However , a brief review of the recent models is presented in this paper . A common objective function for regression is to minimize the least squares error : ||Ŷ − Y ||2 ( 1 ) Where Ŷ is the prediction . According to the Gauss-Markov theorem Gauss ( 1823 ) , the least squares error can be an unbiased linear model of minimum variance of the data under certain assumptions . Ordinary least squares may fail to properly predict outcomes if it is applied to settings where the Gauss-Markov assumptions are not held . Therefore , it is important to understand the assumptions and occasionally apply the proper changes to the objective function of equation 1 to modify the model Tai ( 2021 ) . Manifestations of such changes include imposing regulations or constraints over the objective function . The literature on ordinary least squares estimation has extensively dealt with some of the well-known concerns that might violate the assumptions , such as Ridge Hoerl & Kennard ( 1970a ; b ) , Lasso Tibshirani ( 1996 ) , Elastic Net Zou & Hastie ( 2005 ) , trees Quinlan ( 2014 ) , forest Breiman ( 2001 ) , boosting Bühlmann & Yu ( 2003 ) and others . Common regression models include bagging , boosting , random forest Criminisi & Shotton ( 2013 ) , oblique trees Murthy et al . ( 1994 ) ; Norouzi et al . ( 2015 ) ; Heath et al . ( 1993 ) , and regression SVM Drucker et al . ( 1997 ) . In the context of regression trees , various approaches of inducing a tree are presented in the literature . Most decision tree induction methods are concentrated on the splitting criterion used at the growing phase of the tree Ikonomovska et al . ( 2011 ) ; Levatić et al . ( 2014 ) . Application of decision tree algorithms to multi-response regression has been previously considered in the literature Breiman et al . ( 1984 ) ; De ’ Ath ( 2002 ) . In Breiman et al . ( 1984 ) ; Quinlan ( 1986 ) , authors consider training a decision tree for each individual output response . However , such approach constructs a large model specially if the number of output responses are high . Another approach proposed in De ’ Ath ( 2002 ) consists of constructing a single decision tree for all the output responses . In other words , the model predicts all the output values simultaneously through a single decision tree . However , a model for all the outputs might not be sufficient Kocev et al . ( 2009 ) because they train model for single response rather than the true problem which is a multi-response regression . Authors in Kocev et al . ( 2009 ) have explored two approaches to the multi-response regression problem by comparing learning a model for each output separately ( i.e. , multiple regression trees ) , and learning one model for all outputs simultaneously ( i.e. , a single multi-target regression tree ) . In order to improve predictive performance , Kocev et al . ( 2013 ) has also considered two ensemble learning techniques , namely , bagging Breiman ( 1996 ) ; Liang et al . ( 2011 ) and random forests Breiman ( 2001 ) for regression trees and multi-target regression trees . The results showed that multi-target regression trees created more accurate and compact models . A related topic to the problem of this paper is nearest neighbor regression . Nearest neighbor regression and local estimators are well-established methods in the literature of ordinary univariate location estimators ( Benedetti ( 1977 ) ; Stone ( 1974 ) ; Tukey et al . ( 1977 ) . However , as per our extensive search , there remains a gap in prototype nearest neighbor approaches to regression . The only work that considered a similar model and optimization to SRNN was Huang ( 2017 ) . However , the proposed algorithm does not have guarantee of convergence or achieving some sort of optimum solution . 3 PROPOSED METHOD . 3.1 PRELIMINARIES . Assume a dataset consisting of tuples ( xi , yi ) where x , y and i represent input features , output responses and index number . Each tuple represent a data xi and its corresponding output response yi . Here , xi ∈ R D and yi ∈ R d. The Regression Synthetic Reduced Nearest Neighbor ( Reg . SRNN ) consists of K tuples of synthetically produced centroids/prototypes ( cj , ŷj ) where c , ŷj and j represent the centroid ’ s point in the input space , output prediction and index . At the inference time , the Reg . SRNN operates like a nearest neighbor model where the centroids are used as the samples . The problem of training Reg . SRNN is as follows : min { ( cj , ŷj ) } K1 N ∑ i=1 ||yi − ŷj∗ i ||2 s.t j∗i = argmin { j } K 1 d ( xi − cj ) ( 2 ) where d ( . ) is a distance metric . Through this paper we use the l-2 norm as the distance metric : d ( xi − cj ) = √ ||xi − cj ||2 ( 3 ) Essentially the prediction of the model consists of the output prediction of closest centroid to the input sample . Officially , we define Reg . SRNN as follows : NN ( x ) = K ∑ j=1 yjI ( x ∈ Rj ) ( 4 ) Where , NN ( . ) represents a nearest neighbor function of the K centroids . I ( . ) is an indicator function that produces 1 if the input x is in the region of Rj . Rj represents the region where the closest centroid to the points in that region is cj . 3.2 INITIALIZATION . Numerical optimization algorithms require initialization ( cite num opt ) . In this paper , we propose a novel initialization for the regression SRNN . Previous work on the initialization of SRNN models consisted of learning a K-means model for each class of the data . For example , in case of M classes and K centroids , K M centroids are learned for each class as initialization of the SRNN Kusner et al . ( 2014 ) ; Wang et al . ( 2016 ) ; Zhong et al . ( 2017 ) ; Tavallali et al . ( 2020b ) . However , such approach is not applicable to the regression . This initialization also have close ties with naive Bayes and density estimation Silverman ( 2018 ) . Here , we expand this initialization to the case of Reg . SRNN . Intuitively , the output responses can consist of several modalities . In other words , it is possible that the output responses are generated from several distributions . The clusters of such distributions can be approximated by running a K-means over the output space ( M centroids ) . Assume that Sm represents the set of samples assigned to each output cluster . Next step consists of learning KN|Sm| centroids over the input features of the Sm for all M clusters . In other words , we learn centroids over the input features of each output cluster relative to the population of that cluster . The found centroids at the second step are used as initialization for the Reg . SRNN . At this step , ŷj is found using the following formula : ŷj = mean ( yi ∈ Sj ) ( 5 ) where Sj represents the set of samples that are assigned to j th centroid . Sj essentially consists of samples where jth centroid is the closest centroid to them . mean ( . ) represents the average of its input set . Note that Sj ∈ Rj . | This paper proposed a new method to extend the synthetic reduced nearest neighbor model for regression problems. The idea of the proposed method is based on an initialization approach of creating clusters based on the target variable and an EM approach for minimizing the loss function of the proposed Reg. SRNN model. This paper provides an analysis of the consistency of the proposed method. The experiments results compared the proposed methods with several existing approaches. | SP:4e742c63860ef00798f23d647b46af94de0abad4 |
Patch-Fool: Are Vision Transformers Always Robust Against Adversarial Perturbations? | 1 INTRODUCTION . The recent performance breakthroughs achieved by vision transformers ( ViTs ) Dosovitskiy et al . ( 2020 ) have fueled an increasing enthusiasm towards designing new ViT architectures for different vision tasks , including object detection ( Carion et al. , 2020 ; Beal et al. , 2020 ) , semantic segmentation ( Strudel et al. , 2021 ; Zheng et al. , 2021 ; Wang et al. , 2021 ) , and video recognition ( Arnab et al. , 2021 ; Liu et al. , 2021b ; Li et al. , 2021b ; Fan et al. , 2021 ) . To fulfill the goal of deploying ViTs into real-world vision applications , the security concern of ViTs is of great importance and challenge , especially in the context of adversarial attacks ( Goodfellow et al. , 2014 ) , under which an imperceptible perturbation onto the inputs can mislead the models to malfunction . In response , the robustness of ViTs against adversarial attacks has attracted increasing attention . For example , recent works ( Bhojanapalli et al. , 2021 ; Aldahdooh et al. , 2021 ; Shao et al. , 2021 ) find that in addition to ViTs ’ decent task performance , they are more robust to adversarial attacks compared with convolutional neural networks ( CNNs ) under comparable model complexities . In particular , ( Shao et al. , 2021 ) claims that ViTs focus more on capturing the global interaction among input/feature patches via its self-attention mechanism and the learned features contain less low-level information , leading to superior robustness to the local perturbations introduced by adversarial attacks . A natural response to this seemingly good news would be determining whether ViTs are truly robust against all kinds of adversarial perturbations or if their current winning in robustness is an inevitable result of biased evaluations using existing attack methods that are mostly dedicated to CNNs . To unveil the potential vulnerability of ViTs , this work takes the first step in asking an intriguing question : “ Under what kinds of perturbations do ViTs become weaker learners compared to CNNs ? `` , and makes the following contributions : • We conduct extensive experiments to benchmark the robustness of different ViT variants , which integrate recent advances and new arts in ViTs ’ architecture design , over that of CNNs under existing adversarial attacks ; • We propose a new attack framework , dubbed Patch-Fool , aiming to fool the self-attention mechanism by attacking the basic component ( i.e. , a single patch ) participating in selfattention calculations . Our Patch-Fool attack features a novel objective formulation , which is then solved by Patch-Fool ’ s integrated attention-aware patch selection technique and attention-aware loss design ; • We evaluate the robustness of both ViTs and CNNs against our Patch-Fool attack with extensive experiments and find that ViTs are consistently less robust than CNNs across various attack settings , indicating that ViTs are not always robust learners and their seeming robustness against existing attacks can be overturned under dedicated adversarial attacks ; • We further benchmark the robustness of both ViTs and CNNs under a sparse variant of Patch-Fool , dubbed Sparse Patch-Fool , and discover that the perturbation density , defined as the number of perturbed pixels per patch , highly influences the robustness ranking between ViTs and CNNs , where our Patch-Fool is an extreme case of high perturbation density . We believe our work has opened up a new perspective for exploring ViTs ’ vulnerability and can provide insights to both future architecture designs and training schemes for robustifying ViTs towards their real-world deployment . 2 RELATED WORKS . Vision transformers . Motivated by the great success of Transformers in the natural language processing ( NLP ) field ( Vaswani et al. , 2017 ) , ViTs have been developed by splitting an input image into a series of image patches and adopting self-attention modules for encoding the image ( Dosovitskiy et al. , 2020 ) , and been shown to achieve competitive or superior performance over CNNs via dedicated data augmentation ( Touvron et al. , 2021 ) or self-attention structure ( Yang et al. , 2021 ; Graham et al. , 2021 ; Liu et al. , 2021a ) . As such , there has been tremendously increased attention on applying ViTs to various computer vision applications , such as self-supervised learning ( Caron et al. , 2021 ; Chen et al. , 2021b ; Xie et al. , 2021 ; Li et al. , 2021a ) , object detection ( Carion et al. , 2020 ; Beal et al. , 2020 ) , and semantic segmentation ( Strudel et al. , 2021 ; Zheng et al. , 2021 ; Wang et al. , 2021 ) . The achievable performance of ViTs are continuously refreshed by emerging ViT variants , which provide new arts for designing ViT architectures . For example , convolutional modules have been incorporated into ViTs for capturing low-level features ( Xiao et al. , 2021 ; Wu et al. , 2021 ; Graham et al. , 2021 ; Peng et al. , 2021 ) , and replacing the global self-attention mechanism with local self-attention modules ( Liu et al. , 2021a ; Dong et al. , 2021 ; Liang et al. , 2021 ; Liu et al. , 2021b ; Chu et al. , 2021 ) . Motivated by ViTs ’ impressive performance and the growing interest in deploying ViTs into real-world applications , this work aims to better understand the robustness of ViTs and to develop adversarial attacks dedicated to ViTs . Adversarial attack and defense . Deep neural networks ( DNNs ) are known to be vulnerable to adversarial attacks ( Goodfellow et al. , 2014 ) , i.e. , imperceptible perturbations onto the inputs can mislead DNNs to make wrong predictions . As adversaries , stronger attacks are continuously developed , including both white-box ( Madry et al. , 2017 ; Croce & Hein , 2020 ; Carlini & Wagner , 2017 ; Papernot et al. , 2016 ; Moosavi-Dezfooli et al. , 2016 ) and black-box ones ( Chen et al. , 2017 ; Ilyas et al. , 2018b ; Andriushchenko et al. , 2020 ; Guo et al. , 2019 ; Ilyas et al. , 2018a ) , which aggressively degrade the performances of the target DNN models . In response , various defense methods have been proposed to improve DNNs ’ robustness against those adversarial attacks . For example , purification methods ( Guo et al. , 2017 ; Xie et al. , 2017 ; Xu et al. , 2017 ; Song et al. , 2017 ; Shi et al. , 2021 ) applied onto the inputs purify the adversarial samples back to the distribution of clean ones ; random smoothing ( Cohen et al. , 2019 ; Li et al. , 2018 ) onto the inputs certifiably guarantees DNNs ’ robustness against adversarial perturbations ; detection-based methods ( Metzen et al. , 2017 ; Feinman et al. , 2017 ; Lee et al. , 2018 ) equipped with additional detection modules distinguish adversarial examples from clean ones before feeding them into the target DNN models ; and adversarial training ( Shafahi et al. , 2019 ; Madry et al. , 2017 ; Wong et al. , 2019 ; Tramèr et al. , 2017 ) , which augments the training set with adversarial samples generated on the fly during training , is currently one of the most effective defense methods . There has been a continuous war between adversaries and defenders and the readers are referred to ( Akhtar & Mian , 2018 ; Chakraborty et al. , 2018 ) for more attack and defense methods . Robustness of vision transformers . Driven by the impressive performance recently achieved by ViTs in various vision tasks , their robustness has gained increasing attention . A consistent observation drawn by pioneering works that study ViTs ’ robustness is that ViTs are more robust to adversarial attacks than CNNs since ViTs are more capable of capturing the global interactions among patches , while CNNs focus on local features and thus are more vulnerable to local adversarial perturbations . In particular , ( Bhojanapalli et al. , 2021 ) shows that ViT models pretrained with a sufficient amount of data are at least as robust as their ResNet counterparts on a broad range of perturbations , including natural corruptions , distribution shifts , and adversarial perturbations ; ( Aldahdooh et al. , 2021 ) finds that vanilla ViTs or hybrid-ViTs are more robust than CNNs under 𝐿𝑝-based attacks ; and ( Shao et al. , 2021 ) further explains that ViTs ’ learned features contain less low-level information and are more generalizable , leading to their superior robustness , and introducing convolutional blocks that extract more low-level features will reduce the ViTs ’ adversarial robustness . In addition , ViTs ’ adversarial transferability has also been studied : ( Mahmood et al. , 2021 ) shows that adversarial examples do not readily transfer between CNNs and transformers and ( Naseer et al. , 2021 ; Wei et al. , 2021 ) propose techniques to boost the adversarial transferability between ViTs and from ViTs to CNNs . In parallel , ( Mao et al. , 2021 ) refines ViTs ’ architecture design to improve robustness . In our work , we challenge the common belief that ViTs are more robust than CNNs , which is concluded based on evaluations using existing attack methods , and propose to customize adaptive attacks utilizing ViTs ’ captured patch-wise global interactions to make ViTs weaker learners . 3 EVALUATING ROBUSTNESS OF VITS AND CNNS UNDER EXISTING ATTACKS . Although various comparisons on robustness of ViTs and CNNs have been explored in pioneering works ( Bhojanapalli et al. , 2021 ; Aldahdooh et al. , 2021 ; Shao et al. , 2021 ) , their evaluation suffers from one of the following limitations : ( 1 ) only adopt weak attack methods , ( 2 ) only adopt early ViT designs without considering recently advanced ViT architectures , and ( 3 ) do not adopt the official and latest pretrained models and suffer from inferior clean accuracies . To this end , we extensively evaluate the robustness against common white-box attacks of several representative ViT variants , which cover the popular trends in designing ViT architectures , including ( 1 ) using local self-attention ( Swin ( Liu et al. , 2021a ) ) , which adopts the attention mechanism within a local region instead of the global ones in vanilla ViTs to capture low-level features and reduce the computational cost , and ( 2 ) introducing the inductive bias of CNNs to build hybrid models ( LeViT ( Graham et al. , 2021 ) ) . 3.1 EVALUATION SETUP . Models and datasets . We evaluate the robustness of three ViT families ( i.e. , DeiT ( Touvron et al. , 2021 ) , Swin ( Liu et al. , 2021a ) , and LeViT ( Graham et al. , 2021 ) ) and two CNN families Query Point Adv Img from Patch-Fool Layer 2 Original Img Adv Img from PGD-20 Layer 6 Layer 12 Figure 1 : Comparisons among the attention maps in the intermediate layers of DeiT-S generated by the clean inputs , the adversarial inputs under PGD-20 attacks ( 𝜖 = 0.003 ) , and the proposed Patch-Fool attack , respectively . In particular , we average the attention scores across all the attention heads in each layer and visualize the attention score of each token for a given query token ( the center patch in the red box in our show case ) , following ( Kim et al. , 2021 ) . We can observe that the difference in attention maps between clean and adversarial inputs generated by PGD-20 keeps small across different layers ; In contrast , the proposed Patch-Fool notably enlarges the gap between clean and adversarial attention maps , demonstrating a successful attack for ViTs . ( ResNet ( He et al. , 2016 ) and VGG ( Simonyan & Zisserman , 2014 ) ) on ImageNet using their official implementation and pretrained models . Note that we adopt DeiT models without distillation , which only improves the training schedule over vanilla ViTs , for a fair comparison . Attack settings . We adopt four adversarial attacks ( i.e. , PGD ( Madry et al. , 2017 ) , AutoAttack ( Croce & Hein , 2020 ) , CW-𝐿𝐼𝑛 𝑓 ( Carlini & Wagner , 2017 ) , and CW-𝐿2 ) with different perturbation strengths . In particular , for the CW-𝐿𝐼𝑛 𝑓 and CW-𝐿2 attacks , we adopt the implementation in AdverTorch ( Ding et al. , 2019 ) and the same settings as ( Chen et al. , 2021a ; Rony et al. , 2019 ) ; For AutoAttack , we adopt the official implementation and default settings in ( Croce & Hein , 2020 ) . | This paper propose a new attack on Vision Transformers (ViTs) called Patch-Fool. The attack proceeds by first picking a patch which contributes the most (in the self attention calculation) to other patches and then perturbed it adversarially wrt cross entropy + attention based loss. The results show that this kind of attack degrades performance significantly wrt prior work. The authors then perform various ablation studies to justify their architecture/loss choices. | SP:689c9069c9a3e66b8c7445bb48f67af2b131ce8b |
Patch-Fool: Are Vision Transformers Always Robust Against Adversarial Perturbations? | 1 INTRODUCTION . The recent performance breakthroughs achieved by vision transformers ( ViTs ) Dosovitskiy et al . ( 2020 ) have fueled an increasing enthusiasm towards designing new ViT architectures for different vision tasks , including object detection ( Carion et al. , 2020 ; Beal et al. , 2020 ) , semantic segmentation ( Strudel et al. , 2021 ; Zheng et al. , 2021 ; Wang et al. , 2021 ) , and video recognition ( Arnab et al. , 2021 ; Liu et al. , 2021b ; Li et al. , 2021b ; Fan et al. , 2021 ) . To fulfill the goal of deploying ViTs into real-world vision applications , the security concern of ViTs is of great importance and challenge , especially in the context of adversarial attacks ( Goodfellow et al. , 2014 ) , under which an imperceptible perturbation onto the inputs can mislead the models to malfunction . In response , the robustness of ViTs against adversarial attacks has attracted increasing attention . For example , recent works ( Bhojanapalli et al. , 2021 ; Aldahdooh et al. , 2021 ; Shao et al. , 2021 ) find that in addition to ViTs ’ decent task performance , they are more robust to adversarial attacks compared with convolutional neural networks ( CNNs ) under comparable model complexities . In particular , ( Shao et al. , 2021 ) claims that ViTs focus more on capturing the global interaction among input/feature patches via its self-attention mechanism and the learned features contain less low-level information , leading to superior robustness to the local perturbations introduced by adversarial attacks . A natural response to this seemingly good news would be determining whether ViTs are truly robust against all kinds of adversarial perturbations or if their current winning in robustness is an inevitable result of biased evaluations using existing attack methods that are mostly dedicated to CNNs . To unveil the potential vulnerability of ViTs , this work takes the first step in asking an intriguing question : “ Under what kinds of perturbations do ViTs become weaker learners compared to CNNs ? `` , and makes the following contributions : • We conduct extensive experiments to benchmark the robustness of different ViT variants , which integrate recent advances and new arts in ViTs ’ architecture design , over that of CNNs under existing adversarial attacks ; • We propose a new attack framework , dubbed Patch-Fool , aiming to fool the self-attention mechanism by attacking the basic component ( i.e. , a single patch ) participating in selfattention calculations . Our Patch-Fool attack features a novel objective formulation , which is then solved by Patch-Fool ’ s integrated attention-aware patch selection technique and attention-aware loss design ; • We evaluate the robustness of both ViTs and CNNs against our Patch-Fool attack with extensive experiments and find that ViTs are consistently less robust than CNNs across various attack settings , indicating that ViTs are not always robust learners and their seeming robustness against existing attacks can be overturned under dedicated adversarial attacks ; • We further benchmark the robustness of both ViTs and CNNs under a sparse variant of Patch-Fool , dubbed Sparse Patch-Fool , and discover that the perturbation density , defined as the number of perturbed pixels per patch , highly influences the robustness ranking between ViTs and CNNs , where our Patch-Fool is an extreme case of high perturbation density . We believe our work has opened up a new perspective for exploring ViTs ’ vulnerability and can provide insights to both future architecture designs and training schemes for robustifying ViTs towards their real-world deployment . 2 RELATED WORKS . Vision transformers . Motivated by the great success of Transformers in the natural language processing ( NLP ) field ( Vaswani et al. , 2017 ) , ViTs have been developed by splitting an input image into a series of image patches and adopting self-attention modules for encoding the image ( Dosovitskiy et al. , 2020 ) , and been shown to achieve competitive or superior performance over CNNs via dedicated data augmentation ( Touvron et al. , 2021 ) or self-attention structure ( Yang et al. , 2021 ; Graham et al. , 2021 ; Liu et al. , 2021a ) . As such , there has been tremendously increased attention on applying ViTs to various computer vision applications , such as self-supervised learning ( Caron et al. , 2021 ; Chen et al. , 2021b ; Xie et al. , 2021 ; Li et al. , 2021a ) , object detection ( Carion et al. , 2020 ; Beal et al. , 2020 ) , and semantic segmentation ( Strudel et al. , 2021 ; Zheng et al. , 2021 ; Wang et al. , 2021 ) . The achievable performance of ViTs are continuously refreshed by emerging ViT variants , which provide new arts for designing ViT architectures . For example , convolutional modules have been incorporated into ViTs for capturing low-level features ( Xiao et al. , 2021 ; Wu et al. , 2021 ; Graham et al. , 2021 ; Peng et al. , 2021 ) , and replacing the global self-attention mechanism with local self-attention modules ( Liu et al. , 2021a ; Dong et al. , 2021 ; Liang et al. , 2021 ; Liu et al. , 2021b ; Chu et al. , 2021 ) . Motivated by ViTs ’ impressive performance and the growing interest in deploying ViTs into real-world applications , this work aims to better understand the robustness of ViTs and to develop adversarial attacks dedicated to ViTs . Adversarial attack and defense . Deep neural networks ( DNNs ) are known to be vulnerable to adversarial attacks ( Goodfellow et al. , 2014 ) , i.e. , imperceptible perturbations onto the inputs can mislead DNNs to make wrong predictions . As adversaries , stronger attacks are continuously developed , including both white-box ( Madry et al. , 2017 ; Croce & Hein , 2020 ; Carlini & Wagner , 2017 ; Papernot et al. , 2016 ; Moosavi-Dezfooli et al. , 2016 ) and black-box ones ( Chen et al. , 2017 ; Ilyas et al. , 2018b ; Andriushchenko et al. , 2020 ; Guo et al. , 2019 ; Ilyas et al. , 2018a ) , which aggressively degrade the performances of the target DNN models . In response , various defense methods have been proposed to improve DNNs ’ robustness against those adversarial attacks . For example , purification methods ( Guo et al. , 2017 ; Xie et al. , 2017 ; Xu et al. , 2017 ; Song et al. , 2017 ; Shi et al. , 2021 ) applied onto the inputs purify the adversarial samples back to the distribution of clean ones ; random smoothing ( Cohen et al. , 2019 ; Li et al. , 2018 ) onto the inputs certifiably guarantees DNNs ’ robustness against adversarial perturbations ; detection-based methods ( Metzen et al. , 2017 ; Feinman et al. , 2017 ; Lee et al. , 2018 ) equipped with additional detection modules distinguish adversarial examples from clean ones before feeding them into the target DNN models ; and adversarial training ( Shafahi et al. , 2019 ; Madry et al. , 2017 ; Wong et al. , 2019 ; Tramèr et al. , 2017 ) , which augments the training set with adversarial samples generated on the fly during training , is currently one of the most effective defense methods . There has been a continuous war between adversaries and defenders and the readers are referred to ( Akhtar & Mian , 2018 ; Chakraborty et al. , 2018 ) for more attack and defense methods . Robustness of vision transformers . Driven by the impressive performance recently achieved by ViTs in various vision tasks , their robustness has gained increasing attention . A consistent observation drawn by pioneering works that study ViTs ’ robustness is that ViTs are more robust to adversarial attacks than CNNs since ViTs are more capable of capturing the global interactions among patches , while CNNs focus on local features and thus are more vulnerable to local adversarial perturbations . In particular , ( Bhojanapalli et al. , 2021 ) shows that ViT models pretrained with a sufficient amount of data are at least as robust as their ResNet counterparts on a broad range of perturbations , including natural corruptions , distribution shifts , and adversarial perturbations ; ( Aldahdooh et al. , 2021 ) finds that vanilla ViTs or hybrid-ViTs are more robust than CNNs under 𝐿𝑝-based attacks ; and ( Shao et al. , 2021 ) further explains that ViTs ’ learned features contain less low-level information and are more generalizable , leading to their superior robustness , and introducing convolutional blocks that extract more low-level features will reduce the ViTs ’ adversarial robustness . In addition , ViTs ’ adversarial transferability has also been studied : ( Mahmood et al. , 2021 ) shows that adversarial examples do not readily transfer between CNNs and transformers and ( Naseer et al. , 2021 ; Wei et al. , 2021 ) propose techniques to boost the adversarial transferability between ViTs and from ViTs to CNNs . In parallel , ( Mao et al. , 2021 ) refines ViTs ’ architecture design to improve robustness . In our work , we challenge the common belief that ViTs are more robust than CNNs , which is concluded based on evaluations using existing attack methods , and propose to customize adaptive attacks utilizing ViTs ’ captured patch-wise global interactions to make ViTs weaker learners . 3 EVALUATING ROBUSTNESS OF VITS AND CNNS UNDER EXISTING ATTACKS . Although various comparisons on robustness of ViTs and CNNs have been explored in pioneering works ( Bhojanapalli et al. , 2021 ; Aldahdooh et al. , 2021 ; Shao et al. , 2021 ) , their evaluation suffers from one of the following limitations : ( 1 ) only adopt weak attack methods , ( 2 ) only adopt early ViT designs without considering recently advanced ViT architectures , and ( 3 ) do not adopt the official and latest pretrained models and suffer from inferior clean accuracies . To this end , we extensively evaluate the robustness against common white-box attacks of several representative ViT variants , which cover the popular trends in designing ViT architectures , including ( 1 ) using local self-attention ( Swin ( Liu et al. , 2021a ) ) , which adopts the attention mechanism within a local region instead of the global ones in vanilla ViTs to capture low-level features and reduce the computational cost , and ( 2 ) introducing the inductive bias of CNNs to build hybrid models ( LeViT ( Graham et al. , 2021 ) ) . 3.1 EVALUATION SETUP . Models and datasets . We evaluate the robustness of three ViT families ( i.e. , DeiT ( Touvron et al. , 2021 ) , Swin ( Liu et al. , 2021a ) , and LeViT ( Graham et al. , 2021 ) ) and two CNN families Query Point Adv Img from Patch-Fool Layer 2 Original Img Adv Img from PGD-20 Layer 6 Layer 12 Figure 1 : Comparisons among the attention maps in the intermediate layers of DeiT-S generated by the clean inputs , the adversarial inputs under PGD-20 attacks ( 𝜖 = 0.003 ) , and the proposed Patch-Fool attack , respectively . In particular , we average the attention scores across all the attention heads in each layer and visualize the attention score of each token for a given query token ( the center patch in the red box in our show case ) , following ( Kim et al. , 2021 ) . We can observe that the difference in attention maps between clean and adversarial inputs generated by PGD-20 keeps small across different layers ; In contrast , the proposed Patch-Fool notably enlarges the gap between clean and adversarial attention maps , demonstrating a successful attack for ViTs . ( ResNet ( He et al. , 2016 ) and VGG ( Simonyan & Zisserman , 2014 ) ) on ImageNet using their official implementation and pretrained models . Note that we adopt DeiT models without distillation , which only improves the training schedule over vanilla ViTs , for a fair comparison . Attack settings . We adopt four adversarial attacks ( i.e. , PGD ( Madry et al. , 2017 ) , AutoAttack ( Croce & Hein , 2020 ) , CW-𝐿𝐼𝑛 𝑓 ( Carlini & Wagner , 2017 ) , and CW-𝐿2 ) with different perturbation strengths . In particular , for the CW-𝐿𝐼𝑛 𝑓 and CW-𝐿2 attacks , we adopt the implementation in AdverTorch ( Ding et al. , 2019 ) and the same settings as ( Chen et al. , 2021a ; Rony et al. , 2019 ) ; For AutoAttack , we adopt the official implementation and default settings in ( Croce & Hein , 2020 ) . | Given recent finding shows that ViTs are more robust than CNN, this paper investigates an intriguing question: “Under what kinds of perturbations do ViTs become weaker learners compared to CNNs". They propose a Patch-Fool to fool the attention mechanism. Their investigation leads to some interesting findings and might inspire more interesting future work. | SP:689c9069c9a3e66b8c7445bb48f67af2b131ce8b |
Patch-Fool: Are Vision Transformers Always Robust Against Adversarial Perturbations? | 1 INTRODUCTION . The recent performance breakthroughs achieved by vision transformers ( ViTs ) Dosovitskiy et al . ( 2020 ) have fueled an increasing enthusiasm towards designing new ViT architectures for different vision tasks , including object detection ( Carion et al. , 2020 ; Beal et al. , 2020 ) , semantic segmentation ( Strudel et al. , 2021 ; Zheng et al. , 2021 ; Wang et al. , 2021 ) , and video recognition ( Arnab et al. , 2021 ; Liu et al. , 2021b ; Li et al. , 2021b ; Fan et al. , 2021 ) . To fulfill the goal of deploying ViTs into real-world vision applications , the security concern of ViTs is of great importance and challenge , especially in the context of adversarial attacks ( Goodfellow et al. , 2014 ) , under which an imperceptible perturbation onto the inputs can mislead the models to malfunction . In response , the robustness of ViTs against adversarial attacks has attracted increasing attention . For example , recent works ( Bhojanapalli et al. , 2021 ; Aldahdooh et al. , 2021 ; Shao et al. , 2021 ) find that in addition to ViTs ’ decent task performance , they are more robust to adversarial attacks compared with convolutional neural networks ( CNNs ) under comparable model complexities . In particular , ( Shao et al. , 2021 ) claims that ViTs focus more on capturing the global interaction among input/feature patches via its self-attention mechanism and the learned features contain less low-level information , leading to superior robustness to the local perturbations introduced by adversarial attacks . A natural response to this seemingly good news would be determining whether ViTs are truly robust against all kinds of adversarial perturbations or if their current winning in robustness is an inevitable result of biased evaluations using existing attack methods that are mostly dedicated to CNNs . To unveil the potential vulnerability of ViTs , this work takes the first step in asking an intriguing question : “ Under what kinds of perturbations do ViTs become weaker learners compared to CNNs ? `` , and makes the following contributions : • We conduct extensive experiments to benchmark the robustness of different ViT variants , which integrate recent advances and new arts in ViTs ’ architecture design , over that of CNNs under existing adversarial attacks ; • We propose a new attack framework , dubbed Patch-Fool , aiming to fool the self-attention mechanism by attacking the basic component ( i.e. , a single patch ) participating in selfattention calculations . Our Patch-Fool attack features a novel objective formulation , which is then solved by Patch-Fool ’ s integrated attention-aware patch selection technique and attention-aware loss design ; • We evaluate the robustness of both ViTs and CNNs against our Patch-Fool attack with extensive experiments and find that ViTs are consistently less robust than CNNs across various attack settings , indicating that ViTs are not always robust learners and their seeming robustness against existing attacks can be overturned under dedicated adversarial attacks ; • We further benchmark the robustness of both ViTs and CNNs under a sparse variant of Patch-Fool , dubbed Sparse Patch-Fool , and discover that the perturbation density , defined as the number of perturbed pixels per patch , highly influences the robustness ranking between ViTs and CNNs , where our Patch-Fool is an extreme case of high perturbation density . We believe our work has opened up a new perspective for exploring ViTs ’ vulnerability and can provide insights to both future architecture designs and training schemes for robustifying ViTs towards their real-world deployment . 2 RELATED WORKS . Vision transformers . Motivated by the great success of Transformers in the natural language processing ( NLP ) field ( Vaswani et al. , 2017 ) , ViTs have been developed by splitting an input image into a series of image patches and adopting self-attention modules for encoding the image ( Dosovitskiy et al. , 2020 ) , and been shown to achieve competitive or superior performance over CNNs via dedicated data augmentation ( Touvron et al. , 2021 ) or self-attention structure ( Yang et al. , 2021 ; Graham et al. , 2021 ; Liu et al. , 2021a ) . As such , there has been tremendously increased attention on applying ViTs to various computer vision applications , such as self-supervised learning ( Caron et al. , 2021 ; Chen et al. , 2021b ; Xie et al. , 2021 ; Li et al. , 2021a ) , object detection ( Carion et al. , 2020 ; Beal et al. , 2020 ) , and semantic segmentation ( Strudel et al. , 2021 ; Zheng et al. , 2021 ; Wang et al. , 2021 ) . The achievable performance of ViTs are continuously refreshed by emerging ViT variants , which provide new arts for designing ViT architectures . For example , convolutional modules have been incorporated into ViTs for capturing low-level features ( Xiao et al. , 2021 ; Wu et al. , 2021 ; Graham et al. , 2021 ; Peng et al. , 2021 ) , and replacing the global self-attention mechanism with local self-attention modules ( Liu et al. , 2021a ; Dong et al. , 2021 ; Liang et al. , 2021 ; Liu et al. , 2021b ; Chu et al. , 2021 ) . Motivated by ViTs ’ impressive performance and the growing interest in deploying ViTs into real-world applications , this work aims to better understand the robustness of ViTs and to develop adversarial attacks dedicated to ViTs . Adversarial attack and defense . Deep neural networks ( DNNs ) are known to be vulnerable to adversarial attacks ( Goodfellow et al. , 2014 ) , i.e. , imperceptible perturbations onto the inputs can mislead DNNs to make wrong predictions . As adversaries , stronger attacks are continuously developed , including both white-box ( Madry et al. , 2017 ; Croce & Hein , 2020 ; Carlini & Wagner , 2017 ; Papernot et al. , 2016 ; Moosavi-Dezfooli et al. , 2016 ) and black-box ones ( Chen et al. , 2017 ; Ilyas et al. , 2018b ; Andriushchenko et al. , 2020 ; Guo et al. , 2019 ; Ilyas et al. , 2018a ) , which aggressively degrade the performances of the target DNN models . In response , various defense methods have been proposed to improve DNNs ’ robustness against those adversarial attacks . For example , purification methods ( Guo et al. , 2017 ; Xie et al. , 2017 ; Xu et al. , 2017 ; Song et al. , 2017 ; Shi et al. , 2021 ) applied onto the inputs purify the adversarial samples back to the distribution of clean ones ; random smoothing ( Cohen et al. , 2019 ; Li et al. , 2018 ) onto the inputs certifiably guarantees DNNs ’ robustness against adversarial perturbations ; detection-based methods ( Metzen et al. , 2017 ; Feinman et al. , 2017 ; Lee et al. , 2018 ) equipped with additional detection modules distinguish adversarial examples from clean ones before feeding them into the target DNN models ; and adversarial training ( Shafahi et al. , 2019 ; Madry et al. , 2017 ; Wong et al. , 2019 ; Tramèr et al. , 2017 ) , which augments the training set with adversarial samples generated on the fly during training , is currently one of the most effective defense methods . There has been a continuous war between adversaries and defenders and the readers are referred to ( Akhtar & Mian , 2018 ; Chakraborty et al. , 2018 ) for more attack and defense methods . Robustness of vision transformers . Driven by the impressive performance recently achieved by ViTs in various vision tasks , their robustness has gained increasing attention . A consistent observation drawn by pioneering works that study ViTs ’ robustness is that ViTs are more robust to adversarial attacks than CNNs since ViTs are more capable of capturing the global interactions among patches , while CNNs focus on local features and thus are more vulnerable to local adversarial perturbations . In particular , ( Bhojanapalli et al. , 2021 ) shows that ViT models pretrained with a sufficient amount of data are at least as robust as their ResNet counterparts on a broad range of perturbations , including natural corruptions , distribution shifts , and adversarial perturbations ; ( Aldahdooh et al. , 2021 ) finds that vanilla ViTs or hybrid-ViTs are more robust than CNNs under 𝐿𝑝-based attacks ; and ( Shao et al. , 2021 ) further explains that ViTs ’ learned features contain less low-level information and are more generalizable , leading to their superior robustness , and introducing convolutional blocks that extract more low-level features will reduce the ViTs ’ adversarial robustness . In addition , ViTs ’ adversarial transferability has also been studied : ( Mahmood et al. , 2021 ) shows that adversarial examples do not readily transfer between CNNs and transformers and ( Naseer et al. , 2021 ; Wei et al. , 2021 ) propose techniques to boost the adversarial transferability between ViTs and from ViTs to CNNs . In parallel , ( Mao et al. , 2021 ) refines ViTs ’ architecture design to improve robustness . In our work , we challenge the common belief that ViTs are more robust than CNNs , which is concluded based on evaluations using existing attack methods , and propose to customize adaptive attacks utilizing ViTs ’ captured patch-wise global interactions to make ViTs weaker learners . 3 EVALUATING ROBUSTNESS OF VITS AND CNNS UNDER EXISTING ATTACKS . Although various comparisons on robustness of ViTs and CNNs have been explored in pioneering works ( Bhojanapalli et al. , 2021 ; Aldahdooh et al. , 2021 ; Shao et al. , 2021 ) , their evaluation suffers from one of the following limitations : ( 1 ) only adopt weak attack methods , ( 2 ) only adopt early ViT designs without considering recently advanced ViT architectures , and ( 3 ) do not adopt the official and latest pretrained models and suffer from inferior clean accuracies . To this end , we extensively evaluate the robustness against common white-box attacks of several representative ViT variants , which cover the popular trends in designing ViT architectures , including ( 1 ) using local self-attention ( Swin ( Liu et al. , 2021a ) ) , which adopts the attention mechanism within a local region instead of the global ones in vanilla ViTs to capture low-level features and reduce the computational cost , and ( 2 ) introducing the inductive bias of CNNs to build hybrid models ( LeViT ( Graham et al. , 2021 ) ) . 3.1 EVALUATION SETUP . Models and datasets . We evaluate the robustness of three ViT families ( i.e. , DeiT ( Touvron et al. , 2021 ) , Swin ( Liu et al. , 2021a ) , and LeViT ( Graham et al. , 2021 ) ) and two CNN families Query Point Adv Img from Patch-Fool Layer 2 Original Img Adv Img from PGD-20 Layer 6 Layer 12 Figure 1 : Comparisons among the attention maps in the intermediate layers of DeiT-S generated by the clean inputs , the adversarial inputs under PGD-20 attacks ( 𝜖 = 0.003 ) , and the proposed Patch-Fool attack , respectively . In particular , we average the attention scores across all the attention heads in each layer and visualize the attention score of each token for a given query token ( the center patch in the red box in our show case ) , following ( Kim et al. , 2021 ) . We can observe that the difference in attention maps between clean and adversarial inputs generated by PGD-20 keeps small across different layers ; In contrast , the proposed Patch-Fool notably enlarges the gap between clean and adversarial attention maps , demonstrating a successful attack for ViTs . ( ResNet ( He et al. , 2016 ) and VGG ( Simonyan & Zisserman , 2014 ) ) on ImageNet using their official implementation and pretrained models . Note that we adopt DeiT models without distillation , which only improves the training schedule over vanilla ViTs , for a fair comparison . Attack settings . We adopt four adversarial attacks ( i.e. , PGD ( Madry et al. , 2017 ) , AutoAttack ( Croce & Hein , 2020 ) , CW-𝐿𝐼𝑛 𝑓 ( Carlini & Wagner , 2017 ) , and CW-𝐿2 ) with different perturbation strengths . In particular , for the CW-𝐿𝐼𝑛 𝑓 and CW-𝐿2 attacks , we adopt the implementation in AdverTorch ( Ding et al. , 2019 ) and the same settings as ( Chen et al. , 2021a ; Rony et al. , 2019 ) ; For AutoAttack , we adopt the official implementation and default settings in ( Croce & Hein , 2020 ) . | This paper proposes a Patch-Fool attack which fools the self-attention mechanism of ViTs to evaluate the robustness of ViTs and CNNs-based models. Prior works investigated the robustness of ViTs and CNNs under the adversarial attacks designed mainly for CNNs and conjectured that ViTs are more robust than CNNs. However, in this paper, the authors attack a specific single patch and develop an attention-aware attack framework against which they find that ViTs are weaker learners than CNNs. Moreover, by developing Spare Patch-Fool, the authors find that ViTs are more vulnerable than CNNs under high perturbation density. | SP:689c9069c9a3e66b8c7445bb48f67af2b131ce8b |
PDAML: A Pseudo Domain Adaptation Paradigm for Subject-independent EEG-based Emotion Recognition | Domain adaptation ( DA ) and domain generalization ( DG ) methods have been successfully adopted to alleviate the domain shift problem caused by the subject variability of EEG signals in subject-independent affective brain-computer interfaces ( aBCIs ) . Usually , the DA methods give relatively promising results than the DG methods but require additional computation resources each time a new subject comes . In this paper , we first propose a new paradigm called Pseudo Domain Adaptation ( PDA ) , which is more suitable for subject-independent aBCIs . Then we propose the pseudo domain adaptation via meta-learning ( PDAML ) based on PDA . The PDAML consists of a feature extractor , a classifier , and a sumdecomposable structure called domain shift governor . We prove that a network with a sum-decomposable structure can compute the divergence between different domains effectively in theory . By taking advantage of the adversarial learning and meta-learning , the governor helps PDAML quickly generalize to a new domain using the target data through a few self-adaptation steps in the test phase . Experimental results on the public aBICs dataset demonstrate that our proposed method not only avoids the additional computation resources of the DA methods but also reaches a similar generalization performance of the state-of-the-art DA methods . 1 INTRODUCTION . Affective brain-computer interfaces ( aBCIs ) , which focus on developing machines to recognize human emotion automatically and provide more humanized interaction , have attracted widespread attention from academics and industries ( Mühl et al . ( 2014 ) ; Shanechi ( 2019 ) ) . Various studies have demonstrated that the electroencephalography ( EEG ) signal is especially reliable to recognize human emotion in the subject-dependent emotion model , in which the training and test data are from one subject ( Jenke et al . ( 2014 ) ; Alarcão & Fonseca ( 2019 ) ) . However , due to the non-stationary nature of EEG signal and structural variability between different subjects , subject-independent model based on the assumption of Independent Identity Distribution ( i.i.d . ) usually shows bad generalization performance in real aBCIs applications , which is called the problem of Domain Shift ( Sugiyama et al . ( 2007 ) ; Samek et al . ( 2013 ) ; Sussillo et al . ( 2016 ) ; Zheng & Lu ( 2016 ) ; Li et al . ( 2018d ) ) . Domain adaptation ( DA ) is one of the promising ways to solve this problem . DA uses data from both source and target domain to promote the adaption performance . One of the most sufficient studied DA is mapping the two distributions to one common feature space where they have the same marginal distribution . Though DA has demonstrated significant success in subject-independent EEG-based emotion recognition ( Zheng & Lu ( 2016 ) ; Li et al . ( 2018b ; c ) ; Luo et al . ( 2018 ) ) , the additional computation and time to apply the DA methods is an exasperating problem in real-world scenarios and causes poor user experience . As a consequence , the concept of domain generalization ( DG ) arises in situations where multiple source domains can be accessed but unlabeled target samples are not available . The DG methods have also been successfully adopted to build the subjectindependent emotion model ( Ma et al . ( 2019b ; a ) ) . However , since there is no prior information about the target domain during training , it ’ s challenging for DG to perform as promising as DA . A compromise solution is Fast Domain Adaptation ( FDA ) , which trains the main model in advance and uses a small amount of test samples to adjust efficiently . Even though FDA methods can avoid consuming time cost by adaptation , most of them need to storage both source and target domains in the test phase ( Chai et al . ( 2017 ) ; Zhao et al . ( 2021 ) ) , which needs extra storage space and causes poor portability . However , in real-world applications , whether a EEG-based affective model can adapt to different subjects quickly as well as keep its portability does matter . In this paper , we propose a new paradigm called Pseudo Domain Adaptation ( PDA ) for subjectindependent EEG-based emotion recognition . Compared to typical FDA , only target domain is required in the test phase of PDA , which makes PDA capable to output the prediction more quickly than DA and FDA . From the perspective of the real-world application , PDA is more suited to build subject-independent affective model . To implement PDA , we first prove that with the sum-decomposable structure , a network is equivalent to domain discrepancy metrics in traditional DA methods like MMD orH-divergence . Based on our theory , we propose a method named Pseudo Domain Adaptation via Meta Learning ( PDAML ) . The PDAML consists of a feature extractor , a classifier , and a sum-decomposable structure called domain shift governor . By taking advantage of the adversarial learning and meta-learning , the governor helps PDAML quickly generalize to a new domain using the target data through a few self-adaptive steps in the test phase . The main contributions of this paper can be summarized as follows : • We propose a more suitable paradigm to build subject-independent affective model . • We prove that a sum-decomposable network is equivalent to domain discriminator for rep- resenting any type of domain discrepancy in theory . • We propose PDAML that is portable and can fast adapt to different subjects in EEG-based emotion recognition . • We conduct extensive experiments on the SEED dataset1 , which is a public available EEG- based affective dataset . The experimental results demonstrate that our method has better performance than DG methods . From the perspective of time and storage cost , the PDAML performs as well as DG methods . 2 RELATED WORK . aBCIs have received considerable attention very recent . Mühl et al . presented the definition of aBCIs ( Mühl et al . ( 2014 ) ) by introducing the affective factors into traditional brain-computer interfaces ( BCIs ) ( Zander & Jatzev ( 2011 ) ) . More applied works focus on studying EEG-based emotion recognition . Zheng and Lu recruited 15 subjects to watch 15 selected Chinese movie clips to elicit three emotions : happy , neutral and sad . They developed a public emotion dataset called SEED by recording the EEG signals of the subjects ( Zheng & Lu ( 2015 ) ) . Based on the SEED dataset , researchers have made great progress in developing EEG-based emotion recognition model , especially for subject-dependent model . Due to the non-stationary nature of EEG signal and structural variability between different subjects , it is hard to develop subject-independent EEG-based emotion recognition model by directly using typical machine learning approaches . Researchers have focused on applying DA and DG methods to subject-independent EEG-based emotion recognition . Typical DA methods are discrepancy-based , and they alleviate the domain shift problem by minimizing traditional metrics , such as Maximum Mean Discrepancy ( MMD ) ( Pan et al . ( 2011 ) ; Long et al . ( 2017 ) ; Wang et al . ( 2018 ) ) , KullbackLeibler ( KL ) divergence ( Zhuang et al . ( 2015 ) ) , andH-divergence ( Bendavid et al . ( 2010 ) ) , between the different domains . Zheng and Lu adopted transfer component analysis ( TCA ) ( Pan et al . ( 2011 ) ) , which minimizes MMD ( Gretton et al . ( 2007 ) ) between two domains by constructing kernel matrix , and successfully built personalized EEG-based emotion models ( Zheng & Lu ( 2016 ) ) . Recently , adversarial DA methods have made great successes in different fields ( Ganin & Lempitsky ( 2015 ) ; Tzeng et al . ( 2017 ) ; Shen et al . ( 2018 ) ) . The basic idea of the adversarial training is similar to generative adversarial networks ( GANs ) ( Goodfellow et al . ( 2014 ) ) , which play an adversarial 1http : //bcmi.sjtu.edu.cn/∼seed/index.html game to make the generated distribution approximate to the real distribution . After the adversarial training , the data distribution of the target domain is similar to the source domain , and the domain shift is diminished . Researchers have successfully adopted adversarial DA methods to aBCIs . Li et al . ( Li et al . ( 2018b ) ) adopted domain-adversarial neural networks ( DANN ) ( Ganin & Lempitsky ( 2015 ) ) to EEG-based emotion recognition and improved the recognition accuracies of subjectindependent models . And they ( Li et al . ( 2018c ) ) also achieved significant performance in subjectindependent vigilance estimation by implementing DANN and adversarial discriminative domain adaptation ( ADDA ) ( Tzeng et al . ( 2017 ) ) . Luo et al . proposed Wasserstein GAN ( Arjovsky et al . ( 2017 ) ) adversarial domain adaptation ( WGANDA ) and successfully adopted it to build subjectindependent emotion recognition models ( Luo et al . ( 2018 ) ) . In real-world aBCIs applications , each subject can be viewed as an individual domain . DA methods , which require high additional computation resources for each new domain , hinder the development of aBCIs from lab to real scenarios . DG methods , which can be utilized by data manipulation , representation learning , or meta-learning ( Wang et al . ( 2021 ) ) , aim to generalize to unseeing target domains without additional data collection from target domains . Researchers have focused on adopting DG methods to aBCIs very recently . Ma et al . generalized the structure of DANN into DG and proposed an adversarial structure called Domain Residual Network ( DResNet ) . They adopted DResNet to subject-independent EEG-based vigilance estimation and emotion recognition ( Ma et al . ( 2019b ; a ) ) . The experimental result demonstrated that the DG method could improve the generalization ability without data collection from the target domains . However , the DA methods usually give relatively promising results than DG methods in aBCIs applications . As a compromise way , FDA was adopted in aBCIs . Zhao et al . proposed a Plug-and-Play Domain Adaptation ( PPDA ) method to fast adapt the model in EEG-based emotion recognition ( Zhao et al . ( 2021 ) ) . Meta-learning , also known as learning to learn , has received a resurgence in interest recently with applications , one of which is domain generalization . Meta-learning aims to learn episodes sampled from the related tasks ( Finn et al . ( 2017 ) ) . Meta-Learning for Domain Generalization ( MLDG ) first introduces meta-learning strategy to DG ( Li et al . ( 2018a ) ) , then MetaReg ( Balaji et al . ( 2018 ) ) and Feature-Critic ( Li et al . ( 2019 ) ) are subsequently proposed to enhance the model ’ s generalizing ability by introducing an auxiliary loss in the training . Compared to most previous DG work that designs a specific model , meta-learning-based DG methods focus on model agnostic training strategy by exposing the model to domain shift in the training phase . To the best of our knowledge , we are the first to introduce meta-learning based methods to aBCIs tasks . 3 THEORY . The motivation of PDAML is using a simple network to compute domain shift , taking only the target domain as input . Traditional DA methods usually taking two domains as input when comparing the target domain with a specific source domain . Thus they need extra storage space for source data and sophisticated methods ( e.g . GAN ) to represent domain shift in the test phase , which is time-consuming and storage-consuming . Either or both of these problems obstruct the practical application of DA method in EEG-based diagnosis . However , in multi-source setting , we will show that minimizing the discrepancy between all pairwise domains is equivalent to minimizing the discrepancy between each domain and an implicit domain . Additionally , we will prove that a network with what is termed sum-decomposition form can represent any domain shift metrics in theory . | The authors present an incremental improvement to previously published approaches by Zhao et al., and their SEED dataset, which is currently not available, since a link provided in the manuscript lands on an error page (see below), so a provided code is also unfortunately useless: http://bcmi.sjtu.edu.cn/∼seed/index.html -> 404，您请求的文件不存在! The authors use also the same hand-engineered features (why not end-to-end data-driven?) by Zhao et al., which is another copy and paste step from previous publications. The authors also seem to mix or not understand the difference between affect and induced emotion. The experimental SEED database used in the project, but not available anymore online, seems to be related to affect recognition from brainwaves (here EEG) in response to emotional videos, but not the subject's emotion itself, which seems to be not reported in the original experiment. Finally, a relation to BCI is not explained well. BCI is about intentional, and hopefully fast, control of computers or machines with modulated brainwaves. This manuscripts concerns only affect recognition in response to emotional videos. | SP:90f1ea2fbd4e8643553f3f811c2d5d405eccac12 |
PDAML: A Pseudo Domain Adaptation Paradigm for Subject-independent EEG-based Emotion Recognition | Domain adaptation ( DA ) and domain generalization ( DG ) methods have been successfully adopted to alleviate the domain shift problem caused by the subject variability of EEG signals in subject-independent affective brain-computer interfaces ( aBCIs ) . Usually , the DA methods give relatively promising results than the DG methods but require additional computation resources each time a new subject comes . In this paper , we first propose a new paradigm called Pseudo Domain Adaptation ( PDA ) , which is more suitable for subject-independent aBCIs . Then we propose the pseudo domain adaptation via meta-learning ( PDAML ) based on PDA . The PDAML consists of a feature extractor , a classifier , and a sumdecomposable structure called domain shift governor . We prove that a network with a sum-decomposable structure can compute the divergence between different domains effectively in theory . By taking advantage of the adversarial learning and meta-learning , the governor helps PDAML quickly generalize to a new domain using the target data through a few self-adaptation steps in the test phase . Experimental results on the public aBICs dataset demonstrate that our proposed method not only avoids the additional computation resources of the DA methods but also reaches a similar generalization performance of the state-of-the-art DA methods . 1 INTRODUCTION . Affective brain-computer interfaces ( aBCIs ) , which focus on developing machines to recognize human emotion automatically and provide more humanized interaction , have attracted widespread attention from academics and industries ( Mühl et al . ( 2014 ) ; Shanechi ( 2019 ) ) . Various studies have demonstrated that the electroencephalography ( EEG ) signal is especially reliable to recognize human emotion in the subject-dependent emotion model , in which the training and test data are from one subject ( Jenke et al . ( 2014 ) ; Alarcão & Fonseca ( 2019 ) ) . However , due to the non-stationary nature of EEG signal and structural variability between different subjects , subject-independent model based on the assumption of Independent Identity Distribution ( i.i.d . ) usually shows bad generalization performance in real aBCIs applications , which is called the problem of Domain Shift ( Sugiyama et al . ( 2007 ) ; Samek et al . ( 2013 ) ; Sussillo et al . ( 2016 ) ; Zheng & Lu ( 2016 ) ; Li et al . ( 2018d ) ) . Domain adaptation ( DA ) is one of the promising ways to solve this problem . DA uses data from both source and target domain to promote the adaption performance . One of the most sufficient studied DA is mapping the two distributions to one common feature space where they have the same marginal distribution . Though DA has demonstrated significant success in subject-independent EEG-based emotion recognition ( Zheng & Lu ( 2016 ) ; Li et al . ( 2018b ; c ) ; Luo et al . ( 2018 ) ) , the additional computation and time to apply the DA methods is an exasperating problem in real-world scenarios and causes poor user experience . As a consequence , the concept of domain generalization ( DG ) arises in situations where multiple source domains can be accessed but unlabeled target samples are not available . The DG methods have also been successfully adopted to build the subjectindependent emotion model ( Ma et al . ( 2019b ; a ) ) . However , since there is no prior information about the target domain during training , it ’ s challenging for DG to perform as promising as DA . A compromise solution is Fast Domain Adaptation ( FDA ) , which trains the main model in advance and uses a small amount of test samples to adjust efficiently . Even though FDA methods can avoid consuming time cost by adaptation , most of them need to storage both source and target domains in the test phase ( Chai et al . ( 2017 ) ; Zhao et al . ( 2021 ) ) , which needs extra storage space and causes poor portability . However , in real-world applications , whether a EEG-based affective model can adapt to different subjects quickly as well as keep its portability does matter . In this paper , we propose a new paradigm called Pseudo Domain Adaptation ( PDA ) for subjectindependent EEG-based emotion recognition . Compared to typical FDA , only target domain is required in the test phase of PDA , which makes PDA capable to output the prediction more quickly than DA and FDA . From the perspective of the real-world application , PDA is more suited to build subject-independent affective model . To implement PDA , we first prove that with the sum-decomposable structure , a network is equivalent to domain discrepancy metrics in traditional DA methods like MMD orH-divergence . Based on our theory , we propose a method named Pseudo Domain Adaptation via Meta Learning ( PDAML ) . The PDAML consists of a feature extractor , a classifier , and a sum-decomposable structure called domain shift governor . By taking advantage of the adversarial learning and meta-learning , the governor helps PDAML quickly generalize to a new domain using the target data through a few self-adaptive steps in the test phase . The main contributions of this paper can be summarized as follows : • We propose a more suitable paradigm to build subject-independent affective model . • We prove that a sum-decomposable network is equivalent to domain discriminator for rep- resenting any type of domain discrepancy in theory . • We propose PDAML that is portable and can fast adapt to different subjects in EEG-based emotion recognition . • We conduct extensive experiments on the SEED dataset1 , which is a public available EEG- based affective dataset . The experimental results demonstrate that our method has better performance than DG methods . From the perspective of time and storage cost , the PDAML performs as well as DG methods . 2 RELATED WORK . aBCIs have received considerable attention very recent . Mühl et al . presented the definition of aBCIs ( Mühl et al . ( 2014 ) ) by introducing the affective factors into traditional brain-computer interfaces ( BCIs ) ( Zander & Jatzev ( 2011 ) ) . More applied works focus on studying EEG-based emotion recognition . Zheng and Lu recruited 15 subjects to watch 15 selected Chinese movie clips to elicit three emotions : happy , neutral and sad . They developed a public emotion dataset called SEED by recording the EEG signals of the subjects ( Zheng & Lu ( 2015 ) ) . Based on the SEED dataset , researchers have made great progress in developing EEG-based emotion recognition model , especially for subject-dependent model . Due to the non-stationary nature of EEG signal and structural variability between different subjects , it is hard to develop subject-independent EEG-based emotion recognition model by directly using typical machine learning approaches . Researchers have focused on applying DA and DG methods to subject-independent EEG-based emotion recognition . Typical DA methods are discrepancy-based , and they alleviate the domain shift problem by minimizing traditional metrics , such as Maximum Mean Discrepancy ( MMD ) ( Pan et al . ( 2011 ) ; Long et al . ( 2017 ) ; Wang et al . ( 2018 ) ) , KullbackLeibler ( KL ) divergence ( Zhuang et al . ( 2015 ) ) , andH-divergence ( Bendavid et al . ( 2010 ) ) , between the different domains . Zheng and Lu adopted transfer component analysis ( TCA ) ( Pan et al . ( 2011 ) ) , which minimizes MMD ( Gretton et al . ( 2007 ) ) between two domains by constructing kernel matrix , and successfully built personalized EEG-based emotion models ( Zheng & Lu ( 2016 ) ) . Recently , adversarial DA methods have made great successes in different fields ( Ganin & Lempitsky ( 2015 ) ; Tzeng et al . ( 2017 ) ; Shen et al . ( 2018 ) ) . The basic idea of the adversarial training is similar to generative adversarial networks ( GANs ) ( Goodfellow et al . ( 2014 ) ) , which play an adversarial 1http : //bcmi.sjtu.edu.cn/∼seed/index.html game to make the generated distribution approximate to the real distribution . After the adversarial training , the data distribution of the target domain is similar to the source domain , and the domain shift is diminished . Researchers have successfully adopted adversarial DA methods to aBCIs . Li et al . ( Li et al . ( 2018b ) ) adopted domain-adversarial neural networks ( DANN ) ( Ganin & Lempitsky ( 2015 ) ) to EEG-based emotion recognition and improved the recognition accuracies of subjectindependent models . And they ( Li et al . ( 2018c ) ) also achieved significant performance in subjectindependent vigilance estimation by implementing DANN and adversarial discriminative domain adaptation ( ADDA ) ( Tzeng et al . ( 2017 ) ) . Luo et al . proposed Wasserstein GAN ( Arjovsky et al . ( 2017 ) ) adversarial domain adaptation ( WGANDA ) and successfully adopted it to build subjectindependent emotion recognition models ( Luo et al . ( 2018 ) ) . In real-world aBCIs applications , each subject can be viewed as an individual domain . DA methods , which require high additional computation resources for each new domain , hinder the development of aBCIs from lab to real scenarios . DG methods , which can be utilized by data manipulation , representation learning , or meta-learning ( Wang et al . ( 2021 ) ) , aim to generalize to unseeing target domains without additional data collection from target domains . Researchers have focused on adopting DG methods to aBCIs very recently . Ma et al . generalized the structure of DANN into DG and proposed an adversarial structure called Domain Residual Network ( DResNet ) . They adopted DResNet to subject-independent EEG-based vigilance estimation and emotion recognition ( Ma et al . ( 2019b ; a ) ) . The experimental result demonstrated that the DG method could improve the generalization ability without data collection from the target domains . However , the DA methods usually give relatively promising results than DG methods in aBCIs applications . As a compromise way , FDA was adopted in aBCIs . Zhao et al . proposed a Plug-and-Play Domain Adaptation ( PPDA ) method to fast adapt the model in EEG-based emotion recognition ( Zhao et al . ( 2021 ) ) . Meta-learning , also known as learning to learn , has received a resurgence in interest recently with applications , one of which is domain generalization . Meta-learning aims to learn episodes sampled from the related tasks ( Finn et al . ( 2017 ) ) . Meta-Learning for Domain Generalization ( MLDG ) first introduces meta-learning strategy to DG ( Li et al . ( 2018a ) ) , then MetaReg ( Balaji et al . ( 2018 ) ) and Feature-Critic ( Li et al . ( 2019 ) ) are subsequently proposed to enhance the model ’ s generalizing ability by introducing an auxiliary loss in the training . Compared to most previous DG work that designs a specific model , meta-learning-based DG methods focus on model agnostic training strategy by exposing the model to domain shift in the training phase . To the best of our knowledge , we are the first to introduce meta-learning based methods to aBCIs tasks . 3 THEORY . The motivation of PDAML is using a simple network to compute domain shift , taking only the target domain as input . Traditional DA methods usually taking two domains as input when comparing the target domain with a specific source domain . Thus they need extra storage space for source data and sophisticated methods ( e.g . GAN ) to represent domain shift in the test phase , which is time-consuming and storage-consuming . Either or both of these problems obstruct the practical application of DA method in EEG-based diagnosis . However , in multi-source setting , we will show that minimizing the discrepancy between all pairwise domains is equivalent to minimizing the discrepancy between each domain and an implicit domain . Additionally , we will prove that a network with what is termed sum-decomposition form can represent any domain shift metrics in theory . | The paper proposes an affective emotion recognition method using EEG data with an aim to perform in the subject independent paradigm. Towards this, domain adaptation and generalisation methods are evaluated and in particular a pseudo domain adaptation method is proposed for the said purpose. Experiments are performed on SEED dataset, and a method that use pseudo domain adaptation and meta learning is proposed to improve upon computational and memory requirements in realising practical aBCIs. | SP:90f1ea2fbd4e8643553f3f811c2d5d405eccac12 |
PDAML: A Pseudo Domain Adaptation Paradigm for Subject-independent EEG-based Emotion Recognition | Domain adaptation ( DA ) and domain generalization ( DG ) methods have been successfully adopted to alleviate the domain shift problem caused by the subject variability of EEG signals in subject-independent affective brain-computer interfaces ( aBCIs ) . Usually , the DA methods give relatively promising results than the DG methods but require additional computation resources each time a new subject comes . In this paper , we first propose a new paradigm called Pseudo Domain Adaptation ( PDA ) , which is more suitable for subject-independent aBCIs . Then we propose the pseudo domain adaptation via meta-learning ( PDAML ) based on PDA . The PDAML consists of a feature extractor , a classifier , and a sumdecomposable structure called domain shift governor . We prove that a network with a sum-decomposable structure can compute the divergence between different domains effectively in theory . By taking advantage of the adversarial learning and meta-learning , the governor helps PDAML quickly generalize to a new domain using the target data through a few self-adaptation steps in the test phase . Experimental results on the public aBICs dataset demonstrate that our proposed method not only avoids the additional computation resources of the DA methods but also reaches a similar generalization performance of the state-of-the-art DA methods . 1 INTRODUCTION . Affective brain-computer interfaces ( aBCIs ) , which focus on developing machines to recognize human emotion automatically and provide more humanized interaction , have attracted widespread attention from academics and industries ( Mühl et al . ( 2014 ) ; Shanechi ( 2019 ) ) . Various studies have demonstrated that the electroencephalography ( EEG ) signal is especially reliable to recognize human emotion in the subject-dependent emotion model , in which the training and test data are from one subject ( Jenke et al . ( 2014 ) ; Alarcão & Fonseca ( 2019 ) ) . However , due to the non-stationary nature of EEG signal and structural variability between different subjects , subject-independent model based on the assumption of Independent Identity Distribution ( i.i.d . ) usually shows bad generalization performance in real aBCIs applications , which is called the problem of Domain Shift ( Sugiyama et al . ( 2007 ) ; Samek et al . ( 2013 ) ; Sussillo et al . ( 2016 ) ; Zheng & Lu ( 2016 ) ; Li et al . ( 2018d ) ) . Domain adaptation ( DA ) is one of the promising ways to solve this problem . DA uses data from both source and target domain to promote the adaption performance . One of the most sufficient studied DA is mapping the two distributions to one common feature space where they have the same marginal distribution . Though DA has demonstrated significant success in subject-independent EEG-based emotion recognition ( Zheng & Lu ( 2016 ) ; Li et al . ( 2018b ; c ) ; Luo et al . ( 2018 ) ) , the additional computation and time to apply the DA methods is an exasperating problem in real-world scenarios and causes poor user experience . As a consequence , the concept of domain generalization ( DG ) arises in situations where multiple source domains can be accessed but unlabeled target samples are not available . The DG methods have also been successfully adopted to build the subjectindependent emotion model ( Ma et al . ( 2019b ; a ) ) . However , since there is no prior information about the target domain during training , it ’ s challenging for DG to perform as promising as DA . A compromise solution is Fast Domain Adaptation ( FDA ) , which trains the main model in advance and uses a small amount of test samples to adjust efficiently . Even though FDA methods can avoid consuming time cost by adaptation , most of them need to storage both source and target domains in the test phase ( Chai et al . ( 2017 ) ; Zhao et al . ( 2021 ) ) , which needs extra storage space and causes poor portability . However , in real-world applications , whether a EEG-based affective model can adapt to different subjects quickly as well as keep its portability does matter . In this paper , we propose a new paradigm called Pseudo Domain Adaptation ( PDA ) for subjectindependent EEG-based emotion recognition . Compared to typical FDA , only target domain is required in the test phase of PDA , which makes PDA capable to output the prediction more quickly than DA and FDA . From the perspective of the real-world application , PDA is more suited to build subject-independent affective model . To implement PDA , we first prove that with the sum-decomposable structure , a network is equivalent to domain discrepancy metrics in traditional DA methods like MMD orH-divergence . Based on our theory , we propose a method named Pseudo Domain Adaptation via Meta Learning ( PDAML ) . The PDAML consists of a feature extractor , a classifier , and a sum-decomposable structure called domain shift governor . By taking advantage of the adversarial learning and meta-learning , the governor helps PDAML quickly generalize to a new domain using the target data through a few self-adaptive steps in the test phase . The main contributions of this paper can be summarized as follows : • We propose a more suitable paradigm to build subject-independent affective model . • We prove that a sum-decomposable network is equivalent to domain discriminator for rep- resenting any type of domain discrepancy in theory . • We propose PDAML that is portable and can fast adapt to different subjects in EEG-based emotion recognition . • We conduct extensive experiments on the SEED dataset1 , which is a public available EEG- based affective dataset . The experimental results demonstrate that our method has better performance than DG methods . From the perspective of time and storage cost , the PDAML performs as well as DG methods . 2 RELATED WORK . aBCIs have received considerable attention very recent . Mühl et al . presented the definition of aBCIs ( Mühl et al . ( 2014 ) ) by introducing the affective factors into traditional brain-computer interfaces ( BCIs ) ( Zander & Jatzev ( 2011 ) ) . More applied works focus on studying EEG-based emotion recognition . Zheng and Lu recruited 15 subjects to watch 15 selected Chinese movie clips to elicit three emotions : happy , neutral and sad . They developed a public emotion dataset called SEED by recording the EEG signals of the subjects ( Zheng & Lu ( 2015 ) ) . Based on the SEED dataset , researchers have made great progress in developing EEG-based emotion recognition model , especially for subject-dependent model . Due to the non-stationary nature of EEG signal and structural variability between different subjects , it is hard to develop subject-independent EEG-based emotion recognition model by directly using typical machine learning approaches . Researchers have focused on applying DA and DG methods to subject-independent EEG-based emotion recognition . Typical DA methods are discrepancy-based , and they alleviate the domain shift problem by minimizing traditional metrics , such as Maximum Mean Discrepancy ( MMD ) ( Pan et al . ( 2011 ) ; Long et al . ( 2017 ) ; Wang et al . ( 2018 ) ) , KullbackLeibler ( KL ) divergence ( Zhuang et al . ( 2015 ) ) , andH-divergence ( Bendavid et al . ( 2010 ) ) , between the different domains . Zheng and Lu adopted transfer component analysis ( TCA ) ( Pan et al . ( 2011 ) ) , which minimizes MMD ( Gretton et al . ( 2007 ) ) between two domains by constructing kernel matrix , and successfully built personalized EEG-based emotion models ( Zheng & Lu ( 2016 ) ) . Recently , adversarial DA methods have made great successes in different fields ( Ganin & Lempitsky ( 2015 ) ; Tzeng et al . ( 2017 ) ; Shen et al . ( 2018 ) ) . The basic idea of the adversarial training is similar to generative adversarial networks ( GANs ) ( Goodfellow et al . ( 2014 ) ) , which play an adversarial 1http : //bcmi.sjtu.edu.cn/∼seed/index.html game to make the generated distribution approximate to the real distribution . After the adversarial training , the data distribution of the target domain is similar to the source domain , and the domain shift is diminished . Researchers have successfully adopted adversarial DA methods to aBCIs . Li et al . ( Li et al . ( 2018b ) ) adopted domain-adversarial neural networks ( DANN ) ( Ganin & Lempitsky ( 2015 ) ) to EEG-based emotion recognition and improved the recognition accuracies of subjectindependent models . And they ( Li et al . ( 2018c ) ) also achieved significant performance in subjectindependent vigilance estimation by implementing DANN and adversarial discriminative domain adaptation ( ADDA ) ( Tzeng et al . ( 2017 ) ) . Luo et al . proposed Wasserstein GAN ( Arjovsky et al . ( 2017 ) ) adversarial domain adaptation ( WGANDA ) and successfully adopted it to build subjectindependent emotion recognition models ( Luo et al . ( 2018 ) ) . In real-world aBCIs applications , each subject can be viewed as an individual domain . DA methods , which require high additional computation resources for each new domain , hinder the development of aBCIs from lab to real scenarios . DG methods , which can be utilized by data manipulation , representation learning , or meta-learning ( Wang et al . ( 2021 ) ) , aim to generalize to unseeing target domains without additional data collection from target domains . Researchers have focused on adopting DG methods to aBCIs very recently . Ma et al . generalized the structure of DANN into DG and proposed an adversarial structure called Domain Residual Network ( DResNet ) . They adopted DResNet to subject-independent EEG-based vigilance estimation and emotion recognition ( Ma et al . ( 2019b ; a ) ) . The experimental result demonstrated that the DG method could improve the generalization ability without data collection from the target domains . However , the DA methods usually give relatively promising results than DG methods in aBCIs applications . As a compromise way , FDA was adopted in aBCIs . Zhao et al . proposed a Plug-and-Play Domain Adaptation ( PPDA ) method to fast adapt the model in EEG-based emotion recognition ( Zhao et al . ( 2021 ) ) . Meta-learning , also known as learning to learn , has received a resurgence in interest recently with applications , one of which is domain generalization . Meta-learning aims to learn episodes sampled from the related tasks ( Finn et al . ( 2017 ) ) . Meta-Learning for Domain Generalization ( MLDG ) first introduces meta-learning strategy to DG ( Li et al . ( 2018a ) ) , then MetaReg ( Balaji et al . ( 2018 ) ) and Feature-Critic ( Li et al . ( 2019 ) ) are subsequently proposed to enhance the model ’ s generalizing ability by introducing an auxiliary loss in the training . Compared to most previous DG work that designs a specific model , meta-learning-based DG methods focus on model agnostic training strategy by exposing the model to domain shift in the training phase . To the best of our knowledge , we are the first to introduce meta-learning based methods to aBCIs tasks . 3 THEORY . The motivation of PDAML is using a simple network to compute domain shift , taking only the target domain as input . Traditional DA methods usually taking two domains as input when comparing the target domain with a specific source domain . Thus they need extra storage space for source data and sophisticated methods ( e.g . GAN ) to represent domain shift in the test phase , which is time-consuming and storage-consuming . Either or both of these problems obstruct the practical application of DA method in EEG-based diagnosis . However , in multi-source setting , we will show that minimizing the discrepancy between all pairwise domains is equivalent to minimizing the discrepancy between each domain and an implicit domain . Additionally , we will prove that a network with what is termed sum-decomposition form can represent any domain shift metrics in theory . | A new paradigm is called Pseudo Domain Adaptation (PDA) is proposed for subject-independent EEG-based affective computing. The method performs the pseudo-domain adaptation via meta-learning based on PDA and uses a feature extractor, a classifier, and a domain shift governor. The results show that the proposed method doesn't require the additional computation resources of DA methods but achieves similar generalization performances. | SP:90f1ea2fbd4e8643553f3f811c2d5d405eccac12 |
Non-reversible Parallel Tempering for Uncertainty Approximation in Deep Learning | 1 INTRODUCTION . Langevin diffusion is a standard sampling algorithm that follows a stochastic differential equation dβt = −∇U ( βt ) dt+ √ 2τdWt , where βt ∈ Rd , ∇U ( · ) is the gradient of the energy function U ( · ) , Wt ∈ Rd is a Brownian motion , and τ is the temperature . The diffusion process converges to a stationary distribution π ( β ) ∝ e− U ( β ) τ and setting τ = 1 yields a Bayesian posterior . When U ( · ) is convex , the rapid convergence has been widely studied in Durmus & Moulines ( 2016 ) ; Dalalyan ( 2017 ) ; however , when U ( · ) is non-convex , a slow mixing rate is inevitable ( Raginsky et al. , 2017 ) . To accelerate the simulation , replica exchange Langevin diffusion ( reLD ) proposes to include a high-temperature particle β ( P ) t , where P ∈ N+\ { 1 } , for exploration . Meanwhile , a low-temperature particle β ( 1 ) t is presented for exploitation : dβ ( P ) t = −∇U ( β ( P ) t ) dt+ √ 2τ ( P ) dW ( P ) t , dβ ( 1 ) t = −∇U ( β ( 1 ) t ) dt+ √ 2τ ( 1 ) dW ( 1 ) t , ( 1 ) where τ ( P ) > τ ( 1 ) and W ( P ) t is independent of W ( 1 ) t . To promote more explorations for the low-temperature particle , the particles at the position ( β ( 1 ) , β ( P ) ) ∈ R2d swap with a probability aS ( β ( 1 ) , β ( P ) ) = a · ( 1 ∧ e ( 1 τ ( 1 ) − 1 τ ( P ) ) ( U ( β ( 1 ) ) −U ( β ( P ) ) ) ) , ( 2 ) where a ∈ ( 0 , ∞ ) is the swap intensity . In specific , the conditional swap rate at time t follows that P ( βt+dt = ( β ( P ) , β ( 1 ) ) |βt = ( β ( 1 ) , β ( P ) ) ) = aS ( β ( 1 ) , β ( P ) ) dt , P ( βt+dt = ( β ( 1 ) , β ( P ) ) |βt = ( β ( 1 ) , β ( P ) ) ) = 1− aS ( β ( 1 ) , β ( P ) ) dt . In the longtime limit , the Markov jump process converges to the joint distribution π ( β ( 1 ) , β ( P ) ) ∝ e −U ( β ( 1 ) ) τ ( 1 ) −U ( β ( P ) ) τ ( P ) . For convenience , we refer to the marginal distribution π ( 1 ) ( β ) ∝ e− U ( β ) τ ( 1 ) and π ( P ) ( β ) ∝ e− U ( β ) τ ( P ) as the target distribution and reference distribution , respectively . 2 PRELIMINARIES . Achieving sufficient explorations requires a large τ ( P ) , which leads to limited accelerations due to a small overlap between π ( 1 ) and π ( P ) . To tackle this issue , one can bring in multiple particles with temperatures ( τ ( 2 ) , · · · , τ ( P−1 ) ) , where τ ( 1 ) < τ ( 2 ) < · · · < τ ( P ) , to hollow out “ tunnels ” . To maintain feasibility , numerous schemes are presented to select candidate pairs to attempt the swaps . APE The all-pairs exchange ( APE ) attempts to swap arbitrary pair of chains ( Brenner et al. , 2007 ; Lingenheil et al. , 2009 ) , however , such a method requires a swap time ( see definition in section A.5 ) of O ( P 3 ) and may not be user-friendly in practice . ADJ In addition to swap arbitrary pairs , one can also swap adjacent ( ADJ ) pairs iteratively from ( 1 , 2 ) , ( 2 , 3 ) , to ( P − 1 , P ) under the Metropolis rule . Despite the convenience , the sequential nature requires to wait for exchange information from previous exchanges , which only works well with a small number of chains and has greatly limited its extension to a multi-core or distributed context . SEO The stochastic even-odd ( SEO ) scheme first divides the adjacent pairs { ( p − 1 , p ) |p = 2 , · · · , P } into E and O , where E and O denote even and odd pairs of forms ( 2p − 1 , 2p ) and ( 2p , 2p + 1 ) , respectively . Then , SEO randomly picks E or O pairs with an equal chance in each iteration to attempt the swaps . Notably , it can be conducted simultaneously without waiting from other chains . The scheme yields a reversible process ( see Figure 1 ( a ) ) , however , the gains in overcoming the sequential obstacle don ’ t offset the O ( P 2 ) round trip time and SEO is still not effective enough . DEO The deterministic even-odd ( DEO ) scheme instead attempts to swap even ( E ) pairs at even ( E ) iterations and odd ( O ) pairs at odd ( O ) iterations alternatingly† ( Okabe et al. , 2001 ) . The asymmetric manner was later interpreted as a non-reversible PT ( Syed et al. , 2021 ) and an ideal index process follows a periodic orbit , as shown in Figure 1 ( b ) . With a large swap rate , Figure 1 ( c ) shows how the scheme yields an almost straight path and a linear round trip time can be expected . Equi-acceptance The power of PT hinges on maximizing the number of round trips , which is equivalent to minimizing ∑P−1 p=1 1 1−rp ( Nadler & Hansmann , 2007b ) , where rp denotes the rejection rate for the chain pair ( p , p + 1 ) . Moreover , ∑P−1 p=1 rp converges to a fixed barrier Λ as P → ∞ ( Predescu et al. , 2004 ; Syed et al. , 2021 ) . Applying Lagrange multiplies to the constrained optimization problem leads to r1 = r2 = · · · = rP−1 : = r , where r is the equi-rejection rate . In general , a quadratic round trip time is required for ADJ and SEO due to the reversible indexes . By contrast , DEO only yields a linear round trip time in terms of P as P →∞ Syed et al . ( 2021 ) . 3 OPTIMAL NON-REVERSIBLE SCHEME FOR PARALLEL TEMPERING . The linear round trip time is appealing for maximizing the algorithmic potential , however , such an advance only occurs given sufficiently many chains . In non-asymptotic settings with limited chains , a pearl of wisdom is to avoid frequent swaps ( Dupuis et al. , 2012 ) and to keep the average acceptance rate from 20 % to 40 % ( Kone & Kofke , 2005 ; Lingenheil et al. , 2009 ; Atchadé et al. , 2011 ) . † E shown in iterations means even iterations ; otherwise , it denotes even pairs for chain indexes . The same logic applies toO . Most importantly , the acceptance rates are severely reduced in big data due to the bias-corrected swaps associated with stochastic energies ( Deng et al. , 2020 ) , see details in section A.1 . As such , maintaining low rejection rates becomes quite challenging and the issue of quadratic costs still exists . 3.1 GENERALIZED DEO SCHEME . As illustrated in Figure.2 ( e ) , the blue particle has a larger chance of ( 1 − r2 ) 2 to move upward 2 steps given W = 2 instead of ( 1− r ) 2 when W = 1 , although the window number is also halved . Such a trade-off inspires us to analyze the expected round trip time based on the window of size W . Although allowing at most one swap introduces the stopping time and may affect the distribution , the bias is rather mild due to the noisy energy estimators in big data . Check section C.2 for the details . 3.2 ANALYSIS OF ROUND TRIP TIME . To bring sufficient interactions between the reference distribution π ( P ) and the target distribution π ( 1 ) , we expect to minimize the expected round trip time T ( defined in section A.5 ) to ensure both efficient exploitation and explorations . Combining the Markov property and the idea of the master equation ( Nadler & Hansmann , 2007a ) , we estimate the expected round trip time E [ T ] as follows Lemma 1 . Under the stationary and weak dependence assumptions B1 and B2 in section B , for P ( P ≥ 2 ) chains with window size W ( W ≥ 1 ) and rejection rates { rp } P−1p=1 , we have E [ T ] = 2WP + 2WP P−1∑ p=1 rWp 1− rWp . ( 4 ) The proof in section B.1 shows that E [ T ] increases as we adopt larger number of chains P and rejection rates { rp } P−1p=1 . In such a case , the round trip rate PE [ T ] is also maximized by the key renewal theorem . In particular , applying W = 1 recovers the vanilla DEO scheme . 3.3 ANALYSIS OF OPTIMAL WINDOW SIZE AND ROUND TRIP TIME . By Lemma 1 , we observe a potential to remove the second quadratic term given an appropriate W . Such a fact motivates us to study the optimal window size W to achieve the best efficiency . Under §The generalized DEO with the optimal window size is denoted by DEO ? and will be studied in section 3.3. the equi-acceptance settings , by treating the window size W as a continuous variable and taking the derivative of E [ T ] with respect to W , we have ∂ ∂W E [ T ] = 2P ( 1− rW ) 2 { ( 1− rW ) 2 + ( P − 1 ) rW ( 1− rW +W log r ) } , ( 5 ) where r is the equi-rejection rate for adjacent chains . Define x : = rW ∈ ( 0 , 1 ) , where W = logr ( x ) = log x log r . The following analysis hinges on the study of the solution g ( x ) = ( 1− x ) 2 + ( P − 1 ) x ( 1− x+ log ( x ) ) = 0 . By analyzing the growth of derivatives and boundary values , we can easily identify the uniqueness of the solution . Then , we proceed to verify that 1P logP yields an asymptotic approximation such that g ( 1P logP ) = − log ( logP ) logP +O ( 1 logP ) → 0 as P →∞ . In the end , we have Theorem 1 . Under Assumptions B1 and B2 based on equi-acceptance settings , if P = 2 , 3 , the maximal round trip time is achieved when W = 1 . If P ≥ 4 , with the optimal window size W ? ≈ ⌈ logP+log logP − log r ⌉ , where d·e is the ceiling function . The round trip time follows O ( P logP− log r ) . The above result yields a remarkable round trip time of O ( P logP ) by setting the optimal window size W ? . By contrast , the vanilla DEO scheme only leads to a longer time of O ( P 2 ) § . Denoting by DEO ? the generalized DEO scheme with the optimal window size W ? , we summarize the popular swap schemes in Table.1 , where the DEO ? scheme performs the best among all the three criteria . | The manuscript considers parallel tempering for sampling from multi-modal distributions. The choice of the swap scheme can great impact the performance of parallel tempering algorithms. The authors propose a modification of the existing deterministic even-odd (DEO) swap scheme. Theoretical results are established which show the proposed scheme improves communication cost from O(P^2) to O(P*logP) for P chains. Simulation studies are performed on a stylised multimodal distribution and on CIFAR100 datasets showing empirical improvements. | SP:9a3e22f7e7f34083bac2b2db7e8fab75b8eddda9 |
Non-reversible Parallel Tempering for Uncertainty Approximation in Deep Learning | 1 INTRODUCTION . Langevin diffusion is a standard sampling algorithm that follows a stochastic differential equation dβt = −∇U ( βt ) dt+ √ 2τdWt , where βt ∈ Rd , ∇U ( · ) is the gradient of the energy function U ( · ) , Wt ∈ Rd is a Brownian motion , and τ is the temperature . The diffusion process converges to a stationary distribution π ( β ) ∝ e− U ( β ) τ and setting τ = 1 yields a Bayesian posterior . When U ( · ) is convex , the rapid convergence has been widely studied in Durmus & Moulines ( 2016 ) ; Dalalyan ( 2017 ) ; however , when U ( · ) is non-convex , a slow mixing rate is inevitable ( Raginsky et al. , 2017 ) . To accelerate the simulation , replica exchange Langevin diffusion ( reLD ) proposes to include a high-temperature particle β ( P ) t , where P ∈ N+\ { 1 } , for exploration . Meanwhile , a low-temperature particle β ( 1 ) t is presented for exploitation : dβ ( P ) t = −∇U ( β ( P ) t ) dt+ √ 2τ ( P ) dW ( P ) t , dβ ( 1 ) t = −∇U ( β ( 1 ) t ) dt+ √ 2τ ( 1 ) dW ( 1 ) t , ( 1 ) where τ ( P ) > τ ( 1 ) and W ( P ) t is independent of W ( 1 ) t . To promote more explorations for the low-temperature particle , the particles at the position ( β ( 1 ) , β ( P ) ) ∈ R2d swap with a probability aS ( β ( 1 ) , β ( P ) ) = a · ( 1 ∧ e ( 1 τ ( 1 ) − 1 τ ( P ) ) ( U ( β ( 1 ) ) −U ( β ( P ) ) ) ) , ( 2 ) where a ∈ ( 0 , ∞ ) is the swap intensity . In specific , the conditional swap rate at time t follows that P ( βt+dt = ( β ( P ) , β ( 1 ) ) |βt = ( β ( 1 ) , β ( P ) ) ) = aS ( β ( 1 ) , β ( P ) ) dt , P ( βt+dt = ( β ( 1 ) , β ( P ) ) |βt = ( β ( 1 ) , β ( P ) ) ) = 1− aS ( β ( 1 ) , β ( P ) ) dt . In the longtime limit , the Markov jump process converges to the joint distribution π ( β ( 1 ) , β ( P ) ) ∝ e −U ( β ( 1 ) ) τ ( 1 ) −U ( β ( P ) ) τ ( P ) . For convenience , we refer to the marginal distribution π ( 1 ) ( β ) ∝ e− U ( β ) τ ( 1 ) and π ( P ) ( β ) ∝ e− U ( β ) τ ( P ) as the target distribution and reference distribution , respectively . 2 PRELIMINARIES . Achieving sufficient explorations requires a large τ ( P ) , which leads to limited accelerations due to a small overlap between π ( 1 ) and π ( P ) . To tackle this issue , one can bring in multiple particles with temperatures ( τ ( 2 ) , · · · , τ ( P−1 ) ) , where τ ( 1 ) < τ ( 2 ) < · · · < τ ( P ) , to hollow out “ tunnels ” . To maintain feasibility , numerous schemes are presented to select candidate pairs to attempt the swaps . APE The all-pairs exchange ( APE ) attempts to swap arbitrary pair of chains ( Brenner et al. , 2007 ; Lingenheil et al. , 2009 ) , however , such a method requires a swap time ( see definition in section A.5 ) of O ( P 3 ) and may not be user-friendly in practice . ADJ In addition to swap arbitrary pairs , one can also swap adjacent ( ADJ ) pairs iteratively from ( 1 , 2 ) , ( 2 , 3 ) , to ( P − 1 , P ) under the Metropolis rule . Despite the convenience , the sequential nature requires to wait for exchange information from previous exchanges , which only works well with a small number of chains and has greatly limited its extension to a multi-core or distributed context . SEO The stochastic even-odd ( SEO ) scheme first divides the adjacent pairs { ( p − 1 , p ) |p = 2 , · · · , P } into E and O , where E and O denote even and odd pairs of forms ( 2p − 1 , 2p ) and ( 2p , 2p + 1 ) , respectively . Then , SEO randomly picks E or O pairs with an equal chance in each iteration to attempt the swaps . Notably , it can be conducted simultaneously without waiting from other chains . The scheme yields a reversible process ( see Figure 1 ( a ) ) , however , the gains in overcoming the sequential obstacle don ’ t offset the O ( P 2 ) round trip time and SEO is still not effective enough . DEO The deterministic even-odd ( DEO ) scheme instead attempts to swap even ( E ) pairs at even ( E ) iterations and odd ( O ) pairs at odd ( O ) iterations alternatingly† ( Okabe et al. , 2001 ) . The asymmetric manner was later interpreted as a non-reversible PT ( Syed et al. , 2021 ) and an ideal index process follows a periodic orbit , as shown in Figure 1 ( b ) . With a large swap rate , Figure 1 ( c ) shows how the scheme yields an almost straight path and a linear round trip time can be expected . Equi-acceptance The power of PT hinges on maximizing the number of round trips , which is equivalent to minimizing ∑P−1 p=1 1 1−rp ( Nadler & Hansmann , 2007b ) , where rp denotes the rejection rate for the chain pair ( p , p + 1 ) . Moreover , ∑P−1 p=1 rp converges to a fixed barrier Λ as P → ∞ ( Predescu et al. , 2004 ; Syed et al. , 2021 ) . Applying Lagrange multiplies to the constrained optimization problem leads to r1 = r2 = · · · = rP−1 : = r , where r is the equi-rejection rate . In general , a quadratic round trip time is required for ADJ and SEO due to the reversible indexes . By contrast , DEO only yields a linear round trip time in terms of P as P →∞ Syed et al . ( 2021 ) . 3 OPTIMAL NON-REVERSIBLE SCHEME FOR PARALLEL TEMPERING . The linear round trip time is appealing for maximizing the algorithmic potential , however , such an advance only occurs given sufficiently many chains . In non-asymptotic settings with limited chains , a pearl of wisdom is to avoid frequent swaps ( Dupuis et al. , 2012 ) and to keep the average acceptance rate from 20 % to 40 % ( Kone & Kofke , 2005 ; Lingenheil et al. , 2009 ; Atchadé et al. , 2011 ) . † E shown in iterations means even iterations ; otherwise , it denotes even pairs for chain indexes . The same logic applies toO . Most importantly , the acceptance rates are severely reduced in big data due to the bias-corrected swaps associated with stochastic energies ( Deng et al. , 2020 ) , see details in section A.1 . As such , maintaining low rejection rates becomes quite challenging and the issue of quadratic costs still exists . 3.1 GENERALIZED DEO SCHEME . As illustrated in Figure.2 ( e ) , the blue particle has a larger chance of ( 1 − r2 ) 2 to move upward 2 steps given W = 2 instead of ( 1− r ) 2 when W = 1 , although the window number is also halved . Such a trade-off inspires us to analyze the expected round trip time based on the window of size W . Although allowing at most one swap introduces the stopping time and may affect the distribution , the bias is rather mild due to the noisy energy estimators in big data . Check section C.2 for the details . 3.2 ANALYSIS OF ROUND TRIP TIME . To bring sufficient interactions between the reference distribution π ( P ) and the target distribution π ( 1 ) , we expect to minimize the expected round trip time T ( defined in section A.5 ) to ensure both efficient exploitation and explorations . Combining the Markov property and the idea of the master equation ( Nadler & Hansmann , 2007a ) , we estimate the expected round trip time E [ T ] as follows Lemma 1 . Under the stationary and weak dependence assumptions B1 and B2 in section B , for P ( P ≥ 2 ) chains with window size W ( W ≥ 1 ) and rejection rates { rp } P−1p=1 , we have E [ T ] = 2WP + 2WP P−1∑ p=1 rWp 1− rWp . ( 4 ) The proof in section B.1 shows that E [ T ] increases as we adopt larger number of chains P and rejection rates { rp } P−1p=1 . In such a case , the round trip rate PE [ T ] is also maximized by the key renewal theorem . In particular , applying W = 1 recovers the vanilla DEO scheme . 3.3 ANALYSIS OF OPTIMAL WINDOW SIZE AND ROUND TRIP TIME . By Lemma 1 , we observe a potential to remove the second quadratic term given an appropriate W . Such a fact motivates us to study the optimal window size W to achieve the best efficiency . Under §The generalized DEO with the optimal window size is denoted by DEO ? and will be studied in section 3.3. the equi-acceptance settings , by treating the window size W as a continuous variable and taking the derivative of E [ T ] with respect to W , we have ∂ ∂W E [ T ] = 2P ( 1− rW ) 2 { ( 1− rW ) 2 + ( P − 1 ) rW ( 1− rW +W log r ) } , ( 5 ) where r is the equi-rejection rate for adjacent chains . Define x : = rW ∈ ( 0 , 1 ) , where W = logr ( x ) = log x log r . The following analysis hinges on the study of the solution g ( x ) = ( 1− x ) 2 + ( P − 1 ) x ( 1− x+ log ( x ) ) = 0 . By analyzing the growth of derivatives and boundary values , we can easily identify the uniqueness of the solution . Then , we proceed to verify that 1P logP yields an asymptotic approximation such that g ( 1P logP ) = − log ( logP ) logP +O ( 1 logP ) → 0 as P →∞ . In the end , we have Theorem 1 . Under Assumptions B1 and B2 based on equi-acceptance settings , if P = 2 , 3 , the maximal round trip time is achieved when W = 1 . If P ≥ 4 , with the optimal window size W ? ≈ ⌈ logP+log logP − log r ⌉ , where d·e is the ceiling function . The round trip time follows O ( P logP− log r ) . The above result yields a remarkable round trip time of O ( P logP ) by setting the optimal window size W ? . By contrast , the vanilla DEO scheme only leads to a longer time of O ( P 2 ) § . Denoting by DEO ? the generalized DEO scheme with the optimal window size W ? , we summarize the popular swap schemes in Table.1 , where the DEO ? scheme performs the best among all the three criteria . | The paper studies the communicational complexity of parallel tempering. One example of parallel tempering is when one uses Langevin dynamics with different temperatures to balance exploit and exploration for sampling from a multi-modal distribution. The balance is made by swapping running particles with different temperatures according to a particular schedule. The round trip time (RTT) is a computational complexity metric for measure for the swapping schedule. It is known that the best-known algorithm suffers from $O(P^2)$ RTT for $P$ parallel particles. By adjusting this algorithm, the authors improve the RTT to $O(P\log(P))$. Then, they demonstrate the application of nonconvex with stochastic gradient descent. | SP:9a3e22f7e7f34083bac2b2db7e8fab75b8eddda9 |
Non-reversible Parallel Tempering for Uncertainty Approximation in Deep Learning | 1 INTRODUCTION . Langevin diffusion is a standard sampling algorithm that follows a stochastic differential equation dβt = −∇U ( βt ) dt+ √ 2τdWt , where βt ∈ Rd , ∇U ( · ) is the gradient of the energy function U ( · ) , Wt ∈ Rd is a Brownian motion , and τ is the temperature . The diffusion process converges to a stationary distribution π ( β ) ∝ e− U ( β ) τ and setting τ = 1 yields a Bayesian posterior . When U ( · ) is convex , the rapid convergence has been widely studied in Durmus & Moulines ( 2016 ) ; Dalalyan ( 2017 ) ; however , when U ( · ) is non-convex , a slow mixing rate is inevitable ( Raginsky et al. , 2017 ) . To accelerate the simulation , replica exchange Langevin diffusion ( reLD ) proposes to include a high-temperature particle β ( P ) t , where P ∈ N+\ { 1 } , for exploration . Meanwhile , a low-temperature particle β ( 1 ) t is presented for exploitation : dβ ( P ) t = −∇U ( β ( P ) t ) dt+ √ 2τ ( P ) dW ( P ) t , dβ ( 1 ) t = −∇U ( β ( 1 ) t ) dt+ √ 2τ ( 1 ) dW ( 1 ) t , ( 1 ) where τ ( P ) > τ ( 1 ) and W ( P ) t is independent of W ( 1 ) t . To promote more explorations for the low-temperature particle , the particles at the position ( β ( 1 ) , β ( P ) ) ∈ R2d swap with a probability aS ( β ( 1 ) , β ( P ) ) = a · ( 1 ∧ e ( 1 τ ( 1 ) − 1 τ ( P ) ) ( U ( β ( 1 ) ) −U ( β ( P ) ) ) ) , ( 2 ) where a ∈ ( 0 , ∞ ) is the swap intensity . In specific , the conditional swap rate at time t follows that P ( βt+dt = ( β ( P ) , β ( 1 ) ) |βt = ( β ( 1 ) , β ( P ) ) ) = aS ( β ( 1 ) , β ( P ) ) dt , P ( βt+dt = ( β ( 1 ) , β ( P ) ) |βt = ( β ( 1 ) , β ( P ) ) ) = 1− aS ( β ( 1 ) , β ( P ) ) dt . In the longtime limit , the Markov jump process converges to the joint distribution π ( β ( 1 ) , β ( P ) ) ∝ e −U ( β ( 1 ) ) τ ( 1 ) −U ( β ( P ) ) τ ( P ) . For convenience , we refer to the marginal distribution π ( 1 ) ( β ) ∝ e− U ( β ) τ ( 1 ) and π ( P ) ( β ) ∝ e− U ( β ) τ ( P ) as the target distribution and reference distribution , respectively . 2 PRELIMINARIES . Achieving sufficient explorations requires a large τ ( P ) , which leads to limited accelerations due to a small overlap between π ( 1 ) and π ( P ) . To tackle this issue , one can bring in multiple particles with temperatures ( τ ( 2 ) , · · · , τ ( P−1 ) ) , where τ ( 1 ) < τ ( 2 ) < · · · < τ ( P ) , to hollow out “ tunnels ” . To maintain feasibility , numerous schemes are presented to select candidate pairs to attempt the swaps . APE The all-pairs exchange ( APE ) attempts to swap arbitrary pair of chains ( Brenner et al. , 2007 ; Lingenheil et al. , 2009 ) , however , such a method requires a swap time ( see definition in section A.5 ) of O ( P 3 ) and may not be user-friendly in practice . ADJ In addition to swap arbitrary pairs , one can also swap adjacent ( ADJ ) pairs iteratively from ( 1 , 2 ) , ( 2 , 3 ) , to ( P − 1 , P ) under the Metropolis rule . Despite the convenience , the sequential nature requires to wait for exchange information from previous exchanges , which only works well with a small number of chains and has greatly limited its extension to a multi-core or distributed context . SEO The stochastic even-odd ( SEO ) scheme first divides the adjacent pairs { ( p − 1 , p ) |p = 2 , · · · , P } into E and O , where E and O denote even and odd pairs of forms ( 2p − 1 , 2p ) and ( 2p , 2p + 1 ) , respectively . Then , SEO randomly picks E or O pairs with an equal chance in each iteration to attempt the swaps . Notably , it can be conducted simultaneously without waiting from other chains . The scheme yields a reversible process ( see Figure 1 ( a ) ) , however , the gains in overcoming the sequential obstacle don ’ t offset the O ( P 2 ) round trip time and SEO is still not effective enough . DEO The deterministic even-odd ( DEO ) scheme instead attempts to swap even ( E ) pairs at even ( E ) iterations and odd ( O ) pairs at odd ( O ) iterations alternatingly† ( Okabe et al. , 2001 ) . The asymmetric manner was later interpreted as a non-reversible PT ( Syed et al. , 2021 ) and an ideal index process follows a periodic orbit , as shown in Figure 1 ( b ) . With a large swap rate , Figure 1 ( c ) shows how the scheme yields an almost straight path and a linear round trip time can be expected . Equi-acceptance The power of PT hinges on maximizing the number of round trips , which is equivalent to minimizing ∑P−1 p=1 1 1−rp ( Nadler & Hansmann , 2007b ) , where rp denotes the rejection rate for the chain pair ( p , p + 1 ) . Moreover , ∑P−1 p=1 rp converges to a fixed barrier Λ as P → ∞ ( Predescu et al. , 2004 ; Syed et al. , 2021 ) . Applying Lagrange multiplies to the constrained optimization problem leads to r1 = r2 = · · · = rP−1 : = r , where r is the equi-rejection rate . In general , a quadratic round trip time is required for ADJ and SEO due to the reversible indexes . By contrast , DEO only yields a linear round trip time in terms of P as P →∞ Syed et al . ( 2021 ) . 3 OPTIMAL NON-REVERSIBLE SCHEME FOR PARALLEL TEMPERING . The linear round trip time is appealing for maximizing the algorithmic potential , however , such an advance only occurs given sufficiently many chains . In non-asymptotic settings with limited chains , a pearl of wisdom is to avoid frequent swaps ( Dupuis et al. , 2012 ) and to keep the average acceptance rate from 20 % to 40 % ( Kone & Kofke , 2005 ; Lingenheil et al. , 2009 ; Atchadé et al. , 2011 ) . † E shown in iterations means even iterations ; otherwise , it denotes even pairs for chain indexes . The same logic applies toO . Most importantly , the acceptance rates are severely reduced in big data due to the bias-corrected swaps associated with stochastic energies ( Deng et al. , 2020 ) , see details in section A.1 . As such , maintaining low rejection rates becomes quite challenging and the issue of quadratic costs still exists . 3.1 GENERALIZED DEO SCHEME . As illustrated in Figure.2 ( e ) , the blue particle has a larger chance of ( 1 − r2 ) 2 to move upward 2 steps given W = 2 instead of ( 1− r ) 2 when W = 1 , although the window number is also halved . Such a trade-off inspires us to analyze the expected round trip time based on the window of size W . Although allowing at most one swap introduces the stopping time and may affect the distribution , the bias is rather mild due to the noisy energy estimators in big data . Check section C.2 for the details . 3.2 ANALYSIS OF ROUND TRIP TIME . To bring sufficient interactions between the reference distribution π ( P ) and the target distribution π ( 1 ) , we expect to minimize the expected round trip time T ( defined in section A.5 ) to ensure both efficient exploitation and explorations . Combining the Markov property and the idea of the master equation ( Nadler & Hansmann , 2007a ) , we estimate the expected round trip time E [ T ] as follows Lemma 1 . Under the stationary and weak dependence assumptions B1 and B2 in section B , for P ( P ≥ 2 ) chains with window size W ( W ≥ 1 ) and rejection rates { rp } P−1p=1 , we have E [ T ] = 2WP + 2WP P−1∑ p=1 rWp 1− rWp . ( 4 ) The proof in section B.1 shows that E [ T ] increases as we adopt larger number of chains P and rejection rates { rp } P−1p=1 . In such a case , the round trip rate PE [ T ] is also maximized by the key renewal theorem . In particular , applying W = 1 recovers the vanilla DEO scheme . 3.3 ANALYSIS OF OPTIMAL WINDOW SIZE AND ROUND TRIP TIME . By Lemma 1 , we observe a potential to remove the second quadratic term given an appropriate W . Such a fact motivates us to study the optimal window size W to achieve the best efficiency . Under §The generalized DEO with the optimal window size is denoted by DEO ? and will be studied in section 3.3. the equi-acceptance settings , by treating the window size W as a continuous variable and taking the derivative of E [ T ] with respect to W , we have ∂ ∂W E [ T ] = 2P ( 1− rW ) 2 { ( 1− rW ) 2 + ( P − 1 ) rW ( 1− rW +W log r ) } , ( 5 ) where r is the equi-rejection rate for adjacent chains . Define x : = rW ∈ ( 0 , 1 ) , where W = logr ( x ) = log x log r . The following analysis hinges on the study of the solution g ( x ) = ( 1− x ) 2 + ( P − 1 ) x ( 1− x+ log ( x ) ) = 0 . By analyzing the growth of derivatives and boundary values , we can easily identify the uniqueness of the solution . Then , we proceed to verify that 1P logP yields an asymptotic approximation such that g ( 1P logP ) = − log ( logP ) logP +O ( 1 logP ) → 0 as P →∞ . In the end , we have Theorem 1 . Under Assumptions B1 and B2 based on equi-acceptance settings , if P = 2 , 3 , the maximal round trip time is achieved when W = 1 . If P ≥ 4 , with the optimal window size W ? ≈ ⌈ logP+log logP − log r ⌉ , where d·e is the ceiling function . The round trip time follows O ( P logP− log r ) . The above result yields a remarkable round trip time of O ( P logP ) by setting the optimal window size W ? . By contrast , the vanilla DEO scheme only leads to a longer time of O ( P 2 ) § . Denoting by DEO ? the generalized DEO scheme with the optimal window size W ? , we summarize the popular swap schemes in Table.1 , where the DEO ? scheme performs the best among all the three criteria . | This paper proposes generalized DEO scheme (DEO*) that extends DEO with a window where at most one exchange could happen within one window. This paper also extend existing analysis on expected round trip time into DEO* and authors show that O(P logP) expected round trip time could be achieved even for finite number of chains and non-diminishing rejection rate. This paper also proposes multiple approximation to make DEO* practical: The exploration chains substitute gaussian noise with gradient noise, where noise magnitude is controlled with step size. Deterministic swap condition is used to avoid specific temperature. Adaptive learning rates and adaptive correction buffers are adopted to tune acceptance rate. The authors evaluate DEO* on ResNet model over CIFAR100. | SP:9a3e22f7e7f34083bac2b2db7e8fab75b8eddda9 |
Proving Theorems using Incremental Learning and Hindsight Experience Replay | 1 Introduction . I believe that to achieve human-level performance on hard problems , theorem provers likewise must be equipped with soft knowledge , in particular soft knowledge automatically gained from previous proof experiences . I also suspect that this will be one of the most fruitful areas of research in automated theorem proving . And one of the hardest . Schulz ( 2017 , E ’ s author ) Automated theorem proving ( ATP ) is an important tool both for assisting mathematicians in proving complex theorems as well as for areas such as integrated circuit design , and software and hardware verification ( Leroy , 2009 ; Klein , 2009 ) . Initial research in ATP dates back to 1960s ( e.g. , Robinson ( 1965 ) ; Knuth & Bendix ( 1970 ) ) and was motivated partly by the fact that mathematics is a hallmark of human intelligence . However , despite significant research effort and progress , ATP systems are still far from human capabilities ( Loos et al. , 2017 ) . The highest performing ATP systems ( e.g. , Cruanes et al . ( 2019 ) ; Kovács & Voronkov ( 2013 ) ) are decades old and have grown to use an increasing number of manually designed heuristics , mixed with some machine learning , to obtain a large number of search strategies that are tried sequentially or in parallel . Recent advances ( Loos et al. , 2017 ; Chvalovskỳ et al. , 2019 ) build on top of these provers and used modern machine learning techniques to augment , select or prioritize their heuristics , with some success . However , these machinelearning based provers usually require initial training data in the form of proofs , or positive and negative examples ( provided by the high-performing existing provers ) from which to bootstrap . Recent works do not build on top of other provers , but still require existing proof examples ( e.g. , Goertzel ( 2020 ) ; Polu & Sutskever ( 2020 ) ) . Our perspective is that the reliance of current machine learning techniques on high-end provers limits their potential to consistently surpass human capabilities in this domain . Therefore , in this paper we start with only a basic theorem proving algorithm , and develop machine learning methodology for bootstrapping automatically from this prover . In particular , given a set of conjectures without proofs , our system trains itself , based on its own attempts and ( dis ) proves an increasing number of conjectures , an approach which can be viewed as a form of incremental learning . A particularly interesting recent advance is rlCop ( Kaliszyk et al. , 2018 ; Zombori et al. , 2020 ) , which is based on the minimalistic leanCop theorem prover ( Otten & Bibel , 2003 ) , and is similar in spirit to our approach . It manages to surpass leanCop ’ s performance—but falls short of better competitors such as E , likely because it is based on a ‘ tableau ’ proving style rather than a saturation-based one . This motivated Crouse et al . ( 2021 ) to build on top of a saturation-based theorem prover and indeed see some improvement , while still not quite getting close to E. However , all previous approaches using incremental learning have a blind spot , because they learn exclusively from the proofs of successful attempts : If the given set of conjectures have large gaps in their difficulties , the system may get stuck at a suboptimal level due to the lack of new training data . This could in principle even happen at the very start , if all theorems are too hard to bootstrap from . To tackle this issue , Aygün et al . ( 2020 ) propose to create synthetic theorem generators based on the axioms of the actual conjectures , so as to provide a large initial training set with diverse difficulty . Unfortunately , synthetic theorems can be very different from the target conjectures , making transfer difficult . In this paper , we adapt instead hindsight experience replay ( HER ) ( Andrychowicz et al. , 2017 ) to ATP : clauses reached in proof attempts are turned into goals in hindsight . This generates a large amount of “ auxiliary ” theorems with proofs for the learner , even when no theorem from the original set can be proven . We compare our approach on a subset of TPTP ( Sutcliffe , 2017 ) with the state-of-the-art E prover ( Schulz , 2002 ; Cruanes et al. , 2019 ) , which performs very well on this dataset . Our learning prover eventually reaches equal or better performance on 16 domains out of 20 . In addition , it finds shorter proofs than E in approximately 98 % of the cases . We perform an ablation experiment to highlight specifically the role of hindsight experience replay . In the next sections , we explain our incremental learning methodology with hindsight experience replay , followed by a description of the network architecture and experimental results . 2 Methodology . For the reader unfamiliar with first-order logic , we give a succinct primer in Appendix A . From an abstract viewpoint , in our setting the main object under consideration is the clause , and two operations , factoring and resolution . These operations produce more clauses from one or two parent clauses . Starting from a set of axiom clauses and negated conjecture clauses ( which we will call input clauses ) , the two operations can be composed sequentially to try to reach the empty clause , in which case its ancestors form a refutation proof of the input clauses and correspond to a proof of the non-negated conjecture . We call the tree_size of a clause the number of symbols ( with repetition ) appearing in the clause ; for example tree_size ( p ( X , a , X , b ) ∨ q ( a ) ) is 7 . We start by describing the search algorithm , which allows us then to explain how we integrate machine learning and to describe our overall incremental learning system . 2.1 Search algorithm . To assess the incremental learning capabilities of recent machine learning advances , we have opted for a simple base search algorithm ( see also Kaliszyk et al . ( 2018 ) for example ) , instead of jump-starting from an existing high-performance theorem prover . Indeed , E is a state-of-art prover that incorporates a fair number of heuristics and optimizations ( Schulz , 2002 ; Cruanes et al. , 2019 ) , such as : axiom selection , simplifying the axioms and input clauses , more than 60 literal selection strategies , unit clause rewriting , multiple fast indexing techniques , clause evaluation heuristics ( tree size preference , age preference , preference toward symbols present in the conjecture , watch lists , etc . ) , strategy selection based on the analysis of the prover on similar problems , multiple strategy scheduling with 450 strategies tuned on TPTP 7.3.0,1 integration of a PicoSAT solver , etc . Other machine learning provers 1See http : //www.tptp.org/CASC/J10/SystemDescriptions.html # E -- -2.5. based on E ( e.g. , Jakub̊uv et al . ( 2020 ) ; Loos et al . ( 2017 ) ) automatically take advantage of at least some of these improvements ( but see also Goertzel ( 2020 ) ) . Like E and many other provers , we use a variant of the DISCOUNT loop ( Denzinger et al. , 1997 ) , itself a variant of the given-clause algorithm ( McCune & Wos , 1997 ) ( See Algorithm 3 in Appendix B ) . The input clauses are initially part of the candidates , while the active_clauses list starts empty . A candidate clause is selected at each iteration . New factors of the clause , as well as all its resolvents with the active clauses are pushed back into candidates . The given clause is then added to active_clauses , and we say that it has been processed . We also use the standard forward and backward subsumptions , as well as tautology deletion , which allow to remove too-specific clauses that are provably unnecessary for refutation . If candidates becomes empty before the empty clause can be produced , the algorithm returns `` saturated '' , which means that the input clauses are actually countersatisfiable ( the original conjecture is dis-proven ) . Given-clause-based theorem provers often use several priority queues to sort the set of candidates ( e.g. , McCune & Wos ( 1997 ) ; Schulz ( 2002 ) ; Kovács & Voronkov ( 2013 ) ) . We also use three priority queues : the age queue , ordered from the oldest generated clause to the youngest , the weight queue , ordered by increasing tree_size of the clauses , and the learnedcost queue , which uses a neural network to assign a score to each generated clause . The age queue ensures that every generated clause is processed after a number of steps that is at most a constant factor times its age . The weight queue ensures that small clauses are processed early , as they are “ closer ” to the empty clause . The learned-cost queue allows us to integrate machine learning into the search algorithm , as detailed below . 2.2 Clause-scoring network and hindsight experience replay . The clause-scoring network can be trained in many ways so as to find proofs faster . A method utilized by Loos et al . ( 2017 ) and Jakubuv & Urban ( 2019 ) turns the scoring task into a classification task : a network is trained to predict whether the clause to be scored will appear in the proof or not . In other words , the probability predicted by an ‘ in-proofness ’ classifier is used as the score . To train , once a proof is found , the clauses that participate in the proof ( i.e. , the ancestors of the empty clause ) are considered to be positive examples , while all other generated clauses are taken as negative examples.2 Then , given as input one such generated clause x along with the input clauses Cs , the network must learn to predict whether x is part of the ( found ) proof . There are two main drawbacks to this approach . First , if all conjectures are too hard for the initially unoptimized prover , no proof is found and no positive examples are available , making supervised learning impossible . Second , since proofs are often small ( often a few dozen steps ) , only few positive examples are generated . As the number of available conjectures is often small too , there is far too little data to train a modern high-capacity neural network . Moreover , for supervised learning to be successful , the conjectures that can be proven must be sufficiently diverse , so the learner can steadily improve . Unfortunately , there is no guarantee that such a curriculum is available . If the difficulty suddenly jumps , the learner may be unable to improve further . These shortcomings arise because the learner only uses successful proofs , and all the unsuccessful proof attempts are discarded . In particular , the overwhelming majority of the generated clauses become negative examples , and most need to be discarded to maintain a good balance with the positive examples . To leverage the data generated in unsuccessful proof attempts , we adapt the concept of hindsight experience replay ( HER ) ( Andrychowicz et al. , 2017 ) from goal-conditioned reinforcement learning to theorem proving . The core idea of HER is to take any “ unsuccessful ” trajectory in a goal-based task and convert it into a successful one by treating the final state as if it were the goal state , in hindsight . A deep network is then trained with this trajectory , by contextualizing the policy with this state instead of the original goal . This way , even in 2These examples are technically not necessarily negative , as they may be part of another proof . But avoiding these examples during the search still helps the system to attribute more significance to the positive examples . the absence of positive feedback , the network is still able to adapt to the domain , if not to the goal , thus having a better chance to reach the goal on future tries . Inspired by HER , we use the clauses generated during any proof attempt as additional conjectures , which we call hindsight goals , leading to a supply of positive and negative examples . Let D be any non-input clause generated during the refutation attempt of Cs . We call D a hindsight goal.3 Then , the set Cs∪ { ¬D } can be refuted . Furthermore , once the prover reaches D starting from Cs ∪ { ¬D } , only a few more resolution steps are necessary to reach the empty clause ; that is , there exists a refutation proof of Cs ∪ { ¬D } where D is an ancestor of the empty clause . Hence , we can use the ancestors of D as positive examples for the negated conjecture and axioms Cs ∪ { ¬D } . This generates a very large number of examples , allowing us to effectively train the neural network , even with only a few conjectures at hand . Since each domain has its own set of axioms , and a separate network is trained per domain , axioms are not provided as input to the scoring network . Although the set of active clauses is an important factor in determining the usefulness of a clause , we ignore it in the network input to keep the network size smaller . | The paper proposes an incremental learning approach to theorem proving that learns to prove theorems from scratch, using Hindsight Experience Replay (HER) to learn from unsuccessful proofs. The approach adopts ideas from goal-conditioned reinforcement learning and creates more training examples by turning intermediate steps of an unsuccessful proof attempt into proved conjectures and add these into positive examples to learn from. This significantly increases the number of examples to learn from compared to related work that only learns from successful proofs. The approach is shown to be competitive against E prover, a SOTA prover, on a subset of TPTP without equality. | SP:392d14564f38d713580362e7a0569b377e674617 |
Proving Theorems using Incremental Learning and Hindsight Experience Replay | 1 Introduction . I believe that to achieve human-level performance on hard problems , theorem provers likewise must be equipped with soft knowledge , in particular soft knowledge automatically gained from previous proof experiences . I also suspect that this will be one of the most fruitful areas of research in automated theorem proving . And one of the hardest . Schulz ( 2017 , E ’ s author ) Automated theorem proving ( ATP ) is an important tool both for assisting mathematicians in proving complex theorems as well as for areas such as integrated circuit design , and software and hardware verification ( Leroy , 2009 ; Klein , 2009 ) . Initial research in ATP dates back to 1960s ( e.g. , Robinson ( 1965 ) ; Knuth & Bendix ( 1970 ) ) and was motivated partly by the fact that mathematics is a hallmark of human intelligence . However , despite significant research effort and progress , ATP systems are still far from human capabilities ( Loos et al. , 2017 ) . The highest performing ATP systems ( e.g. , Cruanes et al . ( 2019 ) ; Kovács & Voronkov ( 2013 ) ) are decades old and have grown to use an increasing number of manually designed heuristics , mixed with some machine learning , to obtain a large number of search strategies that are tried sequentially or in parallel . Recent advances ( Loos et al. , 2017 ; Chvalovskỳ et al. , 2019 ) build on top of these provers and used modern machine learning techniques to augment , select or prioritize their heuristics , with some success . However , these machinelearning based provers usually require initial training data in the form of proofs , or positive and negative examples ( provided by the high-performing existing provers ) from which to bootstrap . Recent works do not build on top of other provers , but still require existing proof examples ( e.g. , Goertzel ( 2020 ) ; Polu & Sutskever ( 2020 ) ) . Our perspective is that the reliance of current machine learning techniques on high-end provers limits their potential to consistently surpass human capabilities in this domain . Therefore , in this paper we start with only a basic theorem proving algorithm , and develop machine learning methodology for bootstrapping automatically from this prover . In particular , given a set of conjectures without proofs , our system trains itself , based on its own attempts and ( dis ) proves an increasing number of conjectures , an approach which can be viewed as a form of incremental learning . A particularly interesting recent advance is rlCop ( Kaliszyk et al. , 2018 ; Zombori et al. , 2020 ) , which is based on the minimalistic leanCop theorem prover ( Otten & Bibel , 2003 ) , and is similar in spirit to our approach . It manages to surpass leanCop ’ s performance—but falls short of better competitors such as E , likely because it is based on a ‘ tableau ’ proving style rather than a saturation-based one . This motivated Crouse et al . ( 2021 ) to build on top of a saturation-based theorem prover and indeed see some improvement , while still not quite getting close to E. However , all previous approaches using incremental learning have a blind spot , because they learn exclusively from the proofs of successful attempts : If the given set of conjectures have large gaps in their difficulties , the system may get stuck at a suboptimal level due to the lack of new training data . This could in principle even happen at the very start , if all theorems are too hard to bootstrap from . To tackle this issue , Aygün et al . ( 2020 ) propose to create synthetic theorem generators based on the axioms of the actual conjectures , so as to provide a large initial training set with diverse difficulty . Unfortunately , synthetic theorems can be very different from the target conjectures , making transfer difficult . In this paper , we adapt instead hindsight experience replay ( HER ) ( Andrychowicz et al. , 2017 ) to ATP : clauses reached in proof attempts are turned into goals in hindsight . This generates a large amount of “ auxiliary ” theorems with proofs for the learner , even when no theorem from the original set can be proven . We compare our approach on a subset of TPTP ( Sutcliffe , 2017 ) with the state-of-the-art E prover ( Schulz , 2002 ; Cruanes et al. , 2019 ) , which performs very well on this dataset . Our learning prover eventually reaches equal or better performance on 16 domains out of 20 . In addition , it finds shorter proofs than E in approximately 98 % of the cases . We perform an ablation experiment to highlight specifically the role of hindsight experience replay . In the next sections , we explain our incremental learning methodology with hindsight experience replay , followed by a description of the network architecture and experimental results . 2 Methodology . For the reader unfamiliar with first-order logic , we give a succinct primer in Appendix A . From an abstract viewpoint , in our setting the main object under consideration is the clause , and two operations , factoring and resolution . These operations produce more clauses from one or two parent clauses . Starting from a set of axiom clauses and negated conjecture clauses ( which we will call input clauses ) , the two operations can be composed sequentially to try to reach the empty clause , in which case its ancestors form a refutation proof of the input clauses and correspond to a proof of the non-negated conjecture . We call the tree_size of a clause the number of symbols ( with repetition ) appearing in the clause ; for example tree_size ( p ( X , a , X , b ) ∨ q ( a ) ) is 7 . We start by describing the search algorithm , which allows us then to explain how we integrate machine learning and to describe our overall incremental learning system . 2.1 Search algorithm . To assess the incremental learning capabilities of recent machine learning advances , we have opted for a simple base search algorithm ( see also Kaliszyk et al . ( 2018 ) for example ) , instead of jump-starting from an existing high-performance theorem prover . Indeed , E is a state-of-art prover that incorporates a fair number of heuristics and optimizations ( Schulz , 2002 ; Cruanes et al. , 2019 ) , such as : axiom selection , simplifying the axioms and input clauses , more than 60 literal selection strategies , unit clause rewriting , multiple fast indexing techniques , clause evaluation heuristics ( tree size preference , age preference , preference toward symbols present in the conjecture , watch lists , etc . ) , strategy selection based on the analysis of the prover on similar problems , multiple strategy scheduling with 450 strategies tuned on TPTP 7.3.0,1 integration of a PicoSAT solver , etc . Other machine learning provers 1See http : //www.tptp.org/CASC/J10/SystemDescriptions.html # E -- -2.5. based on E ( e.g. , Jakub̊uv et al . ( 2020 ) ; Loos et al . ( 2017 ) ) automatically take advantage of at least some of these improvements ( but see also Goertzel ( 2020 ) ) . Like E and many other provers , we use a variant of the DISCOUNT loop ( Denzinger et al. , 1997 ) , itself a variant of the given-clause algorithm ( McCune & Wos , 1997 ) ( See Algorithm 3 in Appendix B ) . The input clauses are initially part of the candidates , while the active_clauses list starts empty . A candidate clause is selected at each iteration . New factors of the clause , as well as all its resolvents with the active clauses are pushed back into candidates . The given clause is then added to active_clauses , and we say that it has been processed . We also use the standard forward and backward subsumptions , as well as tautology deletion , which allow to remove too-specific clauses that are provably unnecessary for refutation . If candidates becomes empty before the empty clause can be produced , the algorithm returns `` saturated '' , which means that the input clauses are actually countersatisfiable ( the original conjecture is dis-proven ) . Given-clause-based theorem provers often use several priority queues to sort the set of candidates ( e.g. , McCune & Wos ( 1997 ) ; Schulz ( 2002 ) ; Kovács & Voronkov ( 2013 ) ) . We also use three priority queues : the age queue , ordered from the oldest generated clause to the youngest , the weight queue , ordered by increasing tree_size of the clauses , and the learnedcost queue , which uses a neural network to assign a score to each generated clause . The age queue ensures that every generated clause is processed after a number of steps that is at most a constant factor times its age . The weight queue ensures that small clauses are processed early , as they are “ closer ” to the empty clause . The learned-cost queue allows us to integrate machine learning into the search algorithm , as detailed below . 2.2 Clause-scoring network and hindsight experience replay . The clause-scoring network can be trained in many ways so as to find proofs faster . A method utilized by Loos et al . ( 2017 ) and Jakubuv & Urban ( 2019 ) turns the scoring task into a classification task : a network is trained to predict whether the clause to be scored will appear in the proof or not . In other words , the probability predicted by an ‘ in-proofness ’ classifier is used as the score . To train , once a proof is found , the clauses that participate in the proof ( i.e. , the ancestors of the empty clause ) are considered to be positive examples , while all other generated clauses are taken as negative examples.2 Then , given as input one such generated clause x along with the input clauses Cs , the network must learn to predict whether x is part of the ( found ) proof . There are two main drawbacks to this approach . First , if all conjectures are too hard for the initially unoptimized prover , no proof is found and no positive examples are available , making supervised learning impossible . Second , since proofs are often small ( often a few dozen steps ) , only few positive examples are generated . As the number of available conjectures is often small too , there is far too little data to train a modern high-capacity neural network . Moreover , for supervised learning to be successful , the conjectures that can be proven must be sufficiently diverse , so the learner can steadily improve . Unfortunately , there is no guarantee that such a curriculum is available . If the difficulty suddenly jumps , the learner may be unable to improve further . These shortcomings arise because the learner only uses successful proofs , and all the unsuccessful proof attempts are discarded . In particular , the overwhelming majority of the generated clauses become negative examples , and most need to be discarded to maintain a good balance with the positive examples . To leverage the data generated in unsuccessful proof attempts , we adapt the concept of hindsight experience replay ( HER ) ( Andrychowicz et al. , 2017 ) from goal-conditioned reinforcement learning to theorem proving . The core idea of HER is to take any “ unsuccessful ” trajectory in a goal-based task and convert it into a successful one by treating the final state as if it were the goal state , in hindsight . A deep network is then trained with this trajectory , by contextualizing the policy with this state instead of the original goal . This way , even in 2These examples are technically not necessarily negative , as they may be part of another proof . But avoiding these examples during the search still helps the system to attribute more significance to the positive examples . the absence of positive feedback , the network is still able to adapt to the domain , if not to the goal , thus having a better chance to reach the goal on future tries . Inspired by HER , we use the clauses generated during any proof attempt as additional conjectures , which we call hindsight goals , leading to a supply of positive and negative examples . Let D be any non-input clause generated during the refutation attempt of Cs . We call D a hindsight goal.3 Then , the set Cs∪ { ¬D } can be refuted . Furthermore , once the prover reaches D starting from Cs ∪ { ¬D } , only a few more resolution steps are necessary to reach the empty clause ; that is , there exists a refutation proof of Cs ∪ { ¬D } where D is an ancestor of the empty clause . Hence , we can use the ancestors of D as positive examples for the negated conjecture and axioms Cs ∪ { ¬D } . This generates a very large number of examples , allowing us to effectively train the neural network , even with only a few conjectures at hand . Since each domain has its own set of axioms , and a separate network is trained per domain , axioms are not provided as input to the scoring network . Although the set of active clauses is an important factor in determining the usefulness of a clause , we ignore it in the network input to keep the network size smaller . | The paper applies hindsight experience replay (HER) to automatic theorem proving (ATP). ATP attempts to prove logic statements by (roughly speaking) showing that there are no counterexamples, i.e. showing that nothing resolves the negation of a statement. HER is a technique from reinforcement learning which mitigates sparse rewards by treating failed attempts as successful attempts on a different problem, namely that with the end state of the attempt as the goal. The paper is not quite an HER method however ; it does not use reinforcement learning but rather incremental learning, where clauses generated during proof attempts are used as additional data for (incremental) supervised learning. | SP:392d14564f38d713580362e7a0569b377e674617 |
Proving Theorems using Incremental Learning and Hindsight Experience Replay | 1 Introduction . I believe that to achieve human-level performance on hard problems , theorem provers likewise must be equipped with soft knowledge , in particular soft knowledge automatically gained from previous proof experiences . I also suspect that this will be one of the most fruitful areas of research in automated theorem proving . And one of the hardest . Schulz ( 2017 , E ’ s author ) Automated theorem proving ( ATP ) is an important tool both for assisting mathematicians in proving complex theorems as well as for areas such as integrated circuit design , and software and hardware verification ( Leroy , 2009 ; Klein , 2009 ) . Initial research in ATP dates back to 1960s ( e.g. , Robinson ( 1965 ) ; Knuth & Bendix ( 1970 ) ) and was motivated partly by the fact that mathematics is a hallmark of human intelligence . However , despite significant research effort and progress , ATP systems are still far from human capabilities ( Loos et al. , 2017 ) . The highest performing ATP systems ( e.g. , Cruanes et al . ( 2019 ) ; Kovács & Voronkov ( 2013 ) ) are decades old and have grown to use an increasing number of manually designed heuristics , mixed with some machine learning , to obtain a large number of search strategies that are tried sequentially or in parallel . Recent advances ( Loos et al. , 2017 ; Chvalovskỳ et al. , 2019 ) build on top of these provers and used modern machine learning techniques to augment , select or prioritize their heuristics , with some success . However , these machinelearning based provers usually require initial training data in the form of proofs , or positive and negative examples ( provided by the high-performing existing provers ) from which to bootstrap . Recent works do not build on top of other provers , but still require existing proof examples ( e.g. , Goertzel ( 2020 ) ; Polu & Sutskever ( 2020 ) ) . Our perspective is that the reliance of current machine learning techniques on high-end provers limits their potential to consistently surpass human capabilities in this domain . Therefore , in this paper we start with only a basic theorem proving algorithm , and develop machine learning methodology for bootstrapping automatically from this prover . In particular , given a set of conjectures without proofs , our system trains itself , based on its own attempts and ( dis ) proves an increasing number of conjectures , an approach which can be viewed as a form of incremental learning . A particularly interesting recent advance is rlCop ( Kaliszyk et al. , 2018 ; Zombori et al. , 2020 ) , which is based on the minimalistic leanCop theorem prover ( Otten & Bibel , 2003 ) , and is similar in spirit to our approach . It manages to surpass leanCop ’ s performance—but falls short of better competitors such as E , likely because it is based on a ‘ tableau ’ proving style rather than a saturation-based one . This motivated Crouse et al . ( 2021 ) to build on top of a saturation-based theorem prover and indeed see some improvement , while still not quite getting close to E. However , all previous approaches using incremental learning have a blind spot , because they learn exclusively from the proofs of successful attempts : If the given set of conjectures have large gaps in their difficulties , the system may get stuck at a suboptimal level due to the lack of new training data . This could in principle even happen at the very start , if all theorems are too hard to bootstrap from . To tackle this issue , Aygün et al . ( 2020 ) propose to create synthetic theorem generators based on the axioms of the actual conjectures , so as to provide a large initial training set with diverse difficulty . Unfortunately , synthetic theorems can be very different from the target conjectures , making transfer difficult . In this paper , we adapt instead hindsight experience replay ( HER ) ( Andrychowicz et al. , 2017 ) to ATP : clauses reached in proof attempts are turned into goals in hindsight . This generates a large amount of “ auxiliary ” theorems with proofs for the learner , even when no theorem from the original set can be proven . We compare our approach on a subset of TPTP ( Sutcliffe , 2017 ) with the state-of-the-art E prover ( Schulz , 2002 ; Cruanes et al. , 2019 ) , which performs very well on this dataset . Our learning prover eventually reaches equal or better performance on 16 domains out of 20 . In addition , it finds shorter proofs than E in approximately 98 % of the cases . We perform an ablation experiment to highlight specifically the role of hindsight experience replay . In the next sections , we explain our incremental learning methodology with hindsight experience replay , followed by a description of the network architecture and experimental results . 2 Methodology . For the reader unfamiliar with first-order logic , we give a succinct primer in Appendix A . From an abstract viewpoint , in our setting the main object under consideration is the clause , and two operations , factoring and resolution . These operations produce more clauses from one or two parent clauses . Starting from a set of axiom clauses and negated conjecture clauses ( which we will call input clauses ) , the two operations can be composed sequentially to try to reach the empty clause , in which case its ancestors form a refutation proof of the input clauses and correspond to a proof of the non-negated conjecture . We call the tree_size of a clause the number of symbols ( with repetition ) appearing in the clause ; for example tree_size ( p ( X , a , X , b ) ∨ q ( a ) ) is 7 . We start by describing the search algorithm , which allows us then to explain how we integrate machine learning and to describe our overall incremental learning system . 2.1 Search algorithm . To assess the incremental learning capabilities of recent machine learning advances , we have opted for a simple base search algorithm ( see also Kaliszyk et al . ( 2018 ) for example ) , instead of jump-starting from an existing high-performance theorem prover . Indeed , E is a state-of-art prover that incorporates a fair number of heuristics and optimizations ( Schulz , 2002 ; Cruanes et al. , 2019 ) , such as : axiom selection , simplifying the axioms and input clauses , more than 60 literal selection strategies , unit clause rewriting , multiple fast indexing techniques , clause evaluation heuristics ( tree size preference , age preference , preference toward symbols present in the conjecture , watch lists , etc . ) , strategy selection based on the analysis of the prover on similar problems , multiple strategy scheduling with 450 strategies tuned on TPTP 7.3.0,1 integration of a PicoSAT solver , etc . Other machine learning provers 1See http : //www.tptp.org/CASC/J10/SystemDescriptions.html # E -- -2.5. based on E ( e.g. , Jakub̊uv et al . ( 2020 ) ; Loos et al . ( 2017 ) ) automatically take advantage of at least some of these improvements ( but see also Goertzel ( 2020 ) ) . Like E and many other provers , we use a variant of the DISCOUNT loop ( Denzinger et al. , 1997 ) , itself a variant of the given-clause algorithm ( McCune & Wos , 1997 ) ( See Algorithm 3 in Appendix B ) . The input clauses are initially part of the candidates , while the active_clauses list starts empty . A candidate clause is selected at each iteration . New factors of the clause , as well as all its resolvents with the active clauses are pushed back into candidates . The given clause is then added to active_clauses , and we say that it has been processed . We also use the standard forward and backward subsumptions , as well as tautology deletion , which allow to remove too-specific clauses that are provably unnecessary for refutation . If candidates becomes empty before the empty clause can be produced , the algorithm returns `` saturated '' , which means that the input clauses are actually countersatisfiable ( the original conjecture is dis-proven ) . Given-clause-based theorem provers often use several priority queues to sort the set of candidates ( e.g. , McCune & Wos ( 1997 ) ; Schulz ( 2002 ) ; Kovács & Voronkov ( 2013 ) ) . We also use three priority queues : the age queue , ordered from the oldest generated clause to the youngest , the weight queue , ordered by increasing tree_size of the clauses , and the learnedcost queue , which uses a neural network to assign a score to each generated clause . The age queue ensures that every generated clause is processed after a number of steps that is at most a constant factor times its age . The weight queue ensures that small clauses are processed early , as they are “ closer ” to the empty clause . The learned-cost queue allows us to integrate machine learning into the search algorithm , as detailed below . 2.2 Clause-scoring network and hindsight experience replay . The clause-scoring network can be trained in many ways so as to find proofs faster . A method utilized by Loos et al . ( 2017 ) and Jakubuv & Urban ( 2019 ) turns the scoring task into a classification task : a network is trained to predict whether the clause to be scored will appear in the proof or not . In other words , the probability predicted by an ‘ in-proofness ’ classifier is used as the score . To train , once a proof is found , the clauses that participate in the proof ( i.e. , the ancestors of the empty clause ) are considered to be positive examples , while all other generated clauses are taken as negative examples.2 Then , given as input one such generated clause x along with the input clauses Cs , the network must learn to predict whether x is part of the ( found ) proof . There are two main drawbacks to this approach . First , if all conjectures are too hard for the initially unoptimized prover , no proof is found and no positive examples are available , making supervised learning impossible . Second , since proofs are often small ( often a few dozen steps ) , only few positive examples are generated . As the number of available conjectures is often small too , there is far too little data to train a modern high-capacity neural network . Moreover , for supervised learning to be successful , the conjectures that can be proven must be sufficiently diverse , so the learner can steadily improve . Unfortunately , there is no guarantee that such a curriculum is available . If the difficulty suddenly jumps , the learner may be unable to improve further . These shortcomings arise because the learner only uses successful proofs , and all the unsuccessful proof attempts are discarded . In particular , the overwhelming majority of the generated clauses become negative examples , and most need to be discarded to maintain a good balance with the positive examples . To leverage the data generated in unsuccessful proof attempts , we adapt the concept of hindsight experience replay ( HER ) ( Andrychowicz et al. , 2017 ) from goal-conditioned reinforcement learning to theorem proving . The core idea of HER is to take any “ unsuccessful ” trajectory in a goal-based task and convert it into a successful one by treating the final state as if it were the goal state , in hindsight . A deep network is then trained with this trajectory , by contextualizing the policy with this state instead of the original goal . This way , even in 2These examples are technically not necessarily negative , as they may be part of another proof . But avoiding these examples during the search still helps the system to attribute more significance to the positive examples . the absence of positive feedback , the network is still able to adapt to the domain , if not to the goal , thus having a better chance to reach the goal on future tries . Inspired by HER , we use the clauses generated during any proof attempt as additional conjectures , which we call hindsight goals , leading to a supply of positive and negative examples . Let D be any non-input clause generated during the refutation attempt of Cs . We call D a hindsight goal.3 Then , the set Cs∪ { ¬D } can be refuted . Furthermore , once the prover reaches D starting from Cs ∪ { ¬D } , only a few more resolution steps are necessary to reach the empty clause ; that is , there exists a refutation proof of Cs ∪ { ¬D } where D is an ancestor of the empty clause . Hence , we can use the ancestors of D as positive examples for the negated conjecture and axioms Cs ∪ { ¬D } . This generates a very large number of examples , allowing us to effectively train the neural network , even with only a few conjectures at hand . Since each domain has its own set of axioms , and a separate network is trained per domain , axioms are not provided as input to the scoring network . Although the set of active clauses is an important factor in determining the usefulness of a clause , we ignore it in the network input to keep the network size smaller . | This paper proposes an “incremental learning” algorithm for training a distribution (domain)-specific theorem prover for FO formulae without the need of a classical prover to generate training examples (intermediate clauses of a proof). Given a set of axioms and the negation of the (to be proven) conjecture, the prover iteratively 1. finds factors of clauses; and 2. applies resolution, until either an empty clause is reached (conjecture is proven) or no more new clauses can be generated (the conjecture is dis-proven). It is important for the prover to select clauses for factorization and/or resolution, that would lead to an empty clause faster. To that end, the AI-based theorem provers are augmented with a ML model (policy) that assists the prover by scoring the candidate clauses based on the probability of their applicability in building a proof. This in turn leads to construction of shorter proofs. The authors incorporate Hindsight Experience Replay (HER) in order to mitigate the lack of expert (machine and/or human generated) training sets. They show that HER indeed helps their incremental learning scheme to a point that the resulting ML-aided prover achieves comparable performance wrt. to the SOTA conventional prover E. Additionally the authors propose the use of spectral encoding to represent formulae instead of positional encoding of transformers. The resulting architecture is tested on TPTP benchmark against E, along with an ablation to show the effectiveness of HER. The comparison with E is to measure both the number of proven conjectures within a cutoff as well as the proof quality (proof length). | SP:392d14564f38d713580362e7a0569b377e674617 |
Revisiting Over-smoothing in BERT from the Perspective of Graph | 1 INTRODUCTION . Over the past few years , Transformer ( Vaswani et al. , 2017 ) has been widely used in various natural language processing ( NLP ) tasks , including text classification ( Wang et al. , 2018a ) , text translation ( Ott et al. , 2018 ) , question answering ( Rajpurkar et al. , 2016 ; 2018 ) and text generation ( Brown et al. , 2020 ) . The recent application of Transformer in computer vision ( CV ) field also demonstrate the potential capacity of Transformer architecture . For instance , Transformer variants have been successfully used for image classification ( Dosovitskiy et al. , 2021 ) , object detection ( Carion et al. , 2020 ) and semantic segmentation ( Strudel et al. , 2021 ) . Three fundamental descendants from Transformer include BERT ( Devlin et al. , 2019 ) , RoBERTa ( Liu et al. , 2019 ) and ALBERT ( Lan et al. , 2020 ) , which achieve state-of-the-art performance on a wide range of NLP tasks . Recently , Dong et al . ( 2021 ) observes the “ token uniformity ” problem , which reduces the capacity of Transformer-based architectures by making all token representations identical . They claim that pure self-attention ( SAN ) modules cause token uniformity , but they do not discuss whether the token uniformity problem still exists in Transformer blocks . On the other hand , Gong et al . ( 2021 ) observe the “ over-smoothing ” problem for ViT ( Dosovitskiy et al. , 2021 ) , in that different input patches are mapped to a similar latent representation . To prevent loss of information , they introduce additional loss functions to encourage diversity and successfully improve model performance by suppressing over-smoothing . Moreover , “ overthinking ” phenomenon , indicating that shallow representations are better than deep representations , also be observed in ( Zhou et al. , 2020 ; Kaya et al. , 2019 ) . As discussed in Section 3 , this phenomenon has some inherent connection with over-smoothing . In this paper , we use “ over-smoothing ” to unify the above issues , and refer this as the phenomenon that the model performance is deteriorated because different inputs are mapped to a similar representation . As the over-smoothing problem is first studied in the graph neural network ( GNN ) literature ( Li et al. , 2018 ; Xu et al. , 2018 ; Zhao & Akoglu , 2020 ) , in this paper , we attempt to explore the cause of such ∗Equal contribution . problem by building a relationship between Transformer blocks and graphs . Specifically , we consider the self-attention matrix as the normalized adjacency matrix of a weighted graph , whose nodes are the tokens in a sentence . Furthermore , we consider the inherent connection between BERT and graph convolutional networks ( Kipf & Welling , 2017 ) . Inspired by the over-smoothing problem in GNN , we study over-smoothing in BERT from a theoretical view via matrix projection . As opposed to Dong et al . ( 2021 ) , where the authors claim that layer normalization is irrelevant to over-smoothing , we find that layer normalization ( Ba et al. , 2016 ) plays an important role in over-smoothing . Specifically , we theoretically prove that , if the standard deviation in layer normalization is sufficiently large , the outputs of the Transformer stacks will converge to a low-rank subspace , resulting in over-smoothing . Empirically , we verify that the conditions hold for a certain number of samples for a pre-trained and fine-tuned BERT model ( Devlin et al. , 2019 ) , which is consistent with our above observations . To alleviate the over-smoothing problem , we propose a hierarchical fusion strategy that adaptively fuses representations from different layers . Three fusion approaches are used : ( i ) Concat Fusion , ( ii ) Max Fusion , and ( iii ) Gate Fusion . The proposed method reduces the similarity between tokens and outperforms BERT baseline on the GLUE ( Wang et al. , 2018a ) , SWAG ( Zellers et al. , 2018 ) and SQuAD ( Rajpurkar et al. , 2016 ; 2018 ) data sets . In summary , the contributions of this paper are as follows : ( i ) We develop the relationship between self-attention and graph for a better understanding of over-smoothing in BERT . ( ii ) We provide theoretical analysis on over-smoothing in the BERT model , and empirically verify the theoretical results . ( iii ) We propose hierarchical fusion strategies that adaptively combine different layers to alleviate over-smoothing . Extensive experimental results verify our methods ’ effectiveness . 2 RELATED WORK . 2.1 TRANSFORMER BLOCK AND SELF-ATTENTION . Transformer block is a basic component in Transformer model ( Vaswani et al. , 2017 ) . Each Transformer block consists of a self-attention layer and a feed-forward layer . Let X ∈ Rn×d be the input to a Transformer block , where n is the number of input tokens and d is the embedding size . The self-attention layer output can be written as : Attn ( X ) = X + h∑ k=1 σ ( XWQk ( XW K k ) > ) XW Vk W O > k = X + h∑ k=1 ÂkXW V O k , ( 1 ) where h is the number of heads , σ is the softmax function , and WQk , W K k , W V k , W O k ∈ Rd×dh ( where dh = d/h is the dimension of a single-head output ) are weight matrices for the query , key , value , and output , respectively of the kth head . In particular , the self-attention matrix  = σ ( XWQ ( XWK ) > ) = σ ( QK > ) ( 2 ) in ( 1 ) plays a key role in the self-attention layer ( Park et al. , 2019 ; Gong et al. , 2019 ; Kovaleva et al. , 2019 ) . As in ( Yun et al. , 2020 ; Shi et al. , 2021 ; Dong et al. , 2021 ) , we drop the scale product 1/ √ dh to simplify analysis . The feed-forward layer usually has two fully-connected ( FC ) layers with residual connection : FF ( X ) = Attn ( X ) +ReLU ( Attn ( X ) W1 + b1 ) W2 + b2 , where W1 ∈ Rd×dff , W2 ∈ Rdff×d ( dff is the size of the intermediate layer ) are the weight matrices , and b1 , b2 are the biases . Two layer normalization ( Ba et al. , 2016 ) operations are performed after the self-attention layer and fully-connected layer , respectively . 2.2 OVER-SMOOTHING . In graph neural networks , over-smoothing refers to the problem that the performance deteriorates as representations of all the nodes become similar ( Li et al. , 2018 ; Xu et al. , 2018 ; Huang et al. , 2020 ) . Its main cause is the stacked aggregation layer using the same adjacency matrix . Recently , several approaches have been proposed to alleviate the over-smoothing problem . Xu et al . ( 2018 ) propose a jumping knowledge network for better structure-aware representation , which flexibly leverages different neighborhood ranges . ResGCN ( Li et al. , 2019 ) adapts the residual connection and dilated convolution in the graph convolutional network ( GCN ) , and successfully scales the GCN to 56 layers . Zhao & Akoglu ( 2020 ) propose PairNorm , a novel normalization layer , that prevents node embeddings from becoming too similar . DropEdge ( Rong et al. , 2020 ; Huang et al. , 2020 ) randomly removes edges from the input graph at each training epoch , and reduces the effect of over-smoothing . Unlike graph neural networks , over-smoothing in Transformer-based architectures has not been discussed in detail . Dong et al . ( 2021 ) introduce the “ token-uniformity ” problem for self-attention , and show that skip connections and multi-layer perceptron can mitigate this problem . However , Gong et al . ( 2021 ) still observe over-smoothing on the Vision Transformers ( Dosovitskiy et al. , 2021 ) . 3 DOES OVER-SMOOTHING EXIST IN BERT ? . In this section , we first explore the existence of over-smoothing in BERT , by measuring the similarity between tokens in each Transformer layer . Specifically , we use the token-wise cosine similarity ( Gong et al. , 2021 ) as our similarity measure : CosSim = 1 n ( n− 1 ) ∑ i 6=j h > i hj ‖hi‖2‖hj‖2 , where n is the number of tokens , hi and hj are two representations of different tokens , and ‖ · ‖2 is the Euclidean norm . Following Dong et al . ( 2021 ) , we use WikiBio ( Lebret et al. , 2016 ) as input to the following Transformer-based models fine-tuned on the SQuAD data set ( Rajpurkar et al. , 2018 ) : ( i ) BERT ( Devlin et al. , 2019 ) , ( ii ) RoBERTa ( Liu et al. , 2019 ) and ( iii ) ALBERT ( Lan et al. , 2020 ) .1 For comparison , all three models are stacked with 12 blocks . We calculate each CosSim for each data sample and show the average and standard derivation of CosSim values over all WikiBio data . In the figures , layer 0 represents original input token representation , and layer 1-12 represents the corresponding transformer layers . As shown in Figure 1 ( a ) , the original token representations are different from each other , while token similarities are high in the last layer . For instance , the average token-wise cosine similarity of the last layer of ALBERT and RoBERTa are both larger than 90 % . To illustrate the relationship between “ over-thinking ” and “ over-smoothing ” , we compare the tokenwise cosine similarity at each layer with the corresponding error rate . As for the corresponding error rate of layer i , we use the representations from layer i as the final output and fine-tune the classifier . Following Zhou et al . ( 2020 ) , we experiment with ALBERT ( Lan et al. , 2020 ) fine-tuned on the MRPC data set ( Dolan & Brockett , 2005 ) and use their error rate results for convenience . As shown in Figure 1 ( b ) , layer 10 has the lowest cosine similarity and error rate . At layers 11 and 12 , the tokens have larger cosine similarities , making them harder to distinguish and resulting in the performance drop . Thus , “ over-thinking ” can be explained by “ over-smoothing ” . A direct consequence of over-smoothing is that the performance can not be improved when the model gets deeper , since the individual tokens are no longer distinguishable . To illustrate this , we increase the number of layers in BERT to 24 while keeping the other settings . As shown in Figure 1 ( c ) , the 1Our implementation is based on the HuggingFace ’ s Transformers library ( Wolf et al. , 2020 ) . performance of vanilla BERT can not improve as the model gets deeper . In contrast , the proposed hierarchical fusion ( as will be discussed in Section 6 ) consistently outperforms the baseline , and has better and better performance as the model gets deeper . Based on these observations , we conclude that the over-smoothing problem still exists in BERT . 4 RELATIONSHIP BETWEEN SELF-ATTENTION AND GRAPH . Since over-smoothing is first discussed in the graph neural network literature ( Li et al. , 2018 ; Zhao & Akoglu , 2020 ) , we attempt to understand its cause from a graph perspective in this section . | This paper explored the over-smoothing problem in BERT from the perspective of graph. In detail, empirical analysis first has been provided for demonstrating over-smoothing exist in BERT. Then, the authors theoretically prove the observation through the comparison between self-attention and graph. The authors claim that "layer normalization plays a key role in the over-smoothing issue, namely, if the standard deviation of layer normalization is sufficiently large, the output of Transformer stacks will converge to a specific low-rank subspace and results in over-smoothing". To alleviate the over-smoothing issue, layer fusion methods (which have been widely explored in previous studies) have been presented and verified to be effective in downstream tasks. However, the over-smoothing problem is neither well theoretically analyzed nor empirically proved to be addressed via layer fusion methods. | SP:9b0cc58c52854f857f0f7b4b3eaa375b3dfa98e1 |
Revisiting Over-smoothing in BERT from the Perspective of Graph | 1 INTRODUCTION . Over the past few years , Transformer ( Vaswani et al. , 2017 ) has been widely used in various natural language processing ( NLP ) tasks , including text classification ( Wang et al. , 2018a ) , text translation ( Ott et al. , 2018 ) , question answering ( Rajpurkar et al. , 2016 ; 2018 ) and text generation ( Brown et al. , 2020 ) . The recent application of Transformer in computer vision ( CV ) field also demonstrate the potential capacity of Transformer architecture . For instance , Transformer variants have been successfully used for image classification ( Dosovitskiy et al. , 2021 ) , object detection ( Carion et al. , 2020 ) and semantic segmentation ( Strudel et al. , 2021 ) . Three fundamental descendants from Transformer include BERT ( Devlin et al. , 2019 ) , RoBERTa ( Liu et al. , 2019 ) and ALBERT ( Lan et al. , 2020 ) , which achieve state-of-the-art performance on a wide range of NLP tasks . Recently , Dong et al . ( 2021 ) observes the “ token uniformity ” problem , which reduces the capacity of Transformer-based architectures by making all token representations identical . They claim that pure self-attention ( SAN ) modules cause token uniformity , but they do not discuss whether the token uniformity problem still exists in Transformer blocks . On the other hand , Gong et al . ( 2021 ) observe the “ over-smoothing ” problem for ViT ( Dosovitskiy et al. , 2021 ) , in that different input patches are mapped to a similar latent representation . To prevent loss of information , they introduce additional loss functions to encourage diversity and successfully improve model performance by suppressing over-smoothing . Moreover , “ overthinking ” phenomenon , indicating that shallow representations are better than deep representations , also be observed in ( Zhou et al. , 2020 ; Kaya et al. , 2019 ) . As discussed in Section 3 , this phenomenon has some inherent connection with over-smoothing . In this paper , we use “ over-smoothing ” to unify the above issues , and refer this as the phenomenon that the model performance is deteriorated because different inputs are mapped to a similar representation . As the over-smoothing problem is first studied in the graph neural network ( GNN ) literature ( Li et al. , 2018 ; Xu et al. , 2018 ; Zhao & Akoglu , 2020 ) , in this paper , we attempt to explore the cause of such ∗Equal contribution . problem by building a relationship between Transformer blocks and graphs . Specifically , we consider the self-attention matrix as the normalized adjacency matrix of a weighted graph , whose nodes are the tokens in a sentence . Furthermore , we consider the inherent connection between BERT and graph convolutional networks ( Kipf & Welling , 2017 ) . Inspired by the over-smoothing problem in GNN , we study over-smoothing in BERT from a theoretical view via matrix projection . As opposed to Dong et al . ( 2021 ) , where the authors claim that layer normalization is irrelevant to over-smoothing , we find that layer normalization ( Ba et al. , 2016 ) plays an important role in over-smoothing . Specifically , we theoretically prove that , if the standard deviation in layer normalization is sufficiently large , the outputs of the Transformer stacks will converge to a low-rank subspace , resulting in over-smoothing . Empirically , we verify that the conditions hold for a certain number of samples for a pre-trained and fine-tuned BERT model ( Devlin et al. , 2019 ) , which is consistent with our above observations . To alleviate the over-smoothing problem , we propose a hierarchical fusion strategy that adaptively fuses representations from different layers . Three fusion approaches are used : ( i ) Concat Fusion , ( ii ) Max Fusion , and ( iii ) Gate Fusion . The proposed method reduces the similarity between tokens and outperforms BERT baseline on the GLUE ( Wang et al. , 2018a ) , SWAG ( Zellers et al. , 2018 ) and SQuAD ( Rajpurkar et al. , 2016 ; 2018 ) data sets . In summary , the contributions of this paper are as follows : ( i ) We develop the relationship between self-attention and graph for a better understanding of over-smoothing in BERT . ( ii ) We provide theoretical analysis on over-smoothing in the BERT model , and empirically verify the theoretical results . ( iii ) We propose hierarchical fusion strategies that adaptively combine different layers to alleviate over-smoothing . Extensive experimental results verify our methods ’ effectiveness . 2 RELATED WORK . 2.1 TRANSFORMER BLOCK AND SELF-ATTENTION . Transformer block is a basic component in Transformer model ( Vaswani et al. , 2017 ) . Each Transformer block consists of a self-attention layer and a feed-forward layer . Let X ∈ Rn×d be the input to a Transformer block , where n is the number of input tokens and d is the embedding size . The self-attention layer output can be written as : Attn ( X ) = X + h∑ k=1 σ ( XWQk ( XW K k ) > ) XW Vk W O > k = X + h∑ k=1 ÂkXW V O k , ( 1 ) where h is the number of heads , σ is the softmax function , and WQk , W K k , W V k , W O k ∈ Rd×dh ( where dh = d/h is the dimension of a single-head output ) are weight matrices for the query , key , value , and output , respectively of the kth head . In particular , the self-attention matrix  = σ ( XWQ ( XWK ) > ) = σ ( QK > ) ( 2 ) in ( 1 ) plays a key role in the self-attention layer ( Park et al. , 2019 ; Gong et al. , 2019 ; Kovaleva et al. , 2019 ) . As in ( Yun et al. , 2020 ; Shi et al. , 2021 ; Dong et al. , 2021 ) , we drop the scale product 1/ √ dh to simplify analysis . The feed-forward layer usually has two fully-connected ( FC ) layers with residual connection : FF ( X ) = Attn ( X ) +ReLU ( Attn ( X ) W1 + b1 ) W2 + b2 , where W1 ∈ Rd×dff , W2 ∈ Rdff×d ( dff is the size of the intermediate layer ) are the weight matrices , and b1 , b2 are the biases . Two layer normalization ( Ba et al. , 2016 ) operations are performed after the self-attention layer and fully-connected layer , respectively . 2.2 OVER-SMOOTHING . In graph neural networks , over-smoothing refers to the problem that the performance deteriorates as representations of all the nodes become similar ( Li et al. , 2018 ; Xu et al. , 2018 ; Huang et al. , 2020 ) . Its main cause is the stacked aggregation layer using the same adjacency matrix . Recently , several approaches have been proposed to alleviate the over-smoothing problem . Xu et al . ( 2018 ) propose a jumping knowledge network for better structure-aware representation , which flexibly leverages different neighborhood ranges . ResGCN ( Li et al. , 2019 ) adapts the residual connection and dilated convolution in the graph convolutional network ( GCN ) , and successfully scales the GCN to 56 layers . Zhao & Akoglu ( 2020 ) propose PairNorm , a novel normalization layer , that prevents node embeddings from becoming too similar . DropEdge ( Rong et al. , 2020 ; Huang et al. , 2020 ) randomly removes edges from the input graph at each training epoch , and reduces the effect of over-smoothing . Unlike graph neural networks , over-smoothing in Transformer-based architectures has not been discussed in detail . Dong et al . ( 2021 ) introduce the “ token-uniformity ” problem for self-attention , and show that skip connections and multi-layer perceptron can mitigate this problem . However , Gong et al . ( 2021 ) still observe over-smoothing on the Vision Transformers ( Dosovitskiy et al. , 2021 ) . 3 DOES OVER-SMOOTHING EXIST IN BERT ? . In this section , we first explore the existence of over-smoothing in BERT , by measuring the similarity between tokens in each Transformer layer . Specifically , we use the token-wise cosine similarity ( Gong et al. , 2021 ) as our similarity measure : CosSim = 1 n ( n− 1 ) ∑ i 6=j h > i hj ‖hi‖2‖hj‖2 , where n is the number of tokens , hi and hj are two representations of different tokens , and ‖ · ‖2 is the Euclidean norm . Following Dong et al . ( 2021 ) , we use WikiBio ( Lebret et al. , 2016 ) as input to the following Transformer-based models fine-tuned on the SQuAD data set ( Rajpurkar et al. , 2018 ) : ( i ) BERT ( Devlin et al. , 2019 ) , ( ii ) RoBERTa ( Liu et al. , 2019 ) and ( iii ) ALBERT ( Lan et al. , 2020 ) .1 For comparison , all three models are stacked with 12 blocks . We calculate each CosSim for each data sample and show the average and standard derivation of CosSim values over all WikiBio data . In the figures , layer 0 represents original input token representation , and layer 1-12 represents the corresponding transformer layers . As shown in Figure 1 ( a ) , the original token representations are different from each other , while token similarities are high in the last layer . For instance , the average token-wise cosine similarity of the last layer of ALBERT and RoBERTa are both larger than 90 % . To illustrate the relationship between “ over-thinking ” and “ over-smoothing ” , we compare the tokenwise cosine similarity at each layer with the corresponding error rate . As for the corresponding error rate of layer i , we use the representations from layer i as the final output and fine-tune the classifier . Following Zhou et al . ( 2020 ) , we experiment with ALBERT ( Lan et al. , 2020 ) fine-tuned on the MRPC data set ( Dolan & Brockett , 2005 ) and use their error rate results for convenience . As shown in Figure 1 ( b ) , layer 10 has the lowest cosine similarity and error rate . At layers 11 and 12 , the tokens have larger cosine similarities , making them harder to distinguish and resulting in the performance drop . Thus , “ over-thinking ” can be explained by “ over-smoothing ” . A direct consequence of over-smoothing is that the performance can not be improved when the model gets deeper , since the individual tokens are no longer distinguishable . To illustrate this , we increase the number of layers in BERT to 24 while keeping the other settings . As shown in Figure 1 ( c ) , the 1Our implementation is based on the HuggingFace ’ s Transformers library ( Wolf et al. , 2020 ) . performance of vanilla BERT can not improve as the model gets deeper . In contrast , the proposed hierarchical fusion ( as will be discussed in Section 6 ) consistently outperforms the baseline , and has better and better performance as the model gets deeper . Based on these observations , we conclude that the over-smoothing problem still exists in BERT . 4 RELATIONSHIP BETWEEN SELF-ATTENTION AND GRAPH . Since over-smoothing is first discussed in the graph neural network literature ( Li et al. , 2018 ; Zhao & Akoglu , 2020 ) , we attempt to understand its cause from a graph perspective in this section . | The current manuscript explored the token over-smoothing problem in Bert transformer blocks. Empirically, the authors showed that the token over-smooth problem is increasingly severe in deeper layers of a Bert model. The token over-smoothing is potentially harmful to task performance. As the over-smoothing problems have been discussed in graph networks extensively, the authors established a connection between transformer blocks with graph networks and looked at the over-smoothing problem of transformer blocks analogously. The authors revealed that layer normalization in the transformer blocks contributed to the over-smoothing issue both theoretically and empirically. Moreover, the authors proposed to improve the over-smoothing issue by hierarchical fusion strategies, that is utilize token representation from earlier layers. Overall, the author applied a novel approach to shed light on an insufficiently investigated problem. The paper was well written. | SP:9b0cc58c52854f857f0f7b4b3eaa375b3dfa98e1 |
Revisiting Over-smoothing in BERT from the Perspective of Graph | 1 INTRODUCTION . Over the past few years , Transformer ( Vaswani et al. , 2017 ) has been widely used in various natural language processing ( NLP ) tasks , including text classification ( Wang et al. , 2018a ) , text translation ( Ott et al. , 2018 ) , question answering ( Rajpurkar et al. , 2016 ; 2018 ) and text generation ( Brown et al. , 2020 ) . The recent application of Transformer in computer vision ( CV ) field also demonstrate the potential capacity of Transformer architecture . For instance , Transformer variants have been successfully used for image classification ( Dosovitskiy et al. , 2021 ) , object detection ( Carion et al. , 2020 ) and semantic segmentation ( Strudel et al. , 2021 ) . Three fundamental descendants from Transformer include BERT ( Devlin et al. , 2019 ) , RoBERTa ( Liu et al. , 2019 ) and ALBERT ( Lan et al. , 2020 ) , which achieve state-of-the-art performance on a wide range of NLP tasks . Recently , Dong et al . ( 2021 ) observes the “ token uniformity ” problem , which reduces the capacity of Transformer-based architectures by making all token representations identical . They claim that pure self-attention ( SAN ) modules cause token uniformity , but they do not discuss whether the token uniformity problem still exists in Transformer blocks . On the other hand , Gong et al . ( 2021 ) observe the “ over-smoothing ” problem for ViT ( Dosovitskiy et al. , 2021 ) , in that different input patches are mapped to a similar latent representation . To prevent loss of information , they introduce additional loss functions to encourage diversity and successfully improve model performance by suppressing over-smoothing . Moreover , “ overthinking ” phenomenon , indicating that shallow representations are better than deep representations , also be observed in ( Zhou et al. , 2020 ; Kaya et al. , 2019 ) . As discussed in Section 3 , this phenomenon has some inherent connection with over-smoothing . In this paper , we use “ over-smoothing ” to unify the above issues , and refer this as the phenomenon that the model performance is deteriorated because different inputs are mapped to a similar representation . As the over-smoothing problem is first studied in the graph neural network ( GNN ) literature ( Li et al. , 2018 ; Xu et al. , 2018 ; Zhao & Akoglu , 2020 ) , in this paper , we attempt to explore the cause of such ∗Equal contribution . problem by building a relationship between Transformer blocks and graphs . Specifically , we consider the self-attention matrix as the normalized adjacency matrix of a weighted graph , whose nodes are the tokens in a sentence . Furthermore , we consider the inherent connection between BERT and graph convolutional networks ( Kipf & Welling , 2017 ) . Inspired by the over-smoothing problem in GNN , we study over-smoothing in BERT from a theoretical view via matrix projection . As opposed to Dong et al . ( 2021 ) , where the authors claim that layer normalization is irrelevant to over-smoothing , we find that layer normalization ( Ba et al. , 2016 ) plays an important role in over-smoothing . Specifically , we theoretically prove that , if the standard deviation in layer normalization is sufficiently large , the outputs of the Transformer stacks will converge to a low-rank subspace , resulting in over-smoothing . Empirically , we verify that the conditions hold for a certain number of samples for a pre-trained and fine-tuned BERT model ( Devlin et al. , 2019 ) , which is consistent with our above observations . To alleviate the over-smoothing problem , we propose a hierarchical fusion strategy that adaptively fuses representations from different layers . Three fusion approaches are used : ( i ) Concat Fusion , ( ii ) Max Fusion , and ( iii ) Gate Fusion . The proposed method reduces the similarity between tokens and outperforms BERT baseline on the GLUE ( Wang et al. , 2018a ) , SWAG ( Zellers et al. , 2018 ) and SQuAD ( Rajpurkar et al. , 2016 ; 2018 ) data sets . In summary , the contributions of this paper are as follows : ( i ) We develop the relationship between self-attention and graph for a better understanding of over-smoothing in BERT . ( ii ) We provide theoretical analysis on over-smoothing in the BERT model , and empirically verify the theoretical results . ( iii ) We propose hierarchical fusion strategies that adaptively combine different layers to alleviate over-smoothing . Extensive experimental results verify our methods ’ effectiveness . 2 RELATED WORK . 2.1 TRANSFORMER BLOCK AND SELF-ATTENTION . Transformer block is a basic component in Transformer model ( Vaswani et al. , 2017 ) . Each Transformer block consists of a self-attention layer and a feed-forward layer . Let X ∈ Rn×d be the input to a Transformer block , where n is the number of input tokens and d is the embedding size . The self-attention layer output can be written as : Attn ( X ) = X + h∑ k=1 σ ( XWQk ( XW K k ) > ) XW Vk W O > k = X + h∑ k=1 ÂkXW V O k , ( 1 ) where h is the number of heads , σ is the softmax function , and WQk , W K k , W V k , W O k ∈ Rd×dh ( where dh = d/h is the dimension of a single-head output ) are weight matrices for the query , key , value , and output , respectively of the kth head . In particular , the self-attention matrix  = σ ( XWQ ( XWK ) > ) = σ ( QK > ) ( 2 ) in ( 1 ) plays a key role in the self-attention layer ( Park et al. , 2019 ; Gong et al. , 2019 ; Kovaleva et al. , 2019 ) . As in ( Yun et al. , 2020 ; Shi et al. , 2021 ; Dong et al. , 2021 ) , we drop the scale product 1/ √ dh to simplify analysis . The feed-forward layer usually has two fully-connected ( FC ) layers with residual connection : FF ( X ) = Attn ( X ) +ReLU ( Attn ( X ) W1 + b1 ) W2 + b2 , where W1 ∈ Rd×dff , W2 ∈ Rdff×d ( dff is the size of the intermediate layer ) are the weight matrices , and b1 , b2 are the biases . Two layer normalization ( Ba et al. , 2016 ) operations are performed after the self-attention layer and fully-connected layer , respectively . 2.2 OVER-SMOOTHING . In graph neural networks , over-smoothing refers to the problem that the performance deteriorates as representations of all the nodes become similar ( Li et al. , 2018 ; Xu et al. , 2018 ; Huang et al. , 2020 ) . Its main cause is the stacked aggregation layer using the same adjacency matrix . Recently , several approaches have been proposed to alleviate the over-smoothing problem . Xu et al . ( 2018 ) propose a jumping knowledge network for better structure-aware representation , which flexibly leverages different neighborhood ranges . ResGCN ( Li et al. , 2019 ) adapts the residual connection and dilated convolution in the graph convolutional network ( GCN ) , and successfully scales the GCN to 56 layers . Zhao & Akoglu ( 2020 ) propose PairNorm , a novel normalization layer , that prevents node embeddings from becoming too similar . DropEdge ( Rong et al. , 2020 ; Huang et al. , 2020 ) randomly removes edges from the input graph at each training epoch , and reduces the effect of over-smoothing . Unlike graph neural networks , over-smoothing in Transformer-based architectures has not been discussed in detail . Dong et al . ( 2021 ) introduce the “ token-uniformity ” problem for self-attention , and show that skip connections and multi-layer perceptron can mitigate this problem . However , Gong et al . ( 2021 ) still observe over-smoothing on the Vision Transformers ( Dosovitskiy et al. , 2021 ) . 3 DOES OVER-SMOOTHING EXIST IN BERT ? . In this section , we first explore the existence of over-smoothing in BERT , by measuring the similarity between tokens in each Transformer layer . Specifically , we use the token-wise cosine similarity ( Gong et al. , 2021 ) as our similarity measure : CosSim = 1 n ( n− 1 ) ∑ i 6=j h > i hj ‖hi‖2‖hj‖2 , where n is the number of tokens , hi and hj are two representations of different tokens , and ‖ · ‖2 is the Euclidean norm . Following Dong et al . ( 2021 ) , we use WikiBio ( Lebret et al. , 2016 ) as input to the following Transformer-based models fine-tuned on the SQuAD data set ( Rajpurkar et al. , 2018 ) : ( i ) BERT ( Devlin et al. , 2019 ) , ( ii ) RoBERTa ( Liu et al. , 2019 ) and ( iii ) ALBERT ( Lan et al. , 2020 ) .1 For comparison , all three models are stacked with 12 blocks . We calculate each CosSim for each data sample and show the average and standard derivation of CosSim values over all WikiBio data . In the figures , layer 0 represents original input token representation , and layer 1-12 represents the corresponding transformer layers . As shown in Figure 1 ( a ) , the original token representations are different from each other , while token similarities are high in the last layer . For instance , the average token-wise cosine similarity of the last layer of ALBERT and RoBERTa are both larger than 90 % . To illustrate the relationship between “ over-thinking ” and “ over-smoothing ” , we compare the tokenwise cosine similarity at each layer with the corresponding error rate . As for the corresponding error rate of layer i , we use the representations from layer i as the final output and fine-tune the classifier . Following Zhou et al . ( 2020 ) , we experiment with ALBERT ( Lan et al. , 2020 ) fine-tuned on the MRPC data set ( Dolan & Brockett , 2005 ) and use their error rate results for convenience . As shown in Figure 1 ( b ) , layer 10 has the lowest cosine similarity and error rate . At layers 11 and 12 , the tokens have larger cosine similarities , making them harder to distinguish and resulting in the performance drop . Thus , “ over-thinking ” can be explained by “ over-smoothing ” . A direct consequence of over-smoothing is that the performance can not be improved when the model gets deeper , since the individual tokens are no longer distinguishable . To illustrate this , we increase the number of layers in BERT to 24 while keeping the other settings . As shown in Figure 1 ( c ) , the 1Our implementation is based on the HuggingFace ’ s Transformers library ( Wolf et al. , 2020 ) . performance of vanilla BERT can not improve as the model gets deeper . In contrast , the proposed hierarchical fusion ( as will be discussed in Section 6 ) consistently outperforms the baseline , and has better and better performance as the model gets deeper . Based on these observations , we conclude that the over-smoothing problem still exists in BERT . 4 RELATIONSHIP BETWEEN SELF-ATTENTION AND GRAPH . Since over-smoothing is first discussed in the graph neural network literature ( Li et al. , 2018 ; Zhao & Akoglu , 2020 ) , we attempt to understand its cause from a graph perspective in this section . | This paper tries to understand the over-smoothing phenomenon of Transformer-based models such as BERT. The analysis is from the perspective of viewing BERT and Transformer as graph neural networks. By analogy and analysis of graph neural networks, authors show some theoretical analysis and find that layer normalization plays a key role in the over-smoothing issue. Specifically, if the standard deviation of layer normalization is sufficiently large, the output of Transformer stacks will converge to a specific low-rank subspace and results in over-smoothing. Then to alleviate the over-smoothing problem, authors consider hierarchical fusion strategies, which combine the representations from different layers adaptively to make the output more diverse. | SP:9b0cc58c52854f857f0f7b4b3eaa375b3dfa98e1 |
Heterogeneous Wasserstein Discrepancy for Incomparable Distributions | 1 INTRODUCTION . Optimal Transport-based data analysis has recently found widespread interest in machine learning community , since its significant usefulness to achieve many tasks arising from designing loss functions in supervised learning ( Frogner et al. , 2015 ) , unsupervised learning ( Arjovsky et al. , 2017 ) , text classification ( Kusner et al. , 2015 ) , domain adaptation ( Courty et al. , 2017 ) , generative models ( Arjovsky et al. , 2017 ; Salimans et al. , 2018 ) , computer vision ( Bonneel et al. , 2011 ; Solomon et al. , 2015 ) among many more applications ( Kolouri et al. , 2017 ; Peyré & Cuturi , 2019 ) . Optimal Transport ( OT ) attempts to match real-world entities through computing distances between distributions , and for that it exploits prior geometric knowledge on the base spaces in which the distributions are valued . Computing OT distance equals to finding the most cost-efficiency way to transport mass from source distribution to target distribution , and it is often referred to as the Monge-Kantorovich or Wasserstein distance ( Monge , 1781 ; Kantorovich , 1942 ; Villani , 2009 ) . Matching distributions using Wasserstein distance relies on the assumption that their base spaces must be the same , or that at least a meaningful pairwise distance between the supports of these distributions can be computed . A variant of Wasserstein distance dealing with heterogeneous distributions and overcoming the lack of intrinsic correspondence between their base spaces is Gromov-Wasserstein ( GW ) distance ( Sturm , 2006 ; Mémoli , 2011 ) . GW distance allows to learn an optimal transport-like plan by measuring how the similarity distances between pairs of supports within each ground space are closed . It is increasingly finding applications for learning problems in shape matching ( Mémoli , 2011 ) , graph partitioning and matching ( Xu et al. , 2019 ) , matching of vocabulary sets between different languages ( Alvarez-Melis & Jaakkola , 2018 ) , generative models ( Bunne et al. , 2019 ) , or matching weighted networks ( Chowdhury & Mémoli , 2018 ) . Due to the heterogeneity of the distributions , GW distance uses only the relational aspects in each domain , such as the pairwise relationships to compare the two distributions . As a consequence , the main disadvantage of GW distance is its computational cost as the associated optimization problem is a non-convex quadratic program ( Peyré & Cuturi , 2019 ) , and as few as thousand samples can be computationally challenging . Based on the approach of regularized OT ( Cuturi , 2013 ) , in which an entropic penalty is added to the original objective function defining the Wasserstein OT problem , Peyré et al . ( 2016 ) propose an entropic version called entropic GW discrepancy , that leads to approximate GW distance . Another approach for scaling up the GW distance is Sliced Gromov-Wasserstein ( SGW ) discrepancy ( Vayer et al. , 2019 ) , which leverages on random projections on 1D and on a closed-form solution of the 1D-Gromov-Wasserstein . In this paper , we take a different approach for measuring the discrepancy between two heterogeneous distributions . Unlike GW distance that compares pairwise distances of elements from each distribution , we consider a method that embeds the metric measure spaces into a one-dimensional space and computes a Wasserstein distance between the two 1D-projected distributions . The key element of our approach is to learn two mappings that transform vectors from the unit-sphere of a latent space to the unit-sphere of the metric space underlying the two distributions of interest , see Figure 1 . In a nutshell , we learn to transform a random direction , sampled under an optimal ( learned ) distribution ( optimality being made clear later ) , from a d-dimensional space to a random direction into the desired spaces . This approach has the benefit of avoiding an ad-hoc padding strategy ( completion of 0 of the smaller dimension distributions to fit the high-dimensional one ) as in SGW method ( Vayer et al. , 2019 ) . Another relevant feature of our approach is that the two resulting 1D distributions are now compared through Wasserstein distance . This point , in conjunction , with other key aspect of the method , will lead to a relevant discrepancy between two distributions , called heterogeneous Wasserstein discrepancy ( HWD ) . Although we lose some properties of a distance , we show that HWD is rotation-invariant , that it is robust enough to be considered as a loss for learning generative models between heterogeneous spaces . We also establish that HWD boils down to the recent distributional sliced Wasserstein distance ( Nguyen et al. , 2020 ) if the two distributions live in the same space and if some mild constraints are imposed on the mappings . In summary , our contributions are as follows : • we propose HWD , a novel slicing-based discrepancy for comparing two distributions living in different spaces . Our chosen formulation is based on comparing 1D random-projected versions of the two distributions using a Wasserstein distance ; • The projection operations are materialized by optimally mapping from one common space to the two spaces of interest . We provide a theoretical analysis of the resulting discrepancy and exhibit its relevant properties ; • Since the discrepancy involves several mappings that need to be optimized , we depict an alternate optimization algorithm for learning them ; • Numerically , we validate the benefits of HWD in terms of comparison between heterogeneous distributions . We show that it can be used as a loss for generative models or shape objects retrieval with better performance and robustness than SGW on those tasks . 2 BACKGROUND OF OT DISTANCES . For the reader ’ s convenience , we provide here a brief review of the notations and definitions , that will be frequently used throughout the paper . We start by introducing Wasserstein and Gromov-Wasserstein distances with their sliced versions SW and SGW , where we consider these distances in the specific case of Euclidean base spaces ( Rp , ‖ · ‖ ) and ( Rq , ‖ · ‖ ) . We denote P ( X ) and P ( Y ) the respective sets of probability measures whose supports are contained on compact sets X ⊆ Rp and Y ⊆ Rq . For r ≥ 1 , we denote Pr ( X ) the subset of measures in P ( X ) with finite r-th moment ( r ≥ 1 ) , i.e. , Pr ( X ) = { η ∈ P ( X ) : ´ X ‖x‖ rdη ( x ) < ∞ } . For µ ∈ P ( X ) and ν ∈ P ( Y ) , we write Π ( µ , ν ) ⊂ P ( X × Y ) for the collection of joint probability distributions with marginals µ and ν , known as couplings , Π ( µ , ν ) = { γ ∈P ( X × Y ) : ∀A ⊂ X , B ⊂ Y , γ ( A× Y ) = µ ( A ) , γ ( X ×B ) = ν ( B ) } . 2.1 OT DISTANCES FOR HOMOGENEOUS DOMAINS . We here assume that the distributions µ and ν lie in the same base space , for instance p = q . Taking this into account , we can define the Wasserstein distance and its sliced variant . Wasserstein distance The r-th Wasserstein distance is defined on Pr ( X ) by Wr ( µ , ν ) = ( inf γ∈Π ( µ , ν ) ˆ X×Y ‖x− y‖rdγ ( x , y ) ) 1 r . ( 1 ) The quantityWr ( µ , ν ) describes the least amount effort to transform one distribution µ into another one ν . Since the cost distance used between sample supports is the Euclidean one , the infimum in ( 1 ) is attained ( Villani , 2009 ) , and any probability γ which realizes the minimum is called an optimal transport plan . In a finite discrete setting , Problem ( 1 ) can be formulated as a linear program , that is challenging to solve algorithmically as its computational cost is of order O ( n5/2 log n ) ( Lee & Sidford , 2014 ) , where n is the number of sample supports . Contrastingly , for the 1D case ( i.e . p = 1 ) of continuous probability measures , the r-th Wasserstein distance has a closed-form solution ( Rachev & Rüschendorf , 1998 ) , namely , Wr ( µ , ν ) = ( ´ 1 0 |F−1µ ( u ) − F−1ν ( u ) |rdt ) 1 r where F−1µ and F −1 ν are the quantile functions of µ and ν . For empirical distributions , the 1D-Wasserstein distance is simply calculated by sorting the supports of the distributions on the real line , resulting to a complexity of order O ( n log n ) . This nice computational property motivates the use of sliced-Wasserstein ( SW ) distance ( Rabin et al. , 2012 ; Bonneel et al. , 2015 ) , where one calculates an ( infinity ) of 1D-Wasserstein distances between linear projection pushforwards of distributions in question and then computes their average . To precisely define SW distance , we consider the following notation . Let Sp−1 : = { u ∈ Rp : ‖u‖ = 1 } be the unit sphere in p dimension in ` 2-norm , and for any vector θ in Sp−1 , we define Pθ the orthogonal projection onto the real line Rθ = { αθ : α ∈ R } , that is Pθ ( x ) = 〈θ , x〉 , where 〈· , ·〉 stands for the Euclidean inner-product . Let µθ = Pθ # µ the measure on the real line called pushforward of µ by Pθ , that is µθ ( A ) = µ ( P−1θ ( A ) ) for all Borel set A ⊆ R. We may now define the SW distance . Sliced Wasserstein distance The r-th order sliced Wasserstein distance between two probability distributions µ , ν ∈Pr ( X ) is given by SWr ( µ , ν ) = ( 1 Ap ˆ Sp−1 Wrr ( µθ , νθ ) dθ ) 1 r , ( 2 ) where Ap is the area of the surface of Sp−1 , i.e. , Ap = 2π p/2 Γ ( p/2 ) with Γ : R → R , the Gamma function given as Γ ( u ) = ´∞ 0 tu−1e−tdt . Thanks to its computational benefits and its valid metric property ( Bonnotte , 2013 ) , the SW distance has recently been used for OT-based deep generative modeling ( Kolouri et al. , 2019 ; Deshpande et al. , 2019 ; Wu et al. , 2019 ) . Note that the normalized integral in ( 2 ) can be seen as the expectation for θ ∼ σp−1 , the uniform surface measure on Sp−1 , that is SWr ( µ , ν ) = ( Eθ∼σp−1 [ Wrr ( µθ , νθ ) ] ) 1 r . Therefore , the SW distance can be easily approximated via a Monte Carlo sampling scheme by drawing uniform random samples from Sp−1 : SWrr ( µ , ν ) ≈ 1K ∑K k=1Wrr ( µθk , νθk ) where θ1 , . . . , θK i.i.d.∼ σp−1 and K is the number of random projections . 2.2 OT DISTANCES FOR HETEROGENEOUS DOMAINS . To get benefit from the advantages of OT in many machine learning applications involving heterogeneous and incomparable domains ( p 6= q ) , the Gromov-Wasserstein distance ( Mémoli , 2011 ) stands for the basic OT distance dealing with this setting . Gromov-Wasserstein distance The r-th Gromov-Wasserstein distance between two probability distributions µ ∈Pr ( X ) and ν ∈Pr ( Y ) is defined by GWr ( µ , ν ) = inf γ∈Π ( µ , ν ) Jr ( γ ) def . = 1 2 ( ¨ X 2×Y2 |‖x− x′‖ − ‖y − y′‖|rdγ ( x , y ) dγ ( x′ , y′ ) ) 1 r . ( 3 ) Note that GWr ( µ , ν ) is a valid metric endowing the collection of all isomorphism classes metric measure spaces of Pr ( X ) ×Pr ( Y ) , see Theorem 5 in ( Mémoli , 2011 ) . The GW distance learns an optimal transport-like plan which transports samples from a source metric space X into a target metric space Y , by measuring how the similarity distances between pairs of samples within each space are close . Furthermore , GW distance enjoys several geometric properties , particularly translation and rotation invariance . However , its major bottleneck consists in an expensive computational cost , since problem ( 3 ) is non-convex and quadratic . A remedy to such a heavy computational burden lies in an entropic regularized GW discrepancy ( Peyré et al. , 2016 ) , using Sinkhorn iterations algorithm ( Cuturi , 2013 ) . This latter needs a large regularization parameter to guarantee a fast computation , which , unfortunately , entails a poor approximation of the true GW distance value . Another approach to scale up the computation of GW distance is sliced-GW discrepancy ( Vayer et al. , 2019 ) . The definition of SGW shows 1D-GW distances between projected pushforward of an artifact zero padding of µ or ν distribution . We detail this representation in the following paragraph . Sliced Gromov-Wasserstein discrepancy Assume that p < q and let ∆ be an artifact zero padding from X onto Y , i.e . ∆ ( x ) = ( x1 , . . . , xp , 0 , . . . , 0 ) ∈ Rq . The r-th order sliced Gromov-Wasserstein discrepancy between two probability distributions µ ∈Pr ( X ) and ν ∈Pr ( Y ) is given by SGW∆ , r ( µ , ν ) = ( Eθ∼σq−1 [ GWrr ( ( ∆ # µ ) θ , νθ ) ] ) 1r . ( 4 ) It is worthy to note that SGW∆ , r is depending on the ad-hoc operator ∆ , hence the rotation invariance is lost . Vayer et al . ( 2019 ) propose a variant of SGW that does not depend on the choice of ∆ , called Rotation Invariant SGW ( RI-SGW ) for p = q , defined as the minimizer of SGW∆ , r over the Stiefel manifold , see ( Vayer et al. , 2019 , Equation 6 ) . In this work , we are interested in calculating an OT-based discrepancy between distributions over distinct domains using the slicing technique . Our approach is different from the SGW one in many points , specifically ( and most importantly ) we use a 1D-Wasserstein distance between the projected pushforward distributions and not a 1D-GW distance . In the next section , we detail the setup of our approach . | This paper introduces the so-called Heterogeneous Wasserstein Discrepancy (HWD) between two probability measures supported on Euclidean spaces of different dimensions, a case scenario appearing in several applications. The HWD is built on ideas from the sliced Wasserstein distance and measure embedding : the two distributions are projected into a one-dimensional space, in which the classical Wasserstein distance (closed-form in 1D) is computed. Moreover, an algorithm to compute such a divergence is presented, and its usefulness is illustrated on synthetic examples, validated on real-world data and compared to standard others methods. | SP:8cf31d348eb50e20982d7eea5645978f0108eb38 |
Heterogeneous Wasserstein Discrepancy for Incomparable Distributions | 1 INTRODUCTION . Optimal Transport-based data analysis has recently found widespread interest in machine learning community , since its significant usefulness to achieve many tasks arising from designing loss functions in supervised learning ( Frogner et al. , 2015 ) , unsupervised learning ( Arjovsky et al. , 2017 ) , text classification ( Kusner et al. , 2015 ) , domain adaptation ( Courty et al. , 2017 ) , generative models ( Arjovsky et al. , 2017 ; Salimans et al. , 2018 ) , computer vision ( Bonneel et al. , 2011 ; Solomon et al. , 2015 ) among many more applications ( Kolouri et al. , 2017 ; Peyré & Cuturi , 2019 ) . Optimal Transport ( OT ) attempts to match real-world entities through computing distances between distributions , and for that it exploits prior geometric knowledge on the base spaces in which the distributions are valued . Computing OT distance equals to finding the most cost-efficiency way to transport mass from source distribution to target distribution , and it is often referred to as the Monge-Kantorovich or Wasserstein distance ( Monge , 1781 ; Kantorovich , 1942 ; Villani , 2009 ) . Matching distributions using Wasserstein distance relies on the assumption that their base spaces must be the same , or that at least a meaningful pairwise distance between the supports of these distributions can be computed . A variant of Wasserstein distance dealing with heterogeneous distributions and overcoming the lack of intrinsic correspondence between their base spaces is Gromov-Wasserstein ( GW ) distance ( Sturm , 2006 ; Mémoli , 2011 ) . GW distance allows to learn an optimal transport-like plan by measuring how the similarity distances between pairs of supports within each ground space are closed . It is increasingly finding applications for learning problems in shape matching ( Mémoli , 2011 ) , graph partitioning and matching ( Xu et al. , 2019 ) , matching of vocabulary sets between different languages ( Alvarez-Melis & Jaakkola , 2018 ) , generative models ( Bunne et al. , 2019 ) , or matching weighted networks ( Chowdhury & Mémoli , 2018 ) . Due to the heterogeneity of the distributions , GW distance uses only the relational aspects in each domain , such as the pairwise relationships to compare the two distributions . As a consequence , the main disadvantage of GW distance is its computational cost as the associated optimization problem is a non-convex quadratic program ( Peyré & Cuturi , 2019 ) , and as few as thousand samples can be computationally challenging . Based on the approach of regularized OT ( Cuturi , 2013 ) , in which an entropic penalty is added to the original objective function defining the Wasserstein OT problem , Peyré et al . ( 2016 ) propose an entropic version called entropic GW discrepancy , that leads to approximate GW distance . Another approach for scaling up the GW distance is Sliced Gromov-Wasserstein ( SGW ) discrepancy ( Vayer et al. , 2019 ) , which leverages on random projections on 1D and on a closed-form solution of the 1D-Gromov-Wasserstein . In this paper , we take a different approach for measuring the discrepancy between two heterogeneous distributions . Unlike GW distance that compares pairwise distances of elements from each distribution , we consider a method that embeds the metric measure spaces into a one-dimensional space and computes a Wasserstein distance between the two 1D-projected distributions . The key element of our approach is to learn two mappings that transform vectors from the unit-sphere of a latent space to the unit-sphere of the metric space underlying the two distributions of interest , see Figure 1 . In a nutshell , we learn to transform a random direction , sampled under an optimal ( learned ) distribution ( optimality being made clear later ) , from a d-dimensional space to a random direction into the desired spaces . This approach has the benefit of avoiding an ad-hoc padding strategy ( completion of 0 of the smaller dimension distributions to fit the high-dimensional one ) as in SGW method ( Vayer et al. , 2019 ) . Another relevant feature of our approach is that the two resulting 1D distributions are now compared through Wasserstein distance . This point , in conjunction , with other key aspect of the method , will lead to a relevant discrepancy between two distributions , called heterogeneous Wasserstein discrepancy ( HWD ) . Although we lose some properties of a distance , we show that HWD is rotation-invariant , that it is robust enough to be considered as a loss for learning generative models between heterogeneous spaces . We also establish that HWD boils down to the recent distributional sliced Wasserstein distance ( Nguyen et al. , 2020 ) if the two distributions live in the same space and if some mild constraints are imposed on the mappings . In summary , our contributions are as follows : • we propose HWD , a novel slicing-based discrepancy for comparing two distributions living in different spaces . Our chosen formulation is based on comparing 1D random-projected versions of the two distributions using a Wasserstein distance ; • The projection operations are materialized by optimally mapping from one common space to the two spaces of interest . We provide a theoretical analysis of the resulting discrepancy and exhibit its relevant properties ; • Since the discrepancy involves several mappings that need to be optimized , we depict an alternate optimization algorithm for learning them ; • Numerically , we validate the benefits of HWD in terms of comparison between heterogeneous distributions . We show that it can be used as a loss for generative models or shape objects retrieval with better performance and robustness than SGW on those tasks . 2 BACKGROUND OF OT DISTANCES . For the reader ’ s convenience , we provide here a brief review of the notations and definitions , that will be frequently used throughout the paper . We start by introducing Wasserstein and Gromov-Wasserstein distances with their sliced versions SW and SGW , where we consider these distances in the specific case of Euclidean base spaces ( Rp , ‖ · ‖ ) and ( Rq , ‖ · ‖ ) . We denote P ( X ) and P ( Y ) the respective sets of probability measures whose supports are contained on compact sets X ⊆ Rp and Y ⊆ Rq . For r ≥ 1 , we denote Pr ( X ) the subset of measures in P ( X ) with finite r-th moment ( r ≥ 1 ) , i.e. , Pr ( X ) = { η ∈ P ( X ) : ´ X ‖x‖ rdη ( x ) < ∞ } . For µ ∈ P ( X ) and ν ∈ P ( Y ) , we write Π ( µ , ν ) ⊂ P ( X × Y ) for the collection of joint probability distributions with marginals µ and ν , known as couplings , Π ( µ , ν ) = { γ ∈P ( X × Y ) : ∀A ⊂ X , B ⊂ Y , γ ( A× Y ) = µ ( A ) , γ ( X ×B ) = ν ( B ) } . 2.1 OT DISTANCES FOR HOMOGENEOUS DOMAINS . We here assume that the distributions µ and ν lie in the same base space , for instance p = q . Taking this into account , we can define the Wasserstein distance and its sliced variant . Wasserstein distance The r-th Wasserstein distance is defined on Pr ( X ) by Wr ( µ , ν ) = ( inf γ∈Π ( µ , ν ) ˆ X×Y ‖x− y‖rdγ ( x , y ) ) 1 r . ( 1 ) The quantityWr ( µ , ν ) describes the least amount effort to transform one distribution µ into another one ν . Since the cost distance used between sample supports is the Euclidean one , the infimum in ( 1 ) is attained ( Villani , 2009 ) , and any probability γ which realizes the minimum is called an optimal transport plan . In a finite discrete setting , Problem ( 1 ) can be formulated as a linear program , that is challenging to solve algorithmically as its computational cost is of order O ( n5/2 log n ) ( Lee & Sidford , 2014 ) , where n is the number of sample supports . Contrastingly , for the 1D case ( i.e . p = 1 ) of continuous probability measures , the r-th Wasserstein distance has a closed-form solution ( Rachev & Rüschendorf , 1998 ) , namely , Wr ( µ , ν ) = ( ´ 1 0 |F−1µ ( u ) − F−1ν ( u ) |rdt ) 1 r where F−1µ and F −1 ν are the quantile functions of µ and ν . For empirical distributions , the 1D-Wasserstein distance is simply calculated by sorting the supports of the distributions on the real line , resulting to a complexity of order O ( n log n ) . This nice computational property motivates the use of sliced-Wasserstein ( SW ) distance ( Rabin et al. , 2012 ; Bonneel et al. , 2015 ) , where one calculates an ( infinity ) of 1D-Wasserstein distances between linear projection pushforwards of distributions in question and then computes their average . To precisely define SW distance , we consider the following notation . Let Sp−1 : = { u ∈ Rp : ‖u‖ = 1 } be the unit sphere in p dimension in ` 2-norm , and for any vector θ in Sp−1 , we define Pθ the orthogonal projection onto the real line Rθ = { αθ : α ∈ R } , that is Pθ ( x ) = 〈θ , x〉 , where 〈· , ·〉 stands for the Euclidean inner-product . Let µθ = Pθ # µ the measure on the real line called pushforward of µ by Pθ , that is µθ ( A ) = µ ( P−1θ ( A ) ) for all Borel set A ⊆ R. We may now define the SW distance . Sliced Wasserstein distance The r-th order sliced Wasserstein distance between two probability distributions µ , ν ∈Pr ( X ) is given by SWr ( µ , ν ) = ( 1 Ap ˆ Sp−1 Wrr ( µθ , νθ ) dθ ) 1 r , ( 2 ) where Ap is the area of the surface of Sp−1 , i.e. , Ap = 2π p/2 Γ ( p/2 ) with Γ : R → R , the Gamma function given as Γ ( u ) = ´∞ 0 tu−1e−tdt . Thanks to its computational benefits and its valid metric property ( Bonnotte , 2013 ) , the SW distance has recently been used for OT-based deep generative modeling ( Kolouri et al. , 2019 ; Deshpande et al. , 2019 ; Wu et al. , 2019 ) . Note that the normalized integral in ( 2 ) can be seen as the expectation for θ ∼ σp−1 , the uniform surface measure on Sp−1 , that is SWr ( µ , ν ) = ( Eθ∼σp−1 [ Wrr ( µθ , νθ ) ] ) 1 r . Therefore , the SW distance can be easily approximated via a Monte Carlo sampling scheme by drawing uniform random samples from Sp−1 : SWrr ( µ , ν ) ≈ 1K ∑K k=1Wrr ( µθk , νθk ) where θ1 , . . . , θK i.i.d.∼ σp−1 and K is the number of random projections . 2.2 OT DISTANCES FOR HETEROGENEOUS DOMAINS . To get benefit from the advantages of OT in many machine learning applications involving heterogeneous and incomparable domains ( p 6= q ) , the Gromov-Wasserstein distance ( Mémoli , 2011 ) stands for the basic OT distance dealing with this setting . Gromov-Wasserstein distance The r-th Gromov-Wasserstein distance between two probability distributions µ ∈Pr ( X ) and ν ∈Pr ( Y ) is defined by GWr ( µ , ν ) = inf γ∈Π ( µ , ν ) Jr ( γ ) def . = 1 2 ( ¨ X 2×Y2 |‖x− x′‖ − ‖y − y′‖|rdγ ( x , y ) dγ ( x′ , y′ ) ) 1 r . ( 3 ) Note that GWr ( µ , ν ) is a valid metric endowing the collection of all isomorphism classes metric measure spaces of Pr ( X ) ×Pr ( Y ) , see Theorem 5 in ( Mémoli , 2011 ) . The GW distance learns an optimal transport-like plan which transports samples from a source metric space X into a target metric space Y , by measuring how the similarity distances between pairs of samples within each space are close . Furthermore , GW distance enjoys several geometric properties , particularly translation and rotation invariance . However , its major bottleneck consists in an expensive computational cost , since problem ( 3 ) is non-convex and quadratic . A remedy to such a heavy computational burden lies in an entropic regularized GW discrepancy ( Peyré et al. , 2016 ) , using Sinkhorn iterations algorithm ( Cuturi , 2013 ) . This latter needs a large regularization parameter to guarantee a fast computation , which , unfortunately , entails a poor approximation of the true GW distance value . Another approach to scale up the computation of GW distance is sliced-GW discrepancy ( Vayer et al. , 2019 ) . The definition of SGW shows 1D-GW distances between projected pushforward of an artifact zero padding of µ or ν distribution . We detail this representation in the following paragraph . Sliced Gromov-Wasserstein discrepancy Assume that p < q and let ∆ be an artifact zero padding from X onto Y , i.e . ∆ ( x ) = ( x1 , . . . , xp , 0 , . . . , 0 ) ∈ Rq . The r-th order sliced Gromov-Wasserstein discrepancy between two probability distributions µ ∈Pr ( X ) and ν ∈Pr ( Y ) is given by SGW∆ , r ( µ , ν ) = ( Eθ∼σq−1 [ GWrr ( ( ∆ # µ ) θ , νθ ) ] ) 1r . ( 4 ) It is worthy to note that SGW∆ , r is depending on the ad-hoc operator ∆ , hence the rotation invariance is lost . Vayer et al . ( 2019 ) propose a variant of SGW that does not depend on the choice of ∆ , called Rotation Invariant SGW ( RI-SGW ) for p = q , defined as the minimizer of SGW∆ , r over the Stiefel manifold , see ( Vayer et al. , 2019 , Equation 6 ) . In this work , we are interested in calculating an OT-based discrepancy between distributions over distinct domains using the slicing technique . Our approach is different from the SGW one in many points , specifically ( and most importantly ) we use a 1D-Wasserstein distance between the projected pushforward distributions and not a 1D-GW distance . In the next section , we detail the setup of our approach . | The authors propose a variant of Wasserstein for distributions supported on different Euclidean spaces (e.g., different dimensions). The main ideas are to combine the embedding (for distributions supported in different spaces -- Alaya et al. 2020) and distributional sliced Wasserstein (Nguyen et al., 2020) to generate random slicing projections into 1d-space, and leverage the 1d-Wasserstein for the comparison. The authors apply the proposed method for generating modeling and in query framework. | SP:8cf31d348eb50e20982d7eea5645978f0108eb38 |
Heterogeneous Wasserstein Discrepancy for Incomparable Distributions | 1 INTRODUCTION . Optimal Transport-based data analysis has recently found widespread interest in machine learning community , since its significant usefulness to achieve many tasks arising from designing loss functions in supervised learning ( Frogner et al. , 2015 ) , unsupervised learning ( Arjovsky et al. , 2017 ) , text classification ( Kusner et al. , 2015 ) , domain adaptation ( Courty et al. , 2017 ) , generative models ( Arjovsky et al. , 2017 ; Salimans et al. , 2018 ) , computer vision ( Bonneel et al. , 2011 ; Solomon et al. , 2015 ) among many more applications ( Kolouri et al. , 2017 ; Peyré & Cuturi , 2019 ) . Optimal Transport ( OT ) attempts to match real-world entities through computing distances between distributions , and for that it exploits prior geometric knowledge on the base spaces in which the distributions are valued . Computing OT distance equals to finding the most cost-efficiency way to transport mass from source distribution to target distribution , and it is often referred to as the Monge-Kantorovich or Wasserstein distance ( Monge , 1781 ; Kantorovich , 1942 ; Villani , 2009 ) . Matching distributions using Wasserstein distance relies on the assumption that their base spaces must be the same , or that at least a meaningful pairwise distance between the supports of these distributions can be computed . A variant of Wasserstein distance dealing with heterogeneous distributions and overcoming the lack of intrinsic correspondence between their base spaces is Gromov-Wasserstein ( GW ) distance ( Sturm , 2006 ; Mémoli , 2011 ) . GW distance allows to learn an optimal transport-like plan by measuring how the similarity distances between pairs of supports within each ground space are closed . It is increasingly finding applications for learning problems in shape matching ( Mémoli , 2011 ) , graph partitioning and matching ( Xu et al. , 2019 ) , matching of vocabulary sets between different languages ( Alvarez-Melis & Jaakkola , 2018 ) , generative models ( Bunne et al. , 2019 ) , or matching weighted networks ( Chowdhury & Mémoli , 2018 ) . Due to the heterogeneity of the distributions , GW distance uses only the relational aspects in each domain , such as the pairwise relationships to compare the two distributions . As a consequence , the main disadvantage of GW distance is its computational cost as the associated optimization problem is a non-convex quadratic program ( Peyré & Cuturi , 2019 ) , and as few as thousand samples can be computationally challenging . Based on the approach of regularized OT ( Cuturi , 2013 ) , in which an entropic penalty is added to the original objective function defining the Wasserstein OT problem , Peyré et al . ( 2016 ) propose an entropic version called entropic GW discrepancy , that leads to approximate GW distance . Another approach for scaling up the GW distance is Sliced Gromov-Wasserstein ( SGW ) discrepancy ( Vayer et al. , 2019 ) , which leverages on random projections on 1D and on a closed-form solution of the 1D-Gromov-Wasserstein . In this paper , we take a different approach for measuring the discrepancy between two heterogeneous distributions . Unlike GW distance that compares pairwise distances of elements from each distribution , we consider a method that embeds the metric measure spaces into a one-dimensional space and computes a Wasserstein distance between the two 1D-projected distributions . The key element of our approach is to learn two mappings that transform vectors from the unit-sphere of a latent space to the unit-sphere of the metric space underlying the two distributions of interest , see Figure 1 . In a nutshell , we learn to transform a random direction , sampled under an optimal ( learned ) distribution ( optimality being made clear later ) , from a d-dimensional space to a random direction into the desired spaces . This approach has the benefit of avoiding an ad-hoc padding strategy ( completion of 0 of the smaller dimension distributions to fit the high-dimensional one ) as in SGW method ( Vayer et al. , 2019 ) . Another relevant feature of our approach is that the two resulting 1D distributions are now compared through Wasserstein distance . This point , in conjunction , with other key aspect of the method , will lead to a relevant discrepancy between two distributions , called heterogeneous Wasserstein discrepancy ( HWD ) . Although we lose some properties of a distance , we show that HWD is rotation-invariant , that it is robust enough to be considered as a loss for learning generative models between heterogeneous spaces . We also establish that HWD boils down to the recent distributional sliced Wasserstein distance ( Nguyen et al. , 2020 ) if the two distributions live in the same space and if some mild constraints are imposed on the mappings . In summary , our contributions are as follows : • we propose HWD , a novel slicing-based discrepancy for comparing two distributions living in different spaces . Our chosen formulation is based on comparing 1D random-projected versions of the two distributions using a Wasserstein distance ; • The projection operations are materialized by optimally mapping from one common space to the two spaces of interest . We provide a theoretical analysis of the resulting discrepancy and exhibit its relevant properties ; • Since the discrepancy involves several mappings that need to be optimized , we depict an alternate optimization algorithm for learning them ; • Numerically , we validate the benefits of HWD in terms of comparison between heterogeneous distributions . We show that it can be used as a loss for generative models or shape objects retrieval with better performance and robustness than SGW on those tasks . 2 BACKGROUND OF OT DISTANCES . For the reader ’ s convenience , we provide here a brief review of the notations and definitions , that will be frequently used throughout the paper . We start by introducing Wasserstein and Gromov-Wasserstein distances with their sliced versions SW and SGW , where we consider these distances in the specific case of Euclidean base spaces ( Rp , ‖ · ‖ ) and ( Rq , ‖ · ‖ ) . We denote P ( X ) and P ( Y ) the respective sets of probability measures whose supports are contained on compact sets X ⊆ Rp and Y ⊆ Rq . For r ≥ 1 , we denote Pr ( X ) the subset of measures in P ( X ) with finite r-th moment ( r ≥ 1 ) , i.e. , Pr ( X ) = { η ∈ P ( X ) : ´ X ‖x‖ rdη ( x ) < ∞ } . For µ ∈ P ( X ) and ν ∈ P ( Y ) , we write Π ( µ , ν ) ⊂ P ( X × Y ) for the collection of joint probability distributions with marginals µ and ν , known as couplings , Π ( µ , ν ) = { γ ∈P ( X × Y ) : ∀A ⊂ X , B ⊂ Y , γ ( A× Y ) = µ ( A ) , γ ( X ×B ) = ν ( B ) } . 2.1 OT DISTANCES FOR HOMOGENEOUS DOMAINS . We here assume that the distributions µ and ν lie in the same base space , for instance p = q . Taking this into account , we can define the Wasserstein distance and its sliced variant . Wasserstein distance The r-th Wasserstein distance is defined on Pr ( X ) by Wr ( µ , ν ) = ( inf γ∈Π ( µ , ν ) ˆ X×Y ‖x− y‖rdγ ( x , y ) ) 1 r . ( 1 ) The quantityWr ( µ , ν ) describes the least amount effort to transform one distribution µ into another one ν . Since the cost distance used between sample supports is the Euclidean one , the infimum in ( 1 ) is attained ( Villani , 2009 ) , and any probability γ which realizes the minimum is called an optimal transport plan . In a finite discrete setting , Problem ( 1 ) can be formulated as a linear program , that is challenging to solve algorithmically as its computational cost is of order O ( n5/2 log n ) ( Lee & Sidford , 2014 ) , where n is the number of sample supports . Contrastingly , for the 1D case ( i.e . p = 1 ) of continuous probability measures , the r-th Wasserstein distance has a closed-form solution ( Rachev & Rüschendorf , 1998 ) , namely , Wr ( µ , ν ) = ( ´ 1 0 |F−1µ ( u ) − F−1ν ( u ) |rdt ) 1 r where F−1µ and F −1 ν are the quantile functions of µ and ν . For empirical distributions , the 1D-Wasserstein distance is simply calculated by sorting the supports of the distributions on the real line , resulting to a complexity of order O ( n log n ) . This nice computational property motivates the use of sliced-Wasserstein ( SW ) distance ( Rabin et al. , 2012 ; Bonneel et al. , 2015 ) , where one calculates an ( infinity ) of 1D-Wasserstein distances between linear projection pushforwards of distributions in question and then computes their average . To precisely define SW distance , we consider the following notation . Let Sp−1 : = { u ∈ Rp : ‖u‖ = 1 } be the unit sphere in p dimension in ` 2-norm , and for any vector θ in Sp−1 , we define Pθ the orthogonal projection onto the real line Rθ = { αθ : α ∈ R } , that is Pθ ( x ) = 〈θ , x〉 , where 〈· , ·〉 stands for the Euclidean inner-product . Let µθ = Pθ # µ the measure on the real line called pushforward of µ by Pθ , that is µθ ( A ) = µ ( P−1θ ( A ) ) for all Borel set A ⊆ R. We may now define the SW distance . Sliced Wasserstein distance The r-th order sliced Wasserstein distance between two probability distributions µ , ν ∈Pr ( X ) is given by SWr ( µ , ν ) = ( 1 Ap ˆ Sp−1 Wrr ( µθ , νθ ) dθ ) 1 r , ( 2 ) where Ap is the area of the surface of Sp−1 , i.e. , Ap = 2π p/2 Γ ( p/2 ) with Γ : R → R , the Gamma function given as Γ ( u ) = ´∞ 0 tu−1e−tdt . Thanks to its computational benefits and its valid metric property ( Bonnotte , 2013 ) , the SW distance has recently been used for OT-based deep generative modeling ( Kolouri et al. , 2019 ; Deshpande et al. , 2019 ; Wu et al. , 2019 ) . Note that the normalized integral in ( 2 ) can be seen as the expectation for θ ∼ σp−1 , the uniform surface measure on Sp−1 , that is SWr ( µ , ν ) = ( Eθ∼σp−1 [ Wrr ( µθ , νθ ) ] ) 1 r . Therefore , the SW distance can be easily approximated via a Monte Carlo sampling scheme by drawing uniform random samples from Sp−1 : SWrr ( µ , ν ) ≈ 1K ∑K k=1Wrr ( µθk , νθk ) where θ1 , . . . , θK i.i.d.∼ σp−1 and K is the number of random projections . 2.2 OT DISTANCES FOR HETEROGENEOUS DOMAINS . To get benefit from the advantages of OT in many machine learning applications involving heterogeneous and incomparable domains ( p 6= q ) , the Gromov-Wasserstein distance ( Mémoli , 2011 ) stands for the basic OT distance dealing with this setting . Gromov-Wasserstein distance The r-th Gromov-Wasserstein distance between two probability distributions µ ∈Pr ( X ) and ν ∈Pr ( Y ) is defined by GWr ( µ , ν ) = inf γ∈Π ( µ , ν ) Jr ( γ ) def . = 1 2 ( ¨ X 2×Y2 |‖x− x′‖ − ‖y − y′‖|rdγ ( x , y ) dγ ( x′ , y′ ) ) 1 r . ( 3 ) Note that GWr ( µ , ν ) is a valid metric endowing the collection of all isomorphism classes metric measure spaces of Pr ( X ) ×Pr ( Y ) , see Theorem 5 in ( Mémoli , 2011 ) . The GW distance learns an optimal transport-like plan which transports samples from a source metric space X into a target metric space Y , by measuring how the similarity distances between pairs of samples within each space are close . Furthermore , GW distance enjoys several geometric properties , particularly translation and rotation invariance . However , its major bottleneck consists in an expensive computational cost , since problem ( 3 ) is non-convex and quadratic . A remedy to such a heavy computational burden lies in an entropic regularized GW discrepancy ( Peyré et al. , 2016 ) , using Sinkhorn iterations algorithm ( Cuturi , 2013 ) . This latter needs a large regularization parameter to guarantee a fast computation , which , unfortunately , entails a poor approximation of the true GW distance value . Another approach to scale up the computation of GW distance is sliced-GW discrepancy ( Vayer et al. , 2019 ) . The definition of SGW shows 1D-GW distances between projected pushforward of an artifact zero padding of µ or ν distribution . We detail this representation in the following paragraph . Sliced Gromov-Wasserstein discrepancy Assume that p < q and let ∆ be an artifact zero padding from X onto Y , i.e . ∆ ( x ) = ( x1 , . . . , xp , 0 , . . . , 0 ) ∈ Rq . The r-th order sliced Gromov-Wasserstein discrepancy between two probability distributions µ ∈Pr ( X ) and ν ∈Pr ( Y ) is given by SGW∆ , r ( µ , ν ) = ( Eθ∼σq−1 [ GWrr ( ( ∆ # µ ) θ , νθ ) ] ) 1r . ( 4 ) It is worthy to note that SGW∆ , r is depending on the ad-hoc operator ∆ , hence the rotation invariance is lost . Vayer et al . ( 2019 ) propose a variant of SGW that does not depend on the choice of ∆ , called Rotation Invariant SGW ( RI-SGW ) for p = q , defined as the minimizer of SGW∆ , r over the Stiefel manifold , see ( Vayer et al. , 2019 , Equation 6 ) . In this work , we are interested in calculating an OT-based discrepancy between distributions over distinct domains using the slicing technique . Our approach is different from the SGW one in many points , specifically ( and most importantly ) we use a 1D-Wasserstein distance between the projected pushforward distributions and not a 1D-GW distance . In the next section , we detail the setup of our approach . | In the paper, the authors proposed heterogeneous Wasserstein discrepancy (HWD) to compare incomparable distributions. The main idea of HWD is via distributional slicing, which had been employed in the work of Nguyen et al. [2020]. | SP:8cf31d348eb50e20982d7eea5645978f0108eb38 |
To Smooth or not to Smooth? On Compatibility between Label Smoothing and Knowledge Distillation | 1 INTRODUCTION . This paper deeply investigates the compatibility between label smoothing ( Szegedy et al. , 2016 ) and knowledge distillation ( Hinton et al. , 2015 ) . Specifically , we aim to explain and resolve the contradictory standpoints of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby establishing a foundational understanding on the compatibility between label smoothing ( LS ) and knowledge distillation ( KD ) . Both LS and KD involve training a model ( i.e . : deep neural networks ) with soft-targets . In LS , instead of computing cross entropy loss with the hard-target ( one-hot encoding ) of a training sample , a soft-target is used , which is a weighted mixture of the one-hot encoding and the uniform distribution . A mixture parameter α is used in LS to specify the extent of mixing . On the other hand , KD involves training a teacher model ( usually a powerful model ) and a student model ( usually a compact model ) . The objective of KD is to transfer knowledge from the teacher model to the student model . In the most common form , the student model is trained to match the soft output of the teacher model . The success of KD has been attributed to the transference of logits ’ information about resemblances between instances of different classes ( logits are the inputs to the final softmax which produces the soft targets ) . In KD ( Hinton et al. , 2015 ) , a temperature T is introduced to facilitate the transference : an increased T may produce more suitable soft targets that have more emphasis on the probabilities of incorrect classes ( or equivalently , logits of the incorrect classes ) . To smooth or not to smooth ? Recently , a fair amount of research has been conducted to understand the relationship between LS and KD ( Müller et al. , 2019 ; Shen et al. , 2021b ; Lukasik et al. , 2020 ; Yuan et al. , 2020 ) . One of the most intriguing and controversial discussion is the compatibility between LS and KD . Particularly , in KD , does label smoothing in a teacher network suppress the effectiveness of the distillation ? Müller et al . ( 2019 ) are the first to investigate this topic , and their findings suggest that applying LS to a teacher network impairs the performance of KD . In particular , they visualize the penultimate layer representations in the teacher network to show that LS erases information in the logits about resemblances between instances of different classes . Since this information is essential for KD , they conclude that applying LS for a teacher network can hurt KD . • ‘ If a teacher network is trained with label smoothing , knowledge distillation into a student network is much less effective. ” ( Müller et al. , 2019 ) • “ Label smoothing can hurt distillation ” ( Müller et al. , 2019 ) The conclusion of Müller et al . ( 2019 ) is widely accepted ( Khosla et al. , 2020 ; Arani et al. , 2021 ; Tang et al. , 2021 ; Mghabbar & Ratnamogan , 2020 ; Shen et al. , 2021a ) . However , very recently , this is questioned by Shen et al . ( 2021b ) . In particular , their work discussed a new finding : information erasure in teacher can actually enlarge the central distance between semantically similar classes , allowing the student to learn to classify these categories easily . Shen et al . ( 2021b ) claim that this benefit of using a LS-trained teacher outweighs the detrimental effect due to information erase . Therefore , they conclude that LS in a teacher network does not suppress the effectiveness of KD . • “ Label smoothing will not impair the predictive performance of students. ” ( Shen et al. , 2021b ) • “ Label smoothing is compatible with knowledge distillation ” ( Shen et al. , 2021b ) LS and KD compatibility remains mysterious . We were perplexed by the seemingly contradictory findings by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . While the latter has shown empirical results to support their own finding , their work does not investigate the opposite standpoint and contradictory results by Müller et al . ( 2019 ) . Critically , there is no effort to understand and resolve the seemingly contradictory arguments and supporting evidences by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . Consequently , for practitioners , it remains unclear as to under what situations LS can be applied to the teacher network in KD , and under what situations it must be avoided . Our contributions . In this work , we conduct an empirical investigation to establish a foundational understanding on the compatibility between LS and KD . We begin by meticulously scrutinizing the opposing findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . In particular , we discover that in the presence of a LS-trained teacher , KD at higher temperatures systematically diffuses penultimate layer representations learnt by the student towards semantically similar classes . This systematic diffusion essentially curtails the benefits ( as claimed by Shen et al . ( 2021b ) ) obtained by distilling from a LS-trained teacher , thereby rendering KD at increased temperatures ineffective . We perform largescale distillation experiments using ImageNet-1K to comprehensively demonstrate this systematic diffusion in the student qualitatively using penultimate layer visualizations , and quantitatively using our proposed relative distance metric called diffusion index ( η ) . Our finding on systematic diffusion is very critical when distilling from a LS-trained teacher . Particularly , we argue that this diffusion maneuvers the penultimate layer representations learnt by the student of a given class in a systematic way that targets in the direction of semantically similar classes . Therefore , this systematic diffusion directly curtails the distance enlargement ( between semantically similar classes ) benefits obtained by distilling from a LS-trained teacher . Our qualitative and quantitative analysis with our proposed relative distance metric ( η ) in Sec 4 aims to establish not only the existence of this diffusion , but also establish that such diffusion is systematic . We further conduct extensive experiments using fine-grained image classification ( CUB200-2011 ) , neural machine translation ( English to German , English to Russian translation using IWSLT ) and compact student network distillation ( using MobileNetV2 ) tasks to support our key finding on systematic diffusion . Importantly , using systematic diffusion analysis , we explain and resolve the contradictory findings by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby establishing a foundational understanding on the compatibility between LS and KD . Finally , using our discovery on systematic diffusion , we provide empirical guidelines for practitioners regarding the combined use of LS and KD . We summarize our key findings in Table 1 . The key takeaway from our work is : • In the presence of a LS-trained teacher , KD at higher temperatures systematically diffuses penultimate layer representations learnt by the student towards semantically similar classes . This systematic diffusion essentially curtails the benefits of distilling from a LS-trained teacher , thereby rendering KD at increased temperatures ineffective . Specifically , systematic diffusion was the missing concept that is instrumental in explaining and resolving the contradictory findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby shedding light on whether to smooth or not to smooth a teacher network . Paper organization . In Sec 2 , we review LS and KD . In Sec 3 , we review key findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) to to emphasize the research gap . Our main contribution is Sec 4 , where we introduce our discovered systematic diffusion , conduct qualitative , quantitative and analytical studies to verify that the diffusion is not isotopic but systematic towards semanticallysimilar classes , and therefore it directly curtails the benefits of using a LS-trained teacher . In Sec 5 , we perform rich empirical studies to support our main finding on Systematic Diffusion . In Sec 6 , we provide our perspective regarding the combined use of LS and KD as empirical guidelines for practitioners , and finally conclude this study . 2 PREREQUISITES . Label Smoothing ( LS ) ( Szegedy et al. , 2016 ) : LS was formulated as a regularization strategy to alleviate models ’ over-confidence . LS replaces the original hard target distribution with a mixture of original hard target distribution and the uniform distribution characterized by the mixture parameter α . Consider the formulation of LS objective with mixture parameter α as follows : Let pk , wk represent the probability and last layer weights ( including biases ) corresponding to the k-th class . Let x , yk , yLSk represent the penultimate layer activations , true targets and LS-targets where yk = 1 for the correct class and 0 for all the incorrect classes1 . xT is the transpose of x . Then for a classification network trained with LS containing K classes , we minimize the cross entropy loss between LS-targets yLSk and model predictions pk given by LLS ( y , p ) = ∑K k=1 −yLSk log ( pk ) , where pk = exp ( xTwk ) / ∑K l=1 exp ( x Twl ) and yLSk = yk ( 1− α ) + αK . Knowledge distillation ( KD ) Hinton et al . ( 2015 ) : KD uses a larger capacity teacher model ( s ) to transfer the knowledge to a compact student model . The success of KD methods is largely at- 1x is concatenated with 1 at the end to include bias as wk includes biases at the end . tributed to the information about incorrect classes encoded in the output distribution produced by the teacher model ( s ) ( Hinton et al. , 2015 ) . Consider KD for a classification objective . Let T indicate the temperature factor that controls the importance of each soft target . Given the k-th class logit xTwk , let the temperature scaled probability be pk ( T ) . For KD training , let the loss be LKD . For LKD , we replace the cross entropy loss H ( y , p ) with a weighted sum ( parametrized by β ) of H ( y , p ) and H ( pt ( T ) , p ( T ) ) where pt ( T ) , p ( T ) correspond to the temperature-scaled teacher and student output probabilities . That is , pk ( T ) = exp ( x Twk T ) / ∑K l=1 exp ( xTwl T ) and LKD = ( 1 − β ) H ( y , p ) + βT 2H ( pt ( T ) , p ( T ) ) . Following Hinton et al . ( 2015 ) T 2 scaling is used for the soft-target optimization as T will scale the gradients approximately by a factor of T 2 . Following Müller et al . ( 2019 ) ; Shen et al . ( 2021b ) , we set β = 1 for this study since we primarily aim to isolate and study the effects of KD . β = 1 achieves good performance ( Shen et al. , 2021b ) . 3 A CLOSER LOOK AT LS AND KD COMPATIBILITY . In this section , we review the contradictory findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) from the perspective of information erase in LS-trained teacher . This discussion is a necessary preamble to discuss our main finding , Systematic Diffusion in the student in Sec 4 . Information Erase in LS-trained teacher . LS objective optimizes the probability of the correct class to be equal to 1 − α + α/K , and incorrect classes to be α/K . This directly encourages the differences between logits of the correct class and incorrect classes to be a constant ( Müller et al. , 2019 ) determined by α . Following Müller et al . ( 2019 ) , the logit xTwk can be approximately measured using the squared Euclidean distance between penultimate layer ’ s activations and the template corresponding to class k. That is , xTwk can be approximately measured by ∥x−wk∥2 . This allows to establish 2 important geometric properties of LS ( Müller et al. , 2019 ) : With LS , penultimate layer activations 1 ) are encouraged to be close to the template of the correct class ( large logit value for the correct class , therefore small distance between the activations and the correct class template ) , and 2 ) are encouraged to be equidistant to the templates of the incorrect classes ( equal logit values for all the incorrect classes ) . This results in penultimate layer activations to tightly cluster around the correct class template compared to the model trained with standard cross entropy objective . We demonstrate this clearly in Figure 1 Observation 1 . With LS applied on the ResNet-50 model , we observe that the penultimate layer representations become much tighter . As a result , substantial information regarding the resemblances of these instances to those of other different classes is lost . This is referred to as the information erase in LS-trained network ( teacher ) ( Müller et al. , 2019 ) . Claim 1 : Information erase in LS-trained teacher cause LS and KD to be Incompatible ( Müller et al. , 2019 ) : Müller et al . ( 2019 ) are the first to investigate this compatibility , and they argue that the information erasure effect due to LS ( shown in Figure 1 Observation 1 ) can impair KD . Given the prominent successes in KD methods being largely attributed to dark knowledge/ inter-class information emerging from the trained-teacher ( Hinton et al. , 2015 ; Tang et al. , 2021 ) , the argument by Müller et al . ( 2019 ) that LS and KD are incompatible due to information loss in the logits is generally convincing and widely accepted ( Khosla et al. , 2020 ; Arani et al. , 2021 ; Tang et al. , 2021 ; Mghabbar & Ratnamogan , 2020 ; Shen et al. , 2021a ) . This is also supported by empirical evidence . Claim 2 : Information erase in LS-trained teacher provides distance enlargement benefits between semantically similar classes , resulting in LS and KD to be Compatible ( Shen et al. , 2021b ) : Recently an interesting finding by Shen et al . ( 2021b ) argue that LS and KD are compatible . Though they agree that information erasure generally happens with LS , their argument focuses more on the effect of LS on semantically similar classes . They argue that information erase in LStrained teacher can promote enlargement of central distance of clusters between semantically similar classes . This allows the student network to easily learn to classify semantically similar classes which are generally difficult to classify in conventional training procedures . We show this increased separation between semantically similar classes with LS in Figure 1 Observation 1 . It can be observed that the central distance between the clusters of standard poodle and miniature poodle increases with using LS on the ResNet-50 teacher . In our work , we further extend to show that this property is inherited by the ResNet-18 student as well in Observation 2 . We remark that this inheritance is not shown by Shen et al . ( 2021b ) . This finding by Shen et al . ( 2021b ) is largely supported by experiments and quantitative results . Though Shen et al . ( 2021b ) claim that the benefit derived from larger separation between semantically similar classes outweigh the drawbacks due to information erase , thereby making LS and KD compatible , their investigation does not address the contradictory findings and empirical results obtained by Müller et al . ( 2019 ) . Research Gap : Studied in isolation , both these contradictory arguments are convincing and are well supported empirically . This has caused serious perplexity among the research community regarding the combined use of LS and KD . | The paper provides a new perspective for the compatibility of label smoothing and knowledge distillation. A new definition named "systematic diffusion" is proposed to analyze the contradiction between the claims of [1] and [2]. By observing the visualization of the penultimate layer representations, the authors claim that KD is compatible with LS under the setting of lower temperature T. The paper conducts a large-scale evaluation of the impact of increasing temperature T in KD in the perspective of the systematic diffusion. The findings are interesting and attractive. The visualization looks good. [1] Rafael Muller, Simon Kornblith, and Geoffrey E Hinton. When does label smoothing help? Advances in Neural Information Processing Systems, 2019 [2] Zhiqiang Shen, Zechun Liu, Dejia Xu, Zitian Chen, Kwang-Ting Cheng, and Marios Savvides. Is label smoothing truly incompatible with knowledge distillation: An empirical study. International Conference on Learning Representations, 2021 | SP:6a5a1110220420e5476929ecb0466bfc018a80bb |
To Smooth or not to Smooth? On Compatibility between Label Smoothing and Knowledge Distillation | 1 INTRODUCTION . This paper deeply investigates the compatibility between label smoothing ( Szegedy et al. , 2016 ) and knowledge distillation ( Hinton et al. , 2015 ) . Specifically , we aim to explain and resolve the contradictory standpoints of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby establishing a foundational understanding on the compatibility between label smoothing ( LS ) and knowledge distillation ( KD ) . Both LS and KD involve training a model ( i.e . : deep neural networks ) with soft-targets . In LS , instead of computing cross entropy loss with the hard-target ( one-hot encoding ) of a training sample , a soft-target is used , which is a weighted mixture of the one-hot encoding and the uniform distribution . A mixture parameter α is used in LS to specify the extent of mixing . On the other hand , KD involves training a teacher model ( usually a powerful model ) and a student model ( usually a compact model ) . The objective of KD is to transfer knowledge from the teacher model to the student model . In the most common form , the student model is trained to match the soft output of the teacher model . The success of KD has been attributed to the transference of logits ’ information about resemblances between instances of different classes ( logits are the inputs to the final softmax which produces the soft targets ) . In KD ( Hinton et al. , 2015 ) , a temperature T is introduced to facilitate the transference : an increased T may produce more suitable soft targets that have more emphasis on the probabilities of incorrect classes ( or equivalently , logits of the incorrect classes ) . To smooth or not to smooth ? Recently , a fair amount of research has been conducted to understand the relationship between LS and KD ( Müller et al. , 2019 ; Shen et al. , 2021b ; Lukasik et al. , 2020 ; Yuan et al. , 2020 ) . One of the most intriguing and controversial discussion is the compatibility between LS and KD . Particularly , in KD , does label smoothing in a teacher network suppress the effectiveness of the distillation ? Müller et al . ( 2019 ) are the first to investigate this topic , and their findings suggest that applying LS to a teacher network impairs the performance of KD . In particular , they visualize the penultimate layer representations in the teacher network to show that LS erases information in the logits about resemblances between instances of different classes . Since this information is essential for KD , they conclude that applying LS for a teacher network can hurt KD . • ‘ If a teacher network is trained with label smoothing , knowledge distillation into a student network is much less effective. ” ( Müller et al. , 2019 ) • “ Label smoothing can hurt distillation ” ( Müller et al. , 2019 ) The conclusion of Müller et al . ( 2019 ) is widely accepted ( Khosla et al. , 2020 ; Arani et al. , 2021 ; Tang et al. , 2021 ; Mghabbar & Ratnamogan , 2020 ; Shen et al. , 2021a ) . However , very recently , this is questioned by Shen et al . ( 2021b ) . In particular , their work discussed a new finding : information erasure in teacher can actually enlarge the central distance between semantically similar classes , allowing the student to learn to classify these categories easily . Shen et al . ( 2021b ) claim that this benefit of using a LS-trained teacher outweighs the detrimental effect due to information erase . Therefore , they conclude that LS in a teacher network does not suppress the effectiveness of KD . • “ Label smoothing will not impair the predictive performance of students. ” ( Shen et al. , 2021b ) • “ Label smoothing is compatible with knowledge distillation ” ( Shen et al. , 2021b ) LS and KD compatibility remains mysterious . We were perplexed by the seemingly contradictory findings by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . While the latter has shown empirical results to support their own finding , their work does not investigate the opposite standpoint and contradictory results by Müller et al . ( 2019 ) . Critically , there is no effort to understand and resolve the seemingly contradictory arguments and supporting evidences by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . Consequently , for practitioners , it remains unclear as to under what situations LS can be applied to the teacher network in KD , and under what situations it must be avoided . Our contributions . In this work , we conduct an empirical investigation to establish a foundational understanding on the compatibility between LS and KD . We begin by meticulously scrutinizing the opposing findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . In particular , we discover that in the presence of a LS-trained teacher , KD at higher temperatures systematically diffuses penultimate layer representations learnt by the student towards semantically similar classes . This systematic diffusion essentially curtails the benefits ( as claimed by Shen et al . ( 2021b ) ) obtained by distilling from a LS-trained teacher , thereby rendering KD at increased temperatures ineffective . We perform largescale distillation experiments using ImageNet-1K to comprehensively demonstrate this systematic diffusion in the student qualitatively using penultimate layer visualizations , and quantitatively using our proposed relative distance metric called diffusion index ( η ) . Our finding on systematic diffusion is very critical when distilling from a LS-trained teacher . Particularly , we argue that this diffusion maneuvers the penultimate layer representations learnt by the student of a given class in a systematic way that targets in the direction of semantically similar classes . Therefore , this systematic diffusion directly curtails the distance enlargement ( between semantically similar classes ) benefits obtained by distilling from a LS-trained teacher . Our qualitative and quantitative analysis with our proposed relative distance metric ( η ) in Sec 4 aims to establish not only the existence of this diffusion , but also establish that such diffusion is systematic . We further conduct extensive experiments using fine-grained image classification ( CUB200-2011 ) , neural machine translation ( English to German , English to Russian translation using IWSLT ) and compact student network distillation ( using MobileNetV2 ) tasks to support our key finding on systematic diffusion . Importantly , using systematic diffusion analysis , we explain and resolve the contradictory findings by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby establishing a foundational understanding on the compatibility between LS and KD . Finally , using our discovery on systematic diffusion , we provide empirical guidelines for practitioners regarding the combined use of LS and KD . We summarize our key findings in Table 1 . The key takeaway from our work is : • In the presence of a LS-trained teacher , KD at higher temperatures systematically diffuses penultimate layer representations learnt by the student towards semantically similar classes . This systematic diffusion essentially curtails the benefits of distilling from a LS-trained teacher , thereby rendering KD at increased temperatures ineffective . Specifically , systematic diffusion was the missing concept that is instrumental in explaining and resolving the contradictory findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby shedding light on whether to smooth or not to smooth a teacher network . Paper organization . In Sec 2 , we review LS and KD . In Sec 3 , we review key findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) to to emphasize the research gap . Our main contribution is Sec 4 , where we introduce our discovered systematic diffusion , conduct qualitative , quantitative and analytical studies to verify that the diffusion is not isotopic but systematic towards semanticallysimilar classes , and therefore it directly curtails the benefits of using a LS-trained teacher . In Sec 5 , we perform rich empirical studies to support our main finding on Systematic Diffusion . In Sec 6 , we provide our perspective regarding the combined use of LS and KD as empirical guidelines for practitioners , and finally conclude this study . 2 PREREQUISITES . Label Smoothing ( LS ) ( Szegedy et al. , 2016 ) : LS was formulated as a regularization strategy to alleviate models ’ over-confidence . LS replaces the original hard target distribution with a mixture of original hard target distribution and the uniform distribution characterized by the mixture parameter α . Consider the formulation of LS objective with mixture parameter α as follows : Let pk , wk represent the probability and last layer weights ( including biases ) corresponding to the k-th class . Let x , yk , yLSk represent the penultimate layer activations , true targets and LS-targets where yk = 1 for the correct class and 0 for all the incorrect classes1 . xT is the transpose of x . Then for a classification network trained with LS containing K classes , we minimize the cross entropy loss between LS-targets yLSk and model predictions pk given by LLS ( y , p ) = ∑K k=1 −yLSk log ( pk ) , where pk = exp ( xTwk ) / ∑K l=1 exp ( x Twl ) and yLSk = yk ( 1− α ) + αK . Knowledge distillation ( KD ) Hinton et al . ( 2015 ) : KD uses a larger capacity teacher model ( s ) to transfer the knowledge to a compact student model . The success of KD methods is largely at- 1x is concatenated with 1 at the end to include bias as wk includes biases at the end . tributed to the information about incorrect classes encoded in the output distribution produced by the teacher model ( s ) ( Hinton et al. , 2015 ) . Consider KD for a classification objective . Let T indicate the temperature factor that controls the importance of each soft target . Given the k-th class logit xTwk , let the temperature scaled probability be pk ( T ) . For KD training , let the loss be LKD . For LKD , we replace the cross entropy loss H ( y , p ) with a weighted sum ( parametrized by β ) of H ( y , p ) and H ( pt ( T ) , p ( T ) ) where pt ( T ) , p ( T ) correspond to the temperature-scaled teacher and student output probabilities . That is , pk ( T ) = exp ( x Twk T ) / ∑K l=1 exp ( xTwl T ) and LKD = ( 1 − β ) H ( y , p ) + βT 2H ( pt ( T ) , p ( T ) ) . Following Hinton et al . ( 2015 ) T 2 scaling is used for the soft-target optimization as T will scale the gradients approximately by a factor of T 2 . Following Müller et al . ( 2019 ) ; Shen et al . ( 2021b ) , we set β = 1 for this study since we primarily aim to isolate and study the effects of KD . β = 1 achieves good performance ( Shen et al. , 2021b ) . 3 A CLOSER LOOK AT LS AND KD COMPATIBILITY . In this section , we review the contradictory findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) from the perspective of information erase in LS-trained teacher . This discussion is a necessary preamble to discuss our main finding , Systematic Diffusion in the student in Sec 4 . Information Erase in LS-trained teacher . LS objective optimizes the probability of the correct class to be equal to 1 − α + α/K , and incorrect classes to be α/K . This directly encourages the differences between logits of the correct class and incorrect classes to be a constant ( Müller et al. , 2019 ) determined by α . Following Müller et al . ( 2019 ) , the logit xTwk can be approximately measured using the squared Euclidean distance between penultimate layer ’ s activations and the template corresponding to class k. That is , xTwk can be approximately measured by ∥x−wk∥2 . This allows to establish 2 important geometric properties of LS ( Müller et al. , 2019 ) : With LS , penultimate layer activations 1 ) are encouraged to be close to the template of the correct class ( large logit value for the correct class , therefore small distance between the activations and the correct class template ) , and 2 ) are encouraged to be equidistant to the templates of the incorrect classes ( equal logit values for all the incorrect classes ) . This results in penultimate layer activations to tightly cluster around the correct class template compared to the model trained with standard cross entropy objective . We demonstrate this clearly in Figure 1 Observation 1 . With LS applied on the ResNet-50 model , we observe that the penultimate layer representations become much tighter . As a result , substantial information regarding the resemblances of these instances to those of other different classes is lost . This is referred to as the information erase in LS-trained network ( teacher ) ( Müller et al. , 2019 ) . Claim 1 : Information erase in LS-trained teacher cause LS and KD to be Incompatible ( Müller et al. , 2019 ) : Müller et al . ( 2019 ) are the first to investigate this compatibility , and they argue that the information erasure effect due to LS ( shown in Figure 1 Observation 1 ) can impair KD . Given the prominent successes in KD methods being largely attributed to dark knowledge/ inter-class information emerging from the trained-teacher ( Hinton et al. , 2015 ; Tang et al. , 2021 ) , the argument by Müller et al . ( 2019 ) that LS and KD are incompatible due to information loss in the logits is generally convincing and widely accepted ( Khosla et al. , 2020 ; Arani et al. , 2021 ; Tang et al. , 2021 ; Mghabbar & Ratnamogan , 2020 ; Shen et al. , 2021a ) . This is also supported by empirical evidence . Claim 2 : Information erase in LS-trained teacher provides distance enlargement benefits between semantically similar classes , resulting in LS and KD to be Compatible ( Shen et al. , 2021b ) : Recently an interesting finding by Shen et al . ( 2021b ) argue that LS and KD are compatible . Though they agree that information erasure generally happens with LS , their argument focuses more on the effect of LS on semantically similar classes . They argue that information erase in LStrained teacher can promote enlargement of central distance of clusters between semantically similar classes . This allows the student network to easily learn to classify semantically similar classes which are generally difficult to classify in conventional training procedures . We show this increased separation between semantically similar classes with LS in Figure 1 Observation 1 . It can be observed that the central distance between the clusters of standard poodle and miniature poodle increases with using LS on the ResNet-50 teacher . In our work , we further extend to show that this property is inherited by the ResNet-18 student as well in Observation 2 . We remark that this inheritance is not shown by Shen et al . ( 2021b ) . This finding by Shen et al . ( 2021b ) is largely supported by experiments and quantitative results . Though Shen et al . ( 2021b ) claim that the benefit derived from larger separation between semantically similar classes outweigh the drawbacks due to information erase , thereby making LS and KD compatible , their investigation does not address the contradictory findings and empirical results obtained by Müller et al . ( 2019 ) . Research Gap : Studied in isolation , both these contradictory arguments are convincing and are well supported empirically . This has caused serious perplexity among the research community regarding the combined use of LS and KD . | This paper investigated whether a label smoothing-trained teacher is compatible with knowledge distillation. It started by reproducing the results of two previous studies, i.e., (Muller et al. 2019) claimed that LS erases information in the logits and therefore is harmful to KD, while (Shen et al. 2021b) argued that LS-trained teacher increases the distance between semantically similar classes and therefore is compatible with KD. After that, the authors tried to draw a unified conclusion from the above contradictory claims by introducing experiments with different temperatures: with an LS-trained teacher, a higher temperature introduces systematic diffusion in the representations of the semantically similar classes, which hurts KD. It is the different temperatures used in (Muller et al. 2019) and (Shen et al. 2021b) that resulted in different conclusions. Experiment results supported the main claims. | SP:6a5a1110220420e5476929ecb0466bfc018a80bb |
To Smooth or not to Smooth? On Compatibility between Label Smoothing and Knowledge Distillation | 1 INTRODUCTION . This paper deeply investigates the compatibility between label smoothing ( Szegedy et al. , 2016 ) and knowledge distillation ( Hinton et al. , 2015 ) . Specifically , we aim to explain and resolve the contradictory standpoints of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby establishing a foundational understanding on the compatibility between label smoothing ( LS ) and knowledge distillation ( KD ) . Both LS and KD involve training a model ( i.e . : deep neural networks ) with soft-targets . In LS , instead of computing cross entropy loss with the hard-target ( one-hot encoding ) of a training sample , a soft-target is used , which is a weighted mixture of the one-hot encoding and the uniform distribution . A mixture parameter α is used in LS to specify the extent of mixing . On the other hand , KD involves training a teacher model ( usually a powerful model ) and a student model ( usually a compact model ) . The objective of KD is to transfer knowledge from the teacher model to the student model . In the most common form , the student model is trained to match the soft output of the teacher model . The success of KD has been attributed to the transference of logits ’ information about resemblances between instances of different classes ( logits are the inputs to the final softmax which produces the soft targets ) . In KD ( Hinton et al. , 2015 ) , a temperature T is introduced to facilitate the transference : an increased T may produce more suitable soft targets that have more emphasis on the probabilities of incorrect classes ( or equivalently , logits of the incorrect classes ) . To smooth or not to smooth ? Recently , a fair amount of research has been conducted to understand the relationship between LS and KD ( Müller et al. , 2019 ; Shen et al. , 2021b ; Lukasik et al. , 2020 ; Yuan et al. , 2020 ) . One of the most intriguing and controversial discussion is the compatibility between LS and KD . Particularly , in KD , does label smoothing in a teacher network suppress the effectiveness of the distillation ? Müller et al . ( 2019 ) are the first to investigate this topic , and their findings suggest that applying LS to a teacher network impairs the performance of KD . In particular , they visualize the penultimate layer representations in the teacher network to show that LS erases information in the logits about resemblances between instances of different classes . Since this information is essential for KD , they conclude that applying LS for a teacher network can hurt KD . • ‘ If a teacher network is trained with label smoothing , knowledge distillation into a student network is much less effective. ” ( Müller et al. , 2019 ) • “ Label smoothing can hurt distillation ” ( Müller et al. , 2019 ) The conclusion of Müller et al . ( 2019 ) is widely accepted ( Khosla et al. , 2020 ; Arani et al. , 2021 ; Tang et al. , 2021 ; Mghabbar & Ratnamogan , 2020 ; Shen et al. , 2021a ) . However , very recently , this is questioned by Shen et al . ( 2021b ) . In particular , their work discussed a new finding : information erasure in teacher can actually enlarge the central distance between semantically similar classes , allowing the student to learn to classify these categories easily . Shen et al . ( 2021b ) claim that this benefit of using a LS-trained teacher outweighs the detrimental effect due to information erase . Therefore , they conclude that LS in a teacher network does not suppress the effectiveness of KD . • “ Label smoothing will not impair the predictive performance of students. ” ( Shen et al. , 2021b ) • “ Label smoothing is compatible with knowledge distillation ” ( Shen et al. , 2021b ) LS and KD compatibility remains mysterious . We were perplexed by the seemingly contradictory findings by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . While the latter has shown empirical results to support their own finding , their work does not investigate the opposite standpoint and contradictory results by Müller et al . ( 2019 ) . Critically , there is no effort to understand and resolve the seemingly contradictory arguments and supporting evidences by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . Consequently , for practitioners , it remains unclear as to under what situations LS can be applied to the teacher network in KD , and under what situations it must be avoided . Our contributions . In this work , we conduct an empirical investigation to establish a foundational understanding on the compatibility between LS and KD . We begin by meticulously scrutinizing the opposing findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) . In particular , we discover that in the presence of a LS-trained teacher , KD at higher temperatures systematically diffuses penultimate layer representations learnt by the student towards semantically similar classes . This systematic diffusion essentially curtails the benefits ( as claimed by Shen et al . ( 2021b ) ) obtained by distilling from a LS-trained teacher , thereby rendering KD at increased temperatures ineffective . We perform largescale distillation experiments using ImageNet-1K to comprehensively demonstrate this systematic diffusion in the student qualitatively using penultimate layer visualizations , and quantitatively using our proposed relative distance metric called diffusion index ( η ) . Our finding on systematic diffusion is very critical when distilling from a LS-trained teacher . Particularly , we argue that this diffusion maneuvers the penultimate layer representations learnt by the student of a given class in a systematic way that targets in the direction of semantically similar classes . Therefore , this systematic diffusion directly curtails the distance enlargement ( between semantically similar classes ) benefits obtained by distilling from a LS-trained teacher . Our qualitative and quantitative analysis with our proposed relative distance metric ( η ) in Sec 4 aims to establish not only the existence of this diffusion , but also establish that such diffusion is systematic . We further conduct extensive experiments using fine-grained image classification ( CUB200-2011 ) , neural machine translation ( English to German , English to Russian translation using IWSLT ) and compact student network distillation ( using MobileNetV2 ) tasks to support our key finding on systematic diffusion . Importantly , using systematic diffusion analysis , we explain and resolve the contradictory findings by Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby establishing a foundational understanding on the compatibility between LS and KD . Finally , using our discovery on systematic diffusion , we provide empirical guidelines for practitioners regarding the combined use of LS and KD . We summarize our key findings in Table 1 . The key takeaway from our work is : • In the presence of a LS-trained teacher , KD at higher temperatures systematically diffuses penultimate layer representations learnt by the student towards semantically similar classes . This systematic diffusion essentially curtails the benefits of distilling from a LS-trained teacher , thereby rendering KD at increased temperatures ineffective . Specifically , systematic diffusion was the missing concept that is instrumental in explaining and resolving the contradictory findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) , thereby shedding light on whether to smooth or not to smooth a teacher network . Paper organization . In Sec 2 , we review LS and KD . In Sec 3 , we review key findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) to to emphasize the research gap . Our main contribution is Sec 4 , where we introduce our discovered systematic diffusion , conduct qualitative , quantitative and analytical studies to verify that the diffusion is not isotopic but systematic towards semanticallysimilar classes , and therefore it directly curtails the benefits of using a LS-trained teacher . In Sec 5 , we perform rich empirical studies to support our main finding on Systematic Diffusion . In Sec 6 , we provide our perspective regarding the combined use of LS and KD as empirical guidelines for practitioners , and finally conclude this study . 2 PREREQUISITES . Label Smoothing ( LS ) ( Szegedy et al. , 2016 ) : LS was formulated as a regularization strategy to alleviate models ’ over-confidence . LS replaces the original hard target distribution with a mixture of original hard target distribution and the uniform distribution characterized by the mixture parameter α . Consider the formulation of LS objective with mixture parameter α as follows : Let pk , wk represent the probability and last layer weights ( including biases ) corresponding to the k-th class . Let x , yk , yLSk represent the penultimate layer activations , true targets and LS-targets where yk = 1 for the correct class and 0 for all the incorrect classes1 . xT is the transpose of x . Then for a classification network trained with LS containing K classes , we minimize the cross entropy loss between LS-targets yLSk and model predictions pk given by LLS ( y , p ) = ∑K k=1 −yLSk log ( pk ) , where pk = exp ( xTwk ) / ∑K l=1 exp ( x Twl ) and yLSk = yk ( 1− α ) + αK . Knowledge distillation ( KD ) Hinton et al . ( 2015 ) : KD uses a larger capacity teacher model ( s ) to transfer the knowledge to a compact student model . The success of KD methods is largely at- 1x is concatenated with 1 at the end to include bias as wk includes biases at the end . tributed to the information about incorrect classes encoded in the output distribution produced by the teacher model ( s ) ( Hinton et al. , 2015 ) . Consider KD for a classification objective . Let T indicate the temperature factor that controls the importance of each soft target . Given the k-th class logit xTwk , let the temperature scaled probability be pk ( T ) . For KD training , let the loss be LKD . For LKD , we replace the cross entropy loss H ( y , p ) with a weighted sum ( parametrized by β ) of H ( y , p ) and H ( pt ( T ) , p ( T ) ) where pt ( T ) , p ( T ) correspond to the temperature-scaled teacher and student output probabilities . That is , pk ( T ) = exp ( x Twk T ) / ∑K l=1 exp ( xTwl T ) and LKD = ( 1 − β ) H ( y , p ) + βT 2H ( pt ( T ) , p ( T ) ) . Following Hinton et al . ( 2015 ) T 2 scaling is used for the soft-target optimization as T will scale the gradients approximately by a factor of T 2 . Following Müller et al . ( 2019 ) ; Shen et al . ( 2021b ) , we set β = 1 for this study since we primarily aim to isolate and study the effects of KD . β = 1 achieves good performance ( Shen et al. , 2021b ) . 3 A CLOSER LOOK AT LS AND KD COMPATIBILITY . In this section , we review the contradictory findings of Müller et al . ( 2019 ) and Shen et al . ( 2021b ) from the perspective of information erase in LS-trained teacher . This discussion is a necessary preamble to discuss our main finding , Systematic Diffusion in the student in Sec 4 . Information Erase in LS-trained teacher . LS objective optimizes the probability of the correct class to be equal to 1 − α + α/K , and incorrect classes to be α/K . This directly encourages the differences between logits of the correct class and incorrect classes to be a constant ( Müller et al. , 2019 ) determined by α . Following Müller et al . ( 2019 ) , the logit xTwk can be approximately measured using the squared Euclidean distance between penultimate layer ’ s activations and the template corresponding to class k. That is , xTwk can be approximately measured by ∥x−wk∥2 . This allows to establish 2 important geometric properties of LS ( Müller et al. , 2019 ) : With LS , penultimate layer activations 1 ) are encouraged to be close to the template of the correct class ( large logit value for the correct class , therefore small distance between the activations and the correct class template ) , and 2 ) are encouraged to be equidistant to the templates of the incorrect classes ( equal logit values for all the incorrect classes ) . This results in penultimate layer activations to tightly cluster around the correct class template compared to the model trained with standard cross entropy objective . We demonstrate this clearly in Figure 1 Observation 1 . With LS applied on the ResNet-50 model , we observe that the penultimate layer representations become much tighter . As a result , substantial information regarding the resemblances of these instances to those of other different classes is lost . This is referred to as the information erase in LS-trained network ( teacher ) ( Müller et al. , 2019 ) . Claim 1 : Information erase in LS-trained teacher cause LS and KD to be Incompatible ( Müller et al. , 2019 ) : Müller et al . ( 2019 ) are the first to investigate this compatibility , and they argue that the information erasure effect due to LS ( shown in Figure 1 Observation 1 ) can impair KD . Given the prominent successes in KD methods being largely attributed to dark knowledge/ inter-class information emerging from the trained-teacher ( Hinton et al. , 2015 ; Tang et al. , 2021 ) , the argument by Müller et al . ( 2019 ) that LS and KD are incompatible due to information loss in the logits is generally convincing and widely accepted ( Khosla et al. , 2020 ; Arani et al. , 2021 ; Tang et al. , 2021 ; Mghabbar & Ratnamogan , 2020 ; Shen et al. , 2021a ) . This is also supported by empirical evidence . Claim 2 : Information erase in LS-trained teacher provides distance enlargement benefits between semantically similar classes , resulting in LS and KD to be Compatible ( Shen et al. , 2021b ) : Recently an interesting finding by Shen et al . ( 2021b ) argue that LS and KD are compatible . Though they agree that information erasure generally happens with LS , their argument focuses more on the effect of LS on semantically similar classes . They argue that information erase in LStrained teacher can promote enlargement of central distance of clusters between semantically similar classes . This allows the student network to easily learn to classify semantically similar classes which are generally difficult to classify in conventional training procedures . We show this increased separation between semantically similar classes with LS in Figure 1 Observation 1 . It can be observed that the central distance between the clusters of standard poodle and miniature poodle increases with using LS on the ResNet-50 teacher . In our work , we further extend to show that this property is inherited by the ResNet-18 student as well in Observation 2 . We remark that this inheritance is not shown by Shen et al . ( 2021b ) . This finding by Shen et al . ( 2021b ) is largely supported by experiments and quantitative results . Though Shen et al . ( 2021b ) claim that the benefit derived from larger separation between semantically similar classes outweigh the drawbacks due to information erase , thereby making LS and KD compatible , their investigation does not address the contradictory findings and empirical results obtained by Müller et al . ( 2019 ) . Research Gap : Studied in isolation , both these contradictory arguments are convincing and are well supported empirically . This has caused serious perplexity among the research community regarding the combined use of LS and KD . | This paper investigates the contradiction of previous findings: (1) LS can hurt KD and (2) the benefit of using LS outweighs its disadvantage for KD. The authors provide ample experiments to verify their claim of "LS is useful for KD with a moderate (low) value of temperature". They interpret the negative or positive influence from LS with respect to systematic diffusion. The conclusion is using LS with a low T (e.g., 1) for KD. To support this argument, the authors analyze the diffusion phenomenon qualitatively (maybe t-SNE) and quantitatively (diffusion index). | SP:6a5a1110220420e5476929ecb0466bfc018a80bb |
Beyond ImageNet Attack: Towards Crafting Adversarial Examples for Black-box Domains | 1 INTRODUCTION . Deep neural networks ( DNNs ) have achieved remarkable success in the image classification task in recent years . Nonetheless , advances in the field of adversarial machine learning ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ) make DNNs no longer reliable . By adding a well-designed perturbation on a benign image ( a.k.a adversarial attack ) , the resulting adversarial examples can easily fool state-of-the-art DNNs . To make the matter worse , the adversarial attack technique can even be applied in the physical world ( Sharif et al. , 2016 ; Kurakin et al. , 2017a ; Xu et al. , 2020 ; Duan et al. , 2021 ) , which inevitably raises concerns about the stability of deployed models . Therefore , exposing as many “ blind spots ” of DNNs as possible is a top priority . Generally , deployed models are mainly challenged with two threat models : white-box and blackbox . For white-box threat model ( Kurakin et al. , 2017b ; Moosavi-Dezfooli et al. , 2016 ; Carlini & Wagner , 2017 ; Shi et al. , 2019 ) , the attacker can obtain complete knowledge of the target model , such as the gradient for any input . However , deployed models are usually opaque to unauthorized users . In this scenario , prior black-box works ( Poursaeed et al. , 2018 ; Dong et al. , 2018 ; Xie et al. , 2019 ; Inkawhich et al. , 2020 ; Gao et al. , 2020b ; Wang et al. , 2021 ) mostly assume that the source data for training the target model is available and mainly explore the cross-model transferability among models trained in the same data distribution . Specifically , perturbations are crafted via accessible white-box model ( a.k.a substitute model ) , and resulting adversarial examples sometimes can fool other black-box models as well . Yet , these works still ignore a pivotal issue : A model owner is ∗Work done when Qilong Zhang interns at Alibaba Group , China †Corresponding author unlikely to leak the relevant information of the deployed model . To overcome this limitation , querybased black-box attacks ( Papernot et al. , 2016 ; Brendel et al. , 2018 ; Chen et al. , 2020 ; Li et al. , 2021 ) are proposed , which adjust adversarial examples just according to the output of the target model . However , the resource-intensive query budget is extremely costly and inevitably alerts the model owner . Therefore , we need a more “ practical ” black-box threat model to address this concern , i.e. , without any clue about the training data distribution as well as the pre-trained model based on it , and even querying is forbidden . Intuitively , this threat model is more challenging to build for attackers and more threatening to model owners . To the best of our knowledge , a recent work called CDA ( Naseer et al. , 2019 ) is the first to attempt such an attack . Specifically , it learns a transferable adversarial function via a generator network against a different domain ( training data and pre-trained model are all from ChestX-ray ( Wang et al. , 2017 ) domain ) . During inference , it directly crafts adversarial examples for benign ImageNet images to fool target ImageNet pre-trained models . However , its cross-domain transfer strength is still moderate . Besides , relying on small-scale datasets to train a generator may limit the generalization of the threat model . Considering that ImageNet is a large-scale dataset containing most of common categories in real life and there are various off-the-shelf pre-trained models , one can easily dig out much useful information to build a strong threat model . Therefore , in this paper , solely relying on the knowledge of the ImageNet domain , we introduce an effective Beyond ImageNet Attack ( BIA ) framework to enhance the cross-domain transferability of adversarial examples . To reflect the applicability of our approach , we consider eight different image classification tasks ( listed in Table 1 ) . Figure 1 illustrates an overview of our method . Particularly , we learn a flexible generator network Gθ against ImageNet domain . Instead of optimizing the domain-specific loss function like CDA , our method focuses on disrupting low-level features following previous literature to ensure the good transferability of our BIA . Furthermore , we propose two variants based on the vanilla BIA to narrow the gap between source and target domains . Specifically , from the data perspective , we propose a random normalization ( RN ) module to simulate different data distributions ; from the model perspective , we propose a domain-agnostic attention ( DA ) module to capture essential features for perturbing . In the inference phase , our Gθ accepts images of any domain as the input and crafts adversarial examples with one forward propagation . Extensive experiments demonstrate the effectiveness of our proposed methods . Towards the coarse-grained and fine-grained domains , we can outperform state-of-the-art approaches by up to 7.71 % and 25.91 % on average , respectively . Besides , our methods can also enhance the cross-model transferability in the source domain . 2 RELATED WORKS . Iterative Optimization Approaches . Under the black-box threat model , iterative attack methods are a popular branch , which usually adopt domain-specific loss or intermediate feature loss to craft adversarial examples . For the former , Madry et al . ( 2018 ) extend Goodfellow et al . ( 2015 ) to perform projected gradient descent from randomly chosen starting points inside -ball . Dong et al . ( 2018 ) introduce momentum term to stable the update direction . Xie et al . ( 2019 ) apply random transformations of the input at each iteration , thus mitigating overfitting . Gao et al . ( 2020a ) propose patch-wise perturbation to better cover the discriminative region . Wu et al . ( 2020a ) explore the security weakness of skip connections ( He et al. , 2016 ; Huang et al. , 2017 ) to boost attacks . Different from the methods mentioned above , intermediate feature-based methods focus on disrupting low-level features . For example , Zhou et al . ( 2018 ) maximize the Euclidean distance between the source image and target image in feature space and introduce regularization on perturbations to reduce variations . Inkawhich et al . ( 2019 ) make the source image close to the target image in feature space . Lu et al . ( 2020 ) propose a dispersion reduction attack to make the low-level features featureless . Naseer et al . ( 2020 ) design a self-supervised perturbation mechanism for enabling a transferable defense approach . Wu et al . ( 2020b ) compute model attention over extracted features to regularize the search of adversarial examples . Generator-oriented Approaches . Compared with iterative optimization approaches , generatororiented attacks are more efficient ( i.e. , only need one inference ) to generate adversarial examples . In this branch , Baluja & Fischer ( 2017 ) propose an adversarial transformation network to modify the output of the classifier given the original input . Poursaeed et al . ( 2018 ) present trainable deep neural networks for producing both image-agnostic and image-dependent perturbations . Naseer et al . ( 2019 ) leverage datasets from other domain instead of ImageNet to train generator networks against pre-trained ImageNet models , and inference is performed on ImageNet domain with the aim of fooling black-box ImageNet model . They also attempt a practical black-box threat model ( from ChestX-ray to ImageNet ) , and the attack success rate can outperform the result of Gaussian noise . 3 TRANSFERABLE ADVERSARIAL EXAMPLES BEYOND IMAGENET . 3.1 PROBLEM FORMULATION . Given a target deep learning classifier ft ( · ) trained in a specific data distribution χt , we aim to craft a human-imperceptible perturbation for the benign image xt ∼ χt from the target domain with the only available knowledge of source ImageNet domain ( including pre-trained model fs ( · ) and data distribution χs ) . Formally , suppose we have a threat modelMθ∗ whose parameter θ∗ is solely derived from the source domain , our goal is to craft adversarial examples for xt from target domain so that they can fool the ft ( · ) successfully : ft ( Mθ∗ ( xt ) ) 6= ft ( xt ) s.t . ||Mθ∗ ( xt ) − xt||∞ ≤ , ( 1 ) where is the maximum perturbation to ensure xt is minimally changed . Intuitively , crafting adversarial examples for the black-box domain is very challenging . As shown in Table 1 and Figure 6 of Appendix , images from different domain vary greatly . 3.2 PRELIMINARY . Iterative/Single-step optimization methods ( Goodfellow et al. , 2015 ; Madry et al. , 2018 ; Zhao et al. , 2020 ; Gao et al. , 2021 ; Mao et al. , 2021 ; Li et al. , 2021 ) and generator-oriented methods ( Baluja & Fischer , 2017 ; Poursaeed et al. , 2018 ; Naseer et al. , 2019 ) are two popular branches for building the threat model . Since the attacker has the large-scale ImageNet training set at hand , there is no reason not to take full advantage of them . Therefore , in this paper , we adopt the generator-oriented framework which learns a transferable adversarial function via a generative model Gθ . Given that the threat model aims at crafting transferable adversarial examples for black-box domains , relying on the last layer with domain-specific loss functions ( e.g. , relativistic cross-entropy loss adopted by Naseer et al . ( 2019 ) ) is less effective since this might lead to overfitting to source domain . In contrast , the intermediate layers of the DNN presumably extract general features ( Yosinski et al. , 2014 ) which may share across different models . Hence , as a baseline for the new black-domain attack problem , our Beyond ImageNet Attack ( BIA ) turns to destroy the low-level features of the substitute model at a specific layer L to generate transferable adversarial examples according to existing literature ( Yosinski et al. , 2014 ; Zhou et al. , 2018 ; Inkawhich et al. , 2019 ) . As illustrated in Figure 1 , Gθ is learned to decrease the cosine similarity between adversarial example x′s and benign image xs ∈ RN×Hs×Ws ( sampled from χs ) to make the feature featureless : θ∗ = argmin θ Lcos ( fLs ( x′s ) , fLs ( xs ) ) . ( 2 ) In the inference phase , our generator Gθ∗ can directly craft adversarial examples for input images xt ∈ RN×Ht×Wt from the target domain : x′t = min ( xt + , max ( Gθ∗ ( xt ) , xt − ) . ( 3 ) The resulting adversarial examples x′t are depicted in Figure 8 of Appendix . Compared with CDA , our BIA is more effective in both source ( white-box ) and target ( black-box ) domains . Yet , as shown in Figure 2 , crafting more transferable adversarial examples still has some challenges : • Data Perspective : The distribution ( i.e. , mean and standard deviation ) of source domain is largely different from the target domain . For example , the standard deviation of ImageNet is about twice that of SVHN . • Model Perspective : Although some feature map of fLs ( · ) can capture the object ( ∈ χt ) for feature representation ( e.g. , the first feature map in Figure 2 ) , there are also some feature maps that are significantly biased ( e.g. , the second feature map in Figure 2 ) . To alleviate the concern of generating poor transferable adversarial examples that may arise from the above limitations , we propose two variants , equipped with random normalization ( RN ) module or domain-agnostic attention ( DA ) module , respectively . | This paper tackles the challenge of generating adversarial perturbation for a target model - with no access to the model, or the model's training data (i.e. target domain). Using a trained model and data from a source domain (ImageNet), the authors train a generator to craft perturbations which maximize the cosine distance between the intermediate features of clean and adversarial images. This generator is then assisted by two techniques - random normalization of the input image, and spatial attention on Intermediate-layer features (used for cosine distance). Experiments show that this method outperforms prior methods in black-box setting (no access to target domain or model) as well as white-box setting. | SP:72e008e0707baeec59f80fdffe5cecaec3d591a7 |
Beyond ImageNet Attack: Towards Crafting Adversarial Examples for Black-box Domains | 1 INTRODUCTION . Deep neural networks ( DNNs ) have achieved remarkable success in the image classification task in recent years . Nonetheless , advances in the field of adversarial machine learning ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ) make DNNs no longer reliable . By adding a well-designed perturbation on a benign image ( a.k.a adversarial attack ) , the resulting adversarial examples can easily fool state-of-the-art DNNs . To make the matter worse , the adversarial attack technique can even be applied in the physical world ( Sharif et al. , 2016 ; Kurakin et al. , 2017a ; Xu et al. , 2020 ; Duan et al. , 2021 ) , which inevitably raises concerns about the stability of deployed models . Therefore , exposing as many “ blind spots ” of DNNs as possible is a top priority . Generally , deployed models are mainly challenged with two threat models : white-box and blackbox . For white-box threat model ( Kurakin et al. , 2017b ; Moosavi-Dezfooli et al. , 2016 ; Carlini & Wagner , 2017 ; Shi et al. , 2019 ) , the attacker can obtain complete knowledge of the target model , such as the gradient for any input . However , deployed models are usually opaque to unauthorized users . In this scenario , prior black-box works ( Poursaeed et al. , 2018 ; Dong et al. , 2018 ; Xie et al. , 2019 ; Inkawhich et al. , 2020 ; Gao et al. , 2020b ; Wang et al. , 2021 ) mostly assume that the source data for training the target model is available and mainly explore the cross-model transferability among models trained in the same data distribution . Specifically , perturbations are crafted via accessible white-box model ( a.k.a substitute model ) , and resulting adversarial examples sometimes can fool other black-box models as well . Yet , these works still ignore a pivotal issue : A model owner is ∗Work done when Qilong Zhang interns at Alibaba Group , China †Corresponding author unlikely to leak the relevant information of the deployed model . To overcome this limitation , querybased black-box attacks ( Papernot et al. , 2016 ; Brendel et al. , 2018 ; Chen et al. , 2020 ; Li et al. , 2021 ) are proposed , which adjust adversarial examples just according to the output of the target model . However , the resource-intensive query budget is extremely costly and inevitably alerts the model owner . Therefore , we need a more “ practical ” black-box threat model to address this concern , i.e. , without any clue about the training data distribution as well as the pre-trained model based on it , and even querying is forbidden . Intuitively , this threat model is more challenging to build for attackers and more threatening to model owners . To the best of our knowledge , a recent work called CDA ( Naseer et al. , 2019 ) is the first to attempt such an attack . Specifically , it learns a transferable adversarial function via a generator network against a different domain ( training data and pre-trained model are all from ChestX-ray ( Wang et al. , 2017 ) domain ) . During inference , it directly crafts adversarial examples for benign ImageNet images to fool target ImageNet pre-trained models . However , its cross-domain transfer strength is still moderate . Besides , relying on small-scale datasets to train a generator may limit the generalization of the threat model . Considering that ImageNet is a large-scale dataset containing most of common categories in real life and there are various off-the-shelf pre-trained models , one can easily dig out much useful information to build a strong threat model . Therefore , in this paper , solely relying on the knowledge of the ImageNet domain , we introduce an effective Beyond ImageNet Attack ( BIA ) framework to enhance the cross-domain transferability of adversarial examples . To reflect the applicability of our approach , we consider eight different image classification tasks ( listed in Table 1 ) . Figure 1 illustrates an overview of our method . Particularly , we learn a flexible generator network Gθ against ImageNet domain . Instead of optimizing the domain-specific loss function like CDA , our method focuses on disrupting low-level features following previous literature to ensure the good transferability of our BIA . Furthermore , we propose two variants based on the vanilla BIA to narrow the gap between source and target domains . Specifically , from the data perspective , we propose a random normalization ( RN ) module to simulate different data distributions ; from the model perspective , we propose a domain-agnostic attention ( DA ) module to capture essential features for perturbing . In the inference phase , our Gθ accepts images of any domain as the input and crafts adversarial examples with one forward propagation . Extensive experiments demonstrate the effectiveness of our proposed methods . Towards the coarse-grained and fine-grained domains , we can outperform state-of-the-art approaches by up to 7.71 % and 25.91 % on average , respectively . Besides , our methods can also enhance the cross-model transferability in the source domain . 2 RELATED WORKS . Iterative Optimization Approaches . Under the black-box threat model , iterative attack methods are a popular branch , which usually adopt domain-specific loss or intermediate feature loss to craft adversarial examples . For the former , Madry et al . ( 2018 ) extend Goodfellow et al . ( 2015 ) to perform projected gradient descent from randomly chosen starting points inside -ball . Dong et al . ( 2018 ) introduce momentum term to stable the update direction . Xie et al . ( 2019 ) apply random transformations of the input at each iteration , thus mitigating overfitting . Gao et al . ( 2020a ) propose patch-wise perturbation to better cover the discriminative region . Wu et al . ( 2020a ) explore the security weakness of skip connections ( He et al. , 2016 ; Huang et al. , 2017 ) to boost attacks . Different from the methods mentioned above , intermediate feature-based methods focus on disrupting low-level features . For example , Zhou et al . ( 2018 ) maximize the Euclidean distance between the source image and target image in feature space and introduce regularization on perturbations to reduce variations . Inkawhich et al . ( 2019 ) make the source image close to the target image in feature space . Lu et al . ( 2020 ) propose a dispersion reduction attack to make the low-level features featureless . Naseer et al . ( 2020 ) design a self-supervised perturbation mechanism for enabling a transferable defense approach . Wu et al . ( 2020b ) compute model attention over extracted features to regularize the search of adversarial examples . Generator-oriented Approaches . Compared with iterative optimization approaches , generatororiented attacks are more efficient ( i.e. , only need one inference ) to generate adversarial examples . In this branch , Baluja & Fischer ( 2017 ) propose an adversarial transformation network to modify the output of the classifier given the original input . Poursaeed et al . ( 2018 ) present trainable deep neural networks for producing both image-agnostic and image-dependent perturbations . Naseer et al . ( 2019 ) leverage datasets from other domain instead of ImageNet to train generator networks against pre-trained ImageNet models , and inference is performed on ImageNet domain with the aim of fooling black-box ImageNet model . They also attempt a practical black-box threat model ( from ChestX-ray to ImageNet ) , and the attack success rate can outperform the result of Gaussian noise . 3 TRANSFERABLE ADVERSARIAL EXAMPLES BEYOND IMAGENET . 3.1 PROBLEM FORMULATION . Given a target deep learning classifier ft ( · ) trained in a specific data distribution χt , we aim to craft a human-imperceptible perturbation for the benign image xt ∼ χt from the target domain with the only available knowledge of source ImageNet domain ( including pre-trained model fs ( · ) and data distribution χs ) . Formally , suppose we have a threat modelMθ∗ whose parameter θ∗ is solely derived from the source domain , our goal is to craft adversarial examples for xt from target domain so that they can fool the ft ( · ) successfully : ft ( Mθ∗ ( xt ) ) 6= ft ( xt ) s.t . ||Mθ∗ ( xt ) − xt||∞ ≤ , ( 1 ) where is the maximum perturbation to ensure xt is minimally changed . Intuitively , crafting adversarial examples for the black-box domain is very challenging . As shown in Table 1 and Figure 6 of Appendix , images from different domain vary greatly . 3.2 PRELIMINARY . Iterative/Single-step optimization methods ( Goodfellow et al. , 2015 ; Madry et al. , 2018 ; Zhao et al. , 2020 ; Gao et al. , 2021 ; Mao et al. , 2021 ; Li et al. , 2021 ) and generator-oriented methods ( Baluja & Fischer , 2017 ; Poursaeed et al. , 2018 ; Naseer et al. , 2019 ) are two popular branches for building the threat model . Since the attacker has the large-scale ImageNet training set at hand , there is no reason not to take full advantage of them . Therefore , in this paper , we adopt the generator-oriented framework which learns a transferable adversarial function via a generative model Gθ . Given that the threat model aims at crafting transferable adversarial examples for black-box domains , relying on the last layer with domain-specific loss functions ( e.g. , relativistic cross-entropy loss adopted by Naseer et al . ( 2019 ) ) is less effective since this might lead to overfitting to source domain . In contrast , the intermediate layers of the DNN presumably extract general features ( Yosinski et al. , 2014 ) which may share across different models . Hence , as a baseline for the new black-domain attack problem , our Beyond ImageNet Attack ( BIA ) turns to destroy the low-level features of the substitute model at a specific layer L to generate transferable adversarial examples according to existing literature ( Yosinski et al. , 2014 ; Zhou et al. , 2018 ; Inkawhich et al. , 2019 ) . As illustrated in Figure 1 , Gθ is learned to decrease the cosine similarity between adversarial example x′s and benign image xs ∈ RN×Hs×Ws ( sampled from χs ) to make the feature featureless : θ∗ = argmin θ Lcos ( fLs ( x′s ) , fLs ( xs ) ) . ( 2 ) In the inference phase , our generator Gθ∗ can directly craft adversarial examples for input images xt ∈ RN×Ht×Wt from the target domain : x′t = min ( xt + , max ( Gθ∗ ( xt ) , xt − ) . ( 3 ) The resulting adversarial examples x′t are depicted in Figure 8 of Appendix . Compared with CDA , our BIA is more effective in both source ( white-box ) and target ( black-box ) domains . Yet , as shown in Figure 2 , crafting more transferable adversarial examples still has some challenges : • Data Perspective : The distribution ( i.e. , mean and standard deviation ) of source domain is largely different from the target domain . For example , the standard deviation of ImageNet is about twice that of SVHN . • Model Perspective : Although some feature map of fLs ( · ) can capture the object ( ∈ χt ) for feature representation ( e.g. , the first feature map in Figure 2 ) , there are also some feature maps that are significantly biased ( e.g. , the second feature map in Figure 2 ) . To alleviate the concern of generating poor transferable adversarial examples that may arise from the above limitations , we propose two variants , equipped with random normalization ( RN ) module or domain-agnostic attention ( DA ) module , respectively . | This work first identifies a more practical threat model for black-box transfer adversarial attack, where the target model's domain remains unknown, and the attacker's surrogate model may be trained in another domain. Then, the BIA attack is proposed to enhance transferability, whose key idea is to distort low-level features captured by DNN's intermediate layers instead of perturbing the domain-specific features in the output layer. Two modules, DA and RN, are further proposed to improve attack success rate. Experimental results demonstrate that BIA is more effective than existing methods. | SP:72e008e0707baeec59f80fdffe5cecaec3d591a7 |
Beyond ImageNet Attack: Towards Crafting Adversarial Examples for Black-box Domains | 1 INTRODUCTION . Deep neural networks ( DNNs ) have achieved remarkable success in the image classification task in recent years . Nonetheless , advances in the field of adversarial machine learning ( Szegedy et al. , 2014 ; Goodfellow et al. , 2015 ) make DNNs no longer reliable . By adding a well-designed perturbation on a benign image ( a.k.a adversarial attack ) , the resulting adversarial examples can easily fool state-of-the-art DNNs . To make the matter worse , the adversarial attack technique can even be applied in the physical world ( Sharif et al. , 2016 ; Kurakin et al. , 2017a ; Xu et al. , 2020 ; Duan et al. , 2021 ) , which inevitably raises concerns about the stability of deployed models . Therefore , exposing as many “ blind spots ” of DNNs as possible is a top priority . Generally , deployed models are mainly challenged with two threat models : white-box and blackbox . For white-box threat model ( Kurakin et al. , 2017b ; Moosavi-Dezfooli et al. , 2016 ; Carlini & Wagner , 2017 ; Shi et al. , 2019 ) , the attacker can obtain complete knowledge of the target model , such as the gradient for any input . However , deployed models are usually opaque to unauthorized users . In this scenario , prior black-box works ( Poursaeed et al. , 2018 ; Dong et al. , 2018 ; Xie et al. , 2019 ; Inkawhich et al. , 2020 ; Gao et al. , 2020b ; Wang et al. , 2021 ) mostly assume that the source data for training the target model is available and mainly explore the cross-model transferability among models trained in the same data distribution . Specifically , perturbations are crafted via accessible white-box model ( a.k.a substitute model ) , and resulting adversarial examples sometimes can fool other black-box models as well . Yet , these works still ignore a pivotal issue : A model owner is ∗Work done when Qilong Zhang interns at Alibaba Group , China †Corresponding author unlikely to leak the relevant information of the deployed model . To overcome this limitation , querybased black-box attacks ( Papernot et al. , 2016 ; Brendel et al. , 2018 ; Chen et al. , 2020 ; Li et al. , 2021 ) are proposed , which adjust adversarial examples just according to the output of the target model . However , the resource-intensive query budget is extremely costly and inevitably alerts the model owner . Therefore , we need a more “ practical ” black-box threat model to address this concern , i.e. , without any clue about the training data distribution as well as the pre-trained model based on it , and even querying is forbidden . Intuitively , this threat model is more challenging to build for attackers and more threatening to model owners . To the best of our knowledge , a recent work called CDA ( Naseer et al. , 2019 ) is the first to attempt such an attack . Specifically , it learns a transferable adversarial function via a generator network against a different domain ( training data and pre-trained model are all from ChestX-ray ( Wang et al. , 2017 ) domain ) . During inference , it directly crafts adversarial examples for benign ImageNet images to fool target ImageNet pre-trained models . However , its cross-domain transfer strength is still moderate . Besides , relying on small-scale datasets to train a generator may limit the generalization of the threat model . Considering that ImageNet is a large-scale dataset containing most of common categories in real life and there are various off-the-shelf pre-trained models , one can easily dig out much useful information to build a strong threat model . Therefore , in this paper , solely relying on the knowledge of the ImageNet domain , we introduce an effective Beyond ImageNet Attack ( BIA ) framework to enhance the cross-domain transferability of adversarial examples . To reflect the applicability of our approach , we consider eight different image classification tasks ( listed in Table 1 ) . Figure 1 illustrates an overview of our method . Particularly , we learn a flexible generator network Gθ against ImageNet domain . Instead of optimizing the domain-specific loss function like CDA , our method focuses on disrupting low-level features following previous literature to ensure the good transferability of our BIA . Furthermore , we propose two variants based on the vanilla BIA to narrow the gap between source and target domains . Specifically , from the data perspective , we propose a random normalization ( RN ) module to simulate different data distributions ; from the model perspective , we propose a domain-agnostic attention ( DA ) module to capture essential features for perturbing . In the inference phase , our Gθ accepts images of any domain as the input and crafts adversarial examples with one forward propagation . Extensive experiments demonstrate the effectiveness of our proposed methods . Towards the coarse-grained and fine-grained domains , we can outperform state-of-the-art approaches by up to 7.71 % and 25.91 % on average , respectively . Besides , our methods can also enhance the cross-model transferability in the source domain . 2 RELATED WORKS . Iterative Optimization Approaches . Under the black-box threat model , iterative attack methods are a popular branch , which usually adopt domain-specific loss or intermediate feature loss to craft adversarial examples . For the former , Madry et al . ( 2018 ) extend Goodfellow et al . ( 2015 ) to perform projected gradient descent from randomly chosen starting points inside -ball . Dong et al . ( 2018 ) introduce momentum term to stable the update direction . Xie et al . ( 2019 ) apply random transformations of the input at each iteration , thus mitigating overfitting . Gao et al . ( 2020a ) propose patch-wise perturbation to better cover the discriminative region . Wu et al . ( 2020a ) explore the security weakness of skip connections ( He et al. , 2016 ; Huang et al. , 2017 ) to boost attacks . Different from the methods mentioned above , intermediate feature-based methods focus on disrupting low-level features . For example , Zhou et al . ( 2018 ) maximize the Euclidean distance between the source image and target image in feature space and introduce regularization on perturbations to reduce variations . Inkawhich et al . ( 2019 ) make the source image close to the target image in feature space . Lu et al . ( 2020 ) propose a dispersion reduction attack to make the low-level features featureless . Naseer et al . ( 2020 ) design a self-supervised perturbation mechanism for enabling a transferable defense approach . Wu et al . ( 2020b ) compute model attention over extracted features to regularize the search of adversarial examples . Generator-oriented Approaches . Compared with iterative optimization approaches , generatororiented attacks are more efficient ( i.e. , only need one inference ) to generate adversarial examples . In this branch , Baluja & Fischer ( 2017 ) propose an adversarial transformation network to modify the output of the classifier given the original input . Poursaeed et al . ( 2018 ) present trainable deep neural networks for producing both image-agnostic and image-dependent perturbations . Naseer et al . ( 2019 ) leverage datasets from other domain instead of ImageNet to train generator networks against pre-trained ImageNet models , and inference is performed on ImageNet domain with the aim of fooling black-box ImageNet model . They also attempt a practical black-box threat model ( from ChestX-ray to ImageNet ) , and the attack success rate can outperform the result of Gaussian noise . 3 TRANSFERABLE ADVERSARIAL EXAMPLES BEYOND IMAGENET . 3.1 PROBLEM FORMULATION . Given a target deep learning classifier ft ( · ) trained in a specific data distribution χt , we aim to craft a human-imperceptible perturbation for the benign image xt ∼ χt from the target domain with the only available knowledge of source ImageNet domain ( including pre-trained model fs ( · ) and data distribution χs ) . Formally , suppose we have a threat modelMθ∗ whose parameter θ∗ is solely derived from the source domain , our goal is to craft adversarial examples for xt from target domain so that they can fool the ft ( · ) successfully : ft ( Mθ∗ ( xt ) ) 6= ft ( xt ) s.t . ||Mθ∗ ( xt ) − xt||∞ ≤ , ( 1 ) where is the maximum perturbation to ensure xt is minimally changed . Intuitively , crafting adversarial examples for the black-box domain is very challenging . As shown in Table 1 and Figure 6 of Appendix , images from different domain vary greatly . 3.2 PRELIMINARY . Iterative/Single-step optimization methods ( Goodfellow et al. , 2015 ; Madry et al. , 2018 ; Zhao et al. , 2020 ; Gao et al. , 2021 ; Mao et al. , 2021 ; Li et al. , 2021 ) and generator-oriented methods ( Baluja & Fischer , 2017 ; Poursaeed et al. , 2018 ; Naseer et al. , 2019 ) are two popular branches for building the threat model . Since the attacker has the large-scale ImageNet training set at hand , there is no reason not to take full advantage of them . Therefore , in this paper , we adopt the generator-oriented framework which learns a transferable adversarial function via a generative model Gθ . Given that the threat model aims at crafting transferable adversarial examples for black-box domains , relying on the last layer with domain-specific loss functions ( e.g. , relativistic cross-entropy loss adopted by Naseer et al . ( 2019 ) ) is less effective since this might lead to overfitting to source domain . In contrast , the intermediate layers of the DNN presumably extract general features ( Yosinski et al. , 2014 ) which may share across different models . Hence , as a baseline for the new black-domain attack problem , our Beyond ImageNet Attack ( BIA ) turns to destroy the low-level features of the substitute model at a specific layer L to generate transferable adversarial examples according to existing literature ( Yosinski et al. , 2014 ; Zhou et al. , 2018 ; Inkawhich et al. , 2019 ) . As illustrated in Figure 1 , Gθ is learned to decrease the cosine similarity between adversarial example x′s and benign image xs ∈ RN×Hs×Ws ( sampled from χs ) to make the feature featureless : θ∗ = argmin θ Lcos ( fLs ( x′s ) , fLs ( xs ) ) . ( 2 ) In the inference phase , our generator Gθ∗ can directly craft adversarial examples for input images xt ∈ RN×Ht×Wt from the target domain : x′t = min ( xt + , max ( Gθ∗ ( xt ) , xt − ) . ( 3 ) The resulting adversarial examples x′t are depicted in Figure 8 of Appendix . Compared with CDA , our BIA is more effective in both source ( white-box ) and target ( black-box ) domains . Yet , as shown in Figure 2 , crafting more transferable adversarial examples still has some challenges : • Data Perspective : The distribution ( i.e. , mean and standard deviation ) of source domain is largely different from the target domain . For example , the standard deviation of ImageNet is about twice that of SVHN . • Model Perspective : Although some feature map of fLs ( · ) can capture the object ( ∈ χt ) for feature representation ( e.g. , the first feature map in Figure 2 ) , there are also some feature maps that are significantly biased ( e.g. , the second feature map in Figure 2 ) . To alleviate the concern of generating poor transferable adversarial examples that may arise from the above limitations , we propose two variants , equipped with random normalization ( RN ) module or domain-agnostic attention ( DA ) module , respectively . | This paper focuses on the transferability of black-box domains. In real life, we do not know the relevant information of the deployed model and transfer attacks on black-box domains can better evaluate the vulnerability of deployed models. Therefore, Beyond ImageNet Attack (BIA) is proposed to investigate the transferability towards black-box domains (unknown classification tasks) with the only knowledge of the ImageNet domain. From the perspective of data and model, the authors propose random normalization (RN) module and domain-agnostic attention (DA) module to narrow the gap between the source and target domains. Finally, BIA achieves state-of-the-art performance in black-box domains settings. | SP:72e008e0707baeec59f80fdffe5cecaec3d591a7 |
THOMAS: Trajectory Heatmap Output with learned Multi-Agent Sampling | In this paper , we propose THOMAS , a joint multi-agent trajectory prediction framework allowing for an efficient and consistent prediction of multi-agent multimodal trajectories . We present a unified model architecture for simultaneous agent future heatmap estimation , in which we leverage hierarchical and sparse image generation for fast and memory-efficient inference . We propose a learnable trajectory recombination model that takes as input a set of predicted trajectories for each agent and outputs its consistent reordered recombination . This recombination module is able to realign the initially independent modalities so that they do no collide and are coherent with each other . We report our results on the Interaction multi-agent prediction challenge and rank 1st on the online test leaderboard . 1 INTRODUCTION . Motion forecasting is an essential step in the pipeline of an autonomous driving vehicle , transforming perception data into future prediction which are then leveraged to plan the future moves of the autonomous cars . The self-driving stacks needs to predict the future trajectories for all the neighbor agents , in a fast and coherent way . The interactivity between agents plays an important role for accurate trajectory prediction . Agents need to be aware of their neighbors in order to adapt their speed , yield right of way and merge in neighbor lanes . To do so , different interaction mechanisms have been developed , such as social pooling ( Alahi et al. , 2016 ; Lee et al. , 2017 ; Deo & Trivedi , 2018 ) , graphs ( Salzmann et al. , 2020 ; Zeng et al. , 2021 ) or attention ( Mercat et al. , 2020 ; Messaoud et al. , 2020 ; Luo et al. , 2020 ; Gao et al. , 2020 ; Liang et al. , 2020 ) , benefiting from the progress of powerful transformer architectures ( Li et al. , 2020 ; Yuan et al. , 2021 ; Girgis et al. , 2021 ; Ngiam et al. , 2021 ) . These mechanisms allow agents to look at and share features with neighbors and to take them into account in their own predictions . Multi-modality is another important aspect of the possible future trajectories . A car can indeed chose to turn right or left , or decide to realise a certain maneuver in various ways . Uncertainty modeled as variance of Gaussians is insufficient to model these multiple cases , as it can only represent a continuous spread and can not show multiple discrete possibilities . Therefore , current state-of-the-art produces not one but K possible trajectories for each agent predicted , and most recent benchmarks ( Caesar et al. , 2020 ; Chang et al. , 2019 ; Zhan et al. , 2019 ; Ettinger et al. , 2021 ) include multimodality in their metrics , taking only the minimum error over a predicted set of K trajectories . However , up until very recently and the opening of multi-agent joint interaction challenges ( Ettinger et al. , 2021 ; Zhan et al. , 2021 ) , no motion forecasting prediction datasets were taking into account the coherence of modalities between different agents predicted at the same time . As a result , the most probable predicted modality of a given agent could crash with the most probable modality of another agent . Our THOMAS model encodes the past trajectories of all the agents present in the scene , as well as the HD-Map lanelet graph , and then predicts for each agent a sparse heatmap representing the future probability distribution at a fixed timestep in the future , using hierarchical refinement for very efficient decoding . A deterministic sampling algorithm then iteratively selects the best K trajectory endpoints according to the heatmap for each agent , in order to represent a wide and diverse spec- trum of modalities . Given this wide spectrum of endpoints , a recombination module takes care of addressing consistency in the scene among agents . Our contributions are summarized as follow : • We propose a hierarchical heatmap decoder allowing for unconstrained heatmap generation with optimized computational costs , enabling efficient simultaneous multi-agent prediction . • We design a novel recombination model able to recombine the sampled endpoints to obtain scene-consistent trajectories across the agents . 2 RELATED WORK . Learning-based models have quickly overtaken physics-based methods for trajectory prediction for several reasons . First , the sequential nature of trajectories is a logical application for recurrent architectures ( Alahi et al. , 2016 ; Altché & de La Fortelle , 2017 ; Lee et al. , 2017 ; Mercat et al. , 2020 ; Khandelwal et al. , 2020 ) . Then , benefiting from the latest progresses in computer vision , convolutional layers can easily be applied to bird-view rasters of the map context ( Lee et al. , 2017 ; Tang & Salakhutdinov , 2019 ; Cui et al. , 2019 ; Hong et al. , 2019 ; Salzmann et al. , 2020 ; Chai et al. , 2020 ; Gilles et al. , 2021b ) . A more compact representation closer to the trajectory space can encode surrounding HD-Maps ( usually formalized as connected lanelets ) , using Graph Neural Networks ( Gao et al. , 2020 ; Liang et al. , 2020 ; Zeng et al. , 2021 ; Gilles et al. , 2021a ) . Finally , some pointbased approaches ( Ye et al. , 2021 ) can be applied in a broader way to trajectory prediction , as both lanes and trajectories can be considered as ordered set of points . Multi-modality in prediction can be obtained through a multiple prediction head in the model ( Cui et al. , 2019 ; Liang et al. , 2020 ; Ngiam et al. , 2021 ; Deo et al. , 2021 ) . However , some methods rather adopt a candidate-based approach where potential endpoints are obtained either from anchor trajectories through clustering ( Chai et al. , 2020 ; Phan-Minh et al. , 2020 ) or from a model-based generator ( Song et al. , 2021 ) . Other approaches use a broader set of candidates from the context graph ( Zhang et al. , 2020 ; Zhao et al. , 2020 ; Zeng et al. , 2021 ; Kim et al. , 2021 ) or a dense grid around the target agent ( Deo & Trivedi , 2020 ; Gu et al. , 2021 ; Gilles et al. , 2021b ; a ) . Another family of approaches use variational inference to generate diverse predictions through latent variables ( Lee et al. , 2017 ; Rhinehart et al. , 2018 ; Tang & Salakhutdinov , 2019 ; Casas et al. , 2020 ) or GAN ( Gupta et al. , 2018 ; Rhinehart et al. , 2018 ; Sadeghian et al. , 2019 ) but the sampling of these trajectories is stochastic and does not provide any probability value for each sample . While very little work has directly tackled multi-agent prediction and evaluation so far , multiple methods hint at the ability to predict multiple agents at the same time ( Liang et al. , 2020 ; Zeng et al. , 2021 ) even if they then focus on a more single-agent oriented framework . Other works ( Alahi et al. , 2016 ; Tang & Salakhutdinov , 2019 ; Rhinehart et al. , 2019 ; Girgis et al. , 2021 ) use autoregressive roll-outs to condition the future step of an agent on the previous steps of all the other agents . SceneTransformer ( Ngiam et al. , 2021 ) repeats each agent features across possible modalities , and performs self-attention operations inside each modality before using a loss computed jointly among agents to train a model and evaluate on the WOMD dataset ( Ettinger et al. , 2021 ) interaction track . ILVM ( Casas et al. , 2020 ) uses scene latent representations conditioned on all agents to generate scene-consistent samples , but its variational inference does not provide a confidence score for each modality , hence LookOut ( Cui et al. , 2021 ) proposes a scenario scoring function and a diverse sampler to improve sample efficiency . AIR2 ( Wu & Wu , 2021 ) extends Multipath ( Chai et al. , 2020 ) and produces a cross-distribution for two agents along all possible trajectory anchors , but it scales exponentially with the number of agents , making impractical for a real-time implementation that could encounter more than 10 agents at the same time . The approach most related to this paper is GOHOME ( Gilles et al. , 2021a ) , which uses a similar graph encoder and then leverage lane rasters to generate a probability heatmap in a sparse manner . However , this lane-generated heatmap remains constrained to the drivable area and to fixed lanewidths . Another close approach is the one of DenseTNT ( Gu et al. , 2021 ) , which also uses attention to a grid of points in order to obtain a dense prediction , but their grid also remains constrained to a neighborhood of the drivable area . Finally , none of these previous methods considers the problems of scene consistency for multi-agent prediction . 3 METHOD . Our goal is to predict the future T timesteps ofA agents using their past history made ofH timesteps and the HD-Map context . Similar to recent works ( Zhao et al. , 2020 ; Zeng et al. , 2021 ; Gu et al. , 2021 ) , we will divide the problem into goal-based prediction followed by full trajectory reconstruction . Our prediction pipeline is displayed in Fig . 1 . We first encode each agent trajectory and the HD-Map context graph into a common representation . We then decode a future probability heatmap for each agent in the scene , which we sample heuristically to maximize coverage . Finally , we recombine the sampled endpoints into scene-consistent modalities across agents and build the full trajectories for each agent . Our pipeline shares the same graph encoder , sampling algorithm and full trajectory generation as GOHOME ( Gilles et al. , 2021a ) , but uses a novel efficient hierarchical heatmap process that enables to scale to simultaneous multi-agent prediction . Furthermore , we add a novel scene-consistency module that recombines the marginal outputs into a joint prediction . 3.1 MODEL BACKBONE . 3.1.1 GRAPH ENCODER . We use the same encoder as the GOHOME model ( Gilles et al. , 2021a ) . The agent trajectories are encoded though TrajEncoder using a 1D CNN followed by a UGRU recurrent layer ( Rozenberg et al. , 2021 ) , and the HD-Map is encoded as a lanelet graph using a GNN GraphEncoder made of graph convolutions . We then run cross-attention Lanes2Agents to add context information to the agent features , followed by self-attention Agents2Agents to observe interaction between agents . The final result is an encoding Fa for each agent , where history , context and interactions have been summarized . This encoding Fa is used in the next decoder operations and is also stored to be potentially used in modality recombination described in Sec . 3.2 . The resulting architecture of these encoding operations is illustrated in the first half of Fig . 2 . 3.1.2 HIERARCHICAL GRID DECODER . Our aim here is to decode each agent encoding into a heatmap representing its future probability distribution at prediction horizon T . Since we create this heatmap for each agent in the scene , the decoding process has to be fast so that it can be applied to a great number of agents in parallel . We use hierarchical predictions at various levels of resolutions so that the decoder has the possibility of predicting over the full surroundings of the agent but learns to refine with more precision only in places where the agent will end up with high probability . This hierarchical process is illustrated in Fig . 3 . Starting from an initial full dense grid probability at low resolutionR0×R0 by pixels , we iteratively refine the resolution by a fixed factor f until we reach the desired final resolution Rfinal × Rfinal . At each iteration i , we select only the Ni highest ranking grid points from the previous iteration , and upsample only these points to the Ri × Ri = Ri−1/f × Ri−1/f . At each step , the grid points features are computed by a 2-layer MLP applied on the point coordinates , these features are then concatenated to the agent encoding and passed through a linear layer , and finally enriched by a 2- layer cross-attention on the graph lane features , before applying a linear layer with sigmoid to get the probability . For a given W output range , this hierarchical process allows the model to only operate on W/R0 × W/R0 + ∑ iNi × f2 grid points instead of the W/Rfinal × W/Rfinal available . In practice , for a final output range of 192 meters with desired Rfinal = 0.5m resolution , we start with an initial resolution of R0 = 8m and use two iterations of ( N1 , N2 ) = ( 16 , 64 ) points each and an upscaling factor f = 4 . This way , we compute only 1856 grid points from the 147 456 available , with no performance loss . The heatmap is trained on each resolution level using as pixel-wise focal loss as in Gilles et al . ( 2021b ) , detailed in Appendix A.3 , with the ground truth being a Gaussian centered at the target agent future position . | This paper addresses the task of multi-agent multi-modal trajectory prediction in the context of autonomous driving. Inspired by previous works, the problem is divided into first estimate the goal (end-points), and then re-construct the full trajectory. For end-point estimation, this paper proposes hierarchical iterative refinement from a probability heatmap, with collision-aware greedy sampling to generate collision-free multi-agent trajectories. To further produce scene-consistent multi-agent multi-modal trajectories, the paper also proposes a modality combination ranking module that re-orders the modality of each agent. The method is validated on the Interaction multi-agent prediction challenge and ranks 2nd on the multi-agent track (the 1st-rank entry was submitted 9 days after submission) and 1st on the conditional multi-agent track. | SP:f9b8beceeac894bbe86cf0d2c93f844454ef7cd3 |
THOMAS: Trajectory Heatmap Output with learned Multi-Agent Sampling | In this paper , we propose THOMAS , a joint multi-agent trajectory prediction framework allowing for an efficient and consistent prediction of multi-agent multimodal trajectories . We present a unified model architecture for simultaneous agent future heatmap estimation , in which we leverage hierarchical and sparse image generation for fast and memory-efficient inference . We propose a learnable trajectory recombination model that takes as input a set of predicted trajectories for each agent and outputs its consistent reordered recombination . This recombination module is able to realign the initially independent modalities so that they do no collide and are coherent with each other . We report our results on the Interaction multi-agent prediction challenge and rank 1st on the online test leaderboard . 1 INTRODUCTION . Motion forecasting is an essential step in the pipeline of an autonomous driving vehicle , transforming perception data into future prediction which are then leveraged to plan the future moves of the autonomous cars . The self-driving stacks needs to predict the future trajectories for all the neighbor agents , in a fast and coherent way . The interactivity between agents plays an important role for accurate trajectory prediction . Agents need to be aware of their neighbors in order to adapt their speed , yield right of way and merge in neighbor lanes . To do so , different interaction mechanisms have been developed , such as social pooling ( Alahi et al. , 2016 ; Lee et al. , 2017 ; Deo & Trivedi , 2018 ) , graphs ( Salzmann et al. , 2020 ; Zeng et al. , 2021 ) or attention ( Mercat et al. , 2020 ; Messaoud et al. , 2020 ; Luo et al. , 2020 ; Gao et al. , 2020 ; Liang et al. , 2020 ) , benefiting from the progress of powerful transformer architectures ( Li et al. , 2020 ; Yuan et al. , 2021 ; Girgis et al. , 2021 ; Ngiam et al. , 2021 ) . These mechanisms allow agents to look at and share features with neighbors and to take them into account in their own predictions . Multi-modality is another important aspect of the possible future trajectories . A car can indeed chose to turn right or left , or decide to realise a certain maneuver in various ways . Uncertainty modeled as variance of Gaussians is insufficient to model these multiple cases , as it can only represent a continuous spread and can not show multiple discrete possibilities . Therefore , current state-of-the-art produces not one but K possible trajectories for each agent predicted , and most recent benchmarks ( Caesar et al. , 2020 ; Chang et al. , 2019 ; Zhan et al. , 2019 ; Ettinger et al. , 2021 ) include multimodality in their metrics , taking only the minimum error over a predicted set of K trajectories . However , up until very recently and the opening of multi-agent joint interaction challenges ( Ettinger et al. , 2021 ; Zhan et al. , 2021 ) , no motion forecasting prediction datasets were taking into account the coherence of modalities between different agents predicted at the same time . As a result , the most probable predicted modality of a given agent could crash with the most probable modality of another agent . Our THOMAS model encodes the past trajectories of all the agents present in the scene , as well as the HD-Map lanelet graph , and then predicts for each agent a sparse heatmap representing the future probability distribution at a fixed timestep in the future , using hierarchical refinement for very efficient decoding . A deterministic sampling algorithm then iteratively selects the best K trajectory endpoints according to the heatmap for each agent , in order to represent a wide and diverse spec- trum of modalities . Given this wide spectrum of endpoints , a recombination module takes care of addressing consistency in the scene among agents . Our contributions are summarized as follow : • We propose a hierarchical heatmap decoder allowing for unconstrained heatmap generation with optimized computational costs , enabling efficient simultaneous multi-agent prediction . • We design a novel recombination model able to recombine the sampled endpoints to obtain scene-consistent trajectories across the agents . 2 RELATED WORK . Learning-based models have quickly overtaken physics-based methods for trajectory prediction for several reasons . First , the sequential nature of trajectories is a logical application for recurrent architectures ( Alahi et al. , 2016 ; Altché & de La Fortelle , 2017 ; Lee et al. , 2017 ; Mercat et al. , 2020 ; Khandelwal et al. , 2020 ) . Then , benefiting from the latest progresses in computer vision , convolutional layers can easily be applied to bird-view rasters of the map context ( Lee et al. , 2017 ; Tang & Salakhutdinov , 2019 ; Cui et al. , 2019 ; Hong et al. , 2019 ; Salzmann et al. , 2020 ; Chai et al. , 2020 ; Gilles et al. , 2021b ) . A more compact representation closer to the trajectory space can encode surrounding HD-Maps ( usually formalized as connected lanelets ) , using Graph Neural Networks ( Gao et al. , 2020 ; Liang et al. , 2020 ; Zeng et al. , 2021 ; Gilles et al. , 2021a ) . Finally , some pointbased approaches ( Ye et al. , 2021 ) can be applied in a broader way to trajectory prediction , as both lanes and trajectories can be considered as ordered set of points . Multi-modality in prediction can be obtained through a multiple prediction head in the model ( Cui et al. , 2019 ; Liang et al. , 2020 ; Ngiam et al. , 2021 ; Deo et al. , 2021 ) . However , some methods rather adopt a candidate-based approach where potential endpoints are obtained either from anchor trajectories through clustering ( Chai et al. , 2020 ; Phan-Minh et al. , 2020 ) or from a model-based generator ( Song et al. , 2021 ) . Other approaches use a broader set of candidates from the context graph ( Zhang et al. , 2020 ; Zhao et al. , 2020 ; Zeng et al. , 2021 ; Kim et al. , 2021 ) or a dense grid around the target agent ( Deo & Trivedi , 2020 ; Gu et al. , 2021 ; Gilles et al. , 2021b ; a ) . Another family of approaches use variational inference to generate diverse predictions through latent variables ( Lee et al. , 2017 ; Rhinehart et al. , 2018 ; Tang & Salakhutdinov , 2019 ; Casas et al. , 2020 ) or GAN ( Gupta et al. , 2018 ; Rhinehart et al. , 2018 ; Sadeghian et al. , 2019 ) but the sampling of these trajectories is stochastic and does not provide any probability value for each sample . While very little work has directly tackled multi-agent prediction and evaluation so far , multiple methods hint at the ability to predict multiple agents at the same time ( Liang et al. , 2020 ; Zeng et al. , 2021 ) even if they then focus on a more single-agent oriented framework . Other works ( Alahi et al. , 2016 ; Tang & Salakhutdinov , 2019 ; Rhinehart et al. , 2019 ; Girgis et al. , 2021 ) use autoregressive roll-outs to condition the future step of an agent on the previous steps of all the other agents . SceneTransformer ( Ngiam et al. , 2021 ) repeats each agent features across possible modalities , and performs self-attention operations inside each modality before using a loss computed jointly among agents to train a model and evaluate on the WOMD dataset ( Ettinger et al. , 2021 ) interaction track . ILVM ( Casas et al. , 2020 ) uses scene latent representations conditioned on all agents to generate scene-consistent samples , but its variational inference does not provide a confidence score for each modality , hence LookOut ( Cui et al. , 2021 ) proposes a scenario scoring function and a diverse sampler to improve sample efficiency . AIR2 ( Wu & Wu , 2021 ) extends Multipath ( Chai et al. , 2020 ) and produces a cross-distribution for two agents along all possible trajectory anchors , but it scales exponentially with the number of agents , making impractical for a real-time implementation that could encounter more than 10 agents at the same time . The approach most related to this paper is GOHOME ( Gilles et al. , 2021a ) , which uses a similar graph encoder and then leverage lane rasters to generate a probability heatmap in a sparse manner . However , this lane-generated heatmap remains constrained to the drivable area and to fixed lanewidths . Another close approach is the one of DenseTNT ( Gu et al. , 2021 ) , which also uses attention to a grid of points in order to obtain a dense prediction , but their grid also remains constrained to a neighborhood of the drivable area . Finally , none of these previous methods considers the problems of scene consistency for multi-agent prediction . 3 METHOD . Our goal is to predict the future T timesteps ofA agents using their past history made ofH timesteps and the HD-Map context . Similar to recent works ( Zhao et al. , 2020 ; Zeng et al. , 2021 ; Gu et al. , 2021 ) , we will divide the problem into goal-based prediction followed by full trajectory reconstruction . Our prediction pipeline is displayed in Fig . 1 . We first encode each agent trajectory and the HD-Map context graph into a common representation . We then decode a future probability heatmap for each agent in the scene , which we sample heuristically to maximize coverage . Finally , we recombine the sampled endpoints into scene-consistent modalities across agents and build the full trajectories for each agent . Our pipeline shares the same graph encoder , sampling algorithm and full trajectory generation as GOHOME ( Gilles et al. , 2021a ) , but uses a novel efficient hierarchical heatmap process that enables to scale to simultaneous multi-agent prediction . Furthermore , we add a novel scene-consistency module that recombines the marginal outputs into a joint prediction . 3.1 MODEL BACKBONE . 3.1.1 GRAPH ENCODER . We use the same encoder as the GOHOME model ( Gilles et al. , 2021a ) . The agent trajectories are encoded though TrajEncoder using a 1D CNN followed by a UGRU recurrent layer ( Rozenberg et al. , 2021 ) , and the HD-Map is encoded as a lanelet graph using a GNN GraphEncoder made of graph convolutions . We then run cross-attention Lanes2Agents to add context information to the agent features , followed by self-attention Agents2Agents to observe interaction between agents . The final result is an encoding Fa for each agent , where history , context and interactions have been summarized . This encoding Fa is used in the next decoder operations and is also stored to be potentially used in modality recombination described in Sec . 3.2 . The resulting architecture of these encoding operations is illustrated in the first half of Fig . 2 . 3.1.2 HIERARCHICAL GRID DECODER . Our aim here is to decode each agent encoding into a heatmap representing its future probability distribution at prediction horizon T . Since we create this heatmap for each agent in the scene , the decoding process has to be fast so that it can be applied to a great number of agents in parallel . We use hierarchical predictions at various levels of resolutions so that the decoder has the possibility of predicting over the full surroundings of the agent but learns to refine with more precision only in places where the agent will end up with high probability . This hierarchical process is illustrated in Fig . 3 . Starting from an initial full dense grid probability at low resolutionR0×R0 by pixels , we iteratively refine the resolution by a fixed factor f until we reach the desired final resolution Rfinal × Rfinal . At each iteration i , we select only the Ni highest ranking grid points from the previous iteration , and upsample only these points to the Ri × Ri = Ri−1/f × Ri−1/f . At each step , the grid points features are computed by a 2-layer MLP applied on the point coordinates , these features are then concatenated to the agent encoding and passed through a linear layer , and finally enriched by a 2- layer cross-attention on the graph lane features , before applying a linear layer with sigmoid to get the probability . For a given W output range , this hierarchical process allows the model to only operate on W/R0 × W/R0 + ∑ iNi × f2 grid points instead of the W/Rfinal × W/Rfinal available . In practice , for a final output range of 192 meters with desired Rfinal = 0.5m resolution , we start with an initial resolution of R0 = 8m and use two iterations of ( N1 , N2 ) = ( 16 , 64 ) points each and an upscaling factor f = 4 . This way , we compute only 1856 grid points from the 147 456 available , with no performance loss . The heatmap is trained on each resolution level using as pixel-wise focal loss as in Gilles et al . ( 2021b ) , detailed in Appendix A.3 , with the ground truth being a Gaussian centered at the target agent future position . | The paper proposes a joint trajectory prediction framework. This means that the model produces multi-modal future predictions at the scene level (i.e., for all actors jointly). There has only been very recent research in this area despite its importance to autonomy and motion planning in particular, so the relevance of the paper to the broader motion forecasting field is high. I identify two main contributions in the paper: (i) hierarchical and sparse heatmap output representation and (ii) a recombination method that is pluggable in any multi-modal marginal prediction model. | SP:f9b8beceeac894bbe86cf0d2c93f844454ef7cd3 |
THOMAS: Trajectory Heatmap Output with learned Multi-Agent Sampling | In this paper , we propose THOMAS , a joint multi-agent trajectory prediction framework allowing for an efficient and consistent prediction of multi-agent multimodal trajectories . We present a unified model architecture for simultaneous agent future heatmap estimation , in which we leverage hierarchical and sparse image generation for fast and memory-efficient inference . We propose a learnable trajectory recombination model that takes as input a set of predicted trajectories for each agent and outputs its consistent reordered recombination . This recombination module is able to realign the initially independent modalities so that they do no collide and are coherent with each other . We report our results on the Interaction multi-agent prediction challenge and rank 1st on the online test leaderboard . 1 INTRODUCTION . Motion forecasting is an essential step in the pipeline of an autonomous driving vehicle , transforming perception data into future prediction which are then leveraged to plan the future moves of the autonomous cars . The self-driving stacks needs to predict the future trajectories for all the neighbor agents , in a fast and coherent way . The interactivity between agents plays an important role for accurate trajectory prediction . Agents need to be aware of their neighbors in order to adapt their speed , yield right of way and merge in neighbor lanes . To do so , different interaction mechanisms have been developed , such as social pooling ( Alahi et al. , 2016 ; Lee et al. , 2017 ; Deo & Trivedi , 2018 ) , graphs ( Salzmann et al. , 2020 ; Zeng et al. , 2021 ) or attention ( Mercat et al. , 2020 ; Messaoud et al. , 2020 ; Luo et al. , 2020 ; Gao et al. , 2020 ; Liang et al. , 2020 ) , benefiting from the progress of powerful transformer architectures ( Li et al. , 2020 ; Yuan et al. , 2021 ; Girgis et al. , 2021 ; Ngiam et al. , 2021 ) . These mechanisms allow agents to look at and share features with neighbors and to take them into account in their own predictions . Multi-modality is another important aspect of the possible future trajectories . A car can indeed chose to turn right or left , or decide to realise a certain maneuver in various ways . Uncertainty modeled as variance of Gaussians is insufficient to model these multiple cases , as it can only represent a continuous spread and can not show multiple discrete possibilities . Therefore , current state-of-the-art produces not one but K possible trajectories for each agent predicted , and most recent benchmarks ( Caesar et al. , 2020 ; Chang et al. , 2019 ; Zhan et al. , 2019 ; Ettinger et al. , 2021 ) include multimodality in their metrics , taking only the minimum error over a predicted set of K trajectories . However , up until very recently and the opening of multi-agent joint interaction challenges ( Ettinger et al. , 2021 ; Zhan et al. , 2021 ) , no motion forecasting prediction datasets were taking into account the coherence of modalities between different agents predicted at the same time . As a result , the most probable predicted modality of a given agent could crash with the most probable modality of another agent . Our THOMAS model encodes the past trajectories of all the agents present in the scene , as well as the HD-Map lanelet graph , and then predicts for each agent a sparse heatmap representing the future probability distribution at a fixed timestep in the future , using hierarchical refinement for very efficient decoding . A deterministic sampling algorithm then iteratively selects the best K trajectory endpoints according to the heatmap for each agent , in order to represent a wide and diverse spec- trum of modalities . Given this wide spectrum of endpoints , a recombination module takes care of addressing consistency in the scene among agents . Our contributions are summarized as follow : • We propose a hierarchical heatmap decoder allowing for unconstrained heatmap generation with optimized computational costs , enabling efficient simultaneous multi-agent prediction . • We design a novel recombination model able to recombine the sampled endpoints to obtain scene-consistent trajectories across the agents . 2 RELATED WORK . Learning-based models have quickly overtaken physics-based methods for trajectory prediction for several reasons . First , the sequential nature of trajectories is a logical application for recurrent architectures ( Alahi et al. , 2016 ; Altché & de La Fortelle , 2017 ; Lee et al. , 2017 ; Mercat et al. , 2020 ; Khandelwal et al. , 2020 ) . Then , benefiting from the latest progresses in computer vision , convolutional layers can easily be applied to bird-view rasters of the map context ( Lee et al. , 2017 ; Tang & Salakhutdinov , 2019 ; Cui et al. , 2019 ; Hong et al. , 2019 ; Salzmann et al. , 2020 ; Chai et al. , 2020 ; Gilles et al. , 2021b ) . A more compact representation closer to the trajectory space can encode surrounding HD-Maps ( usually formalized as connected lanelets ) , using Graph Neural Networks ( Gao et al. , 2020 ; Liang et al. , 2020 ; Zeng et al. , 2021 ; Gilles et al. , 2021a ) . Finally , some pointbased approaches ( Ye et al. , 2021 ) can be applied in a broader way to trajectory prediction , as both lanes and trajectories can be considered as ordered set of points . Multi-modality in prediction can be obtained through a multiple prediction head in the model ( Cui et al. , 2019 ; Liang et al. , 2020 ; Ngiam et al. , 2021 ; Deo et al. , 2021 ) . However , some methods rather adopt a candidate-based approach where potential endpoints are obtained either from anchor trajectories through clustering ( Chai et al. , 2020 ; Phan-Minh et al. , 2020 ) or from a model-based generator ( Song et al. , 2021 ) . Other approaches use a broader set of candidates from the context graph ( Zhang et al. , 2020 ; Zhao et al. , 2020 ; Zeng et al. , 2021 ; Kim et al. , 2021 ) or a dense grid around the target agent ( Deo & Trivedi , 2020 ; Gu et al. , 2021 ; Gilles et al. , 2021b ; a ) . Another family of approaches use variational inference to generate diverse predictions through latent variables ( Lee et al. , 2017 ; Rhinehart et al. , 2018 ; Tang & Salakhutdinov , 2019 ; Casas et al. , 2020 ) or GAN ( Gupta et al. , 2018 ; Rhinehart et al. , 2018 ; Sadeghian et al. , 2019 ) but the sampling of these trajectories is stochastic and does not provide any probability value for each sample . While very little work has directly tackled multi-agent prediction and evaluation so far , multiple methods hint at the ability to predict multiple agents at the same time ( Liang et al. , 2020 ; Zeng et al. , 2021 ) even if they then focus on a more single-agent oriented framework . Other works ( Alahi et al. , 2016 ; Tang & Salakhutdinov , 2019 ; Rhinehart et al. , 2019 ; Girgis et al. , 2021 ) use autoregressive roll-outs to condition the future step of an agent on the previous steps of all the other agents . SceneTransformer ( Ngiam et al. , 2021 ) repeats each agent features across possible modalities , and performs self-attention operations inside each modality before using a loss computed jointly among agents to train a model and evaluate on the WOMD dataset ( Ettinger et al. , 2021 ) interaction track . ILVM ( Casas et al. , 2020 ) uses scene latent representations conditioned on all agents to generate scene-consistent samples , but its variational inference does not provide a confidence score for each modality , hence LookOut ( Cui et al. , 2021 ) proposes a scenario scoring function and a diverse sampler to improve sample efficiency . AIR2 ( Wu & Wu , 2021 ) extends Multipath ( Chai et al. , 2020 ) and produces a cross-distribution for two agents along all possible trajectory anchors , but it scales exponentially with the number of agents , making impractical for a real-time implementation that could encounter more than 10 agents at the same time . The approach most related to this paper is GOHOME ( Gilles et al. , 2021a ) , which uses a similar graph encoder and then leverage lane rasters to generate a probability heatmap in a sparse manner . However , this lane-generated heatmap remains constrained to the drivable area and to fixed lanewidths . Another close approach is the one of DenseTNT ( Gu et al. , 2021 ) , which also uses attention to a grid of points in order to obtain a dense prediction , but their grid also remains constrained to a neighborhood of the drivable area . Finally , none of these previous methods considers the problems of scene consistency for multi-agent prediction . 3 METHOD . Our goal is to predict the future T timesteps ofA agents using their past history made ofH timesteps and the HD-Map context . Similar to recent works ( Zhao et al. , 2020 ; Zeng et al. , 2021 ; Gu et al. , 2021 ) , we will divide the problem into goal-based prediction followed by full trajectory reconstruction . Our prediction pipeline is displayed in Fig . 1 . We first encode each agent trajectory and the HD-Map context graph into a common representation . We then decode a future probability heatmap for each agent in the scene , which we sample heuristically to maximize coverage . Finally , we recombine the sampled endpoints into scene-consistent modalities across agents and build the full trajectories for each agent . Our pipeline shares the same graph encoder , sampling algorithm and full trajectory generation as GOHOME ( Gilles et al. , 2021a ) , but uses a novel efficient hierarchical heatmap process that enables to scale to simultaneous multi-agent prediction . Furthermore , we add a novel scene-consistency module that recombines the marginal outputs into a joint prediction . 3.1 MODEL BACKBONE . 3.1.1 GRAPH ENCODER . We use the same encoder as the GOHOME model ( Gilles et al. , 2021a ) . The agent trajectories are encoded though TrajEncoder using a 1D CNN followed by a UGRU recurrent layer ( Rozenberg et al. , 2021 ) , and the HD-Map is encoded as a lanelet graph using a GNN GraphEncoder made of graph convolutions . We then run cross-attention Lanes2Agents to add context information to the agent features , followed by self-attention Agents2Agents to observe interaction between agents . The final result is an encoding Fa for each agent , where history , context and interactions have been summarized . This encoding Fa is used in the next decoder operations and is also stored to be potentially used in modality recombination described in Sec . 3.2 . The resulting architecture of these encoding operations is illustrated in the first half of Fig . 2 . 3.1.2 HIERARCHICAL GRID DECODER . Our aim here is to decode each agent encoding into a heatmap representing its future probability distribution at prediction horizon T . Since we create this heatmap for each agent in the scene , the decoding process has to be fast so that it can be applied to a great number of agents in parallel . We use hierarchical predictions at various levels of resolutions so that the decoder has the possibility of predicting over the full surroundings of the agent but learns to refine with more precision only in places where the agent will end up with high probability . This hierarchical process is illustrated in Fig . 3 . Starting from an initial full dense grid probability at low resolutionR0×R0 by pixels , we iteratively refine the resolution by a fixed factor f until we reach the desired final resolution Rfinal × Rfinal . At each iteration i , we select only the Ni highest ranking grid points from the previous iteration , and upsample only these points to the Ri × Ri = Ri−1/f × Ri−1/f . At each step , the grid points features are computed by a 2-layer MLP applied on the point coordinates , these features are then concatenated to the agent encoding and passed through a linear layer , and finally enriched by a 2- layer cross-attention on the graph lane features , before applying a linear layer with sigmoid to get the probability . For a given W output range , this hierarchical process allows the model to only operate on W/R0 × W/R0 + ∑ iNi × f2 grid points instead of the W/Rfinal × W/Rfinal available . In practice , for a final output range of 192 meters with desired Rfinal = 0.5m resolution , we start with an initial resolution of R0 = 8m and use two iterations of ( N1 , N2 ) = ( 16 , 64 ) points each and an upscaling factor f = 4 . This way , we compute only 1856 grid points from the 147 456 available , with no performance loss . The heatmap is trained on each resolution level using as pixel-wise focal loss as in Gilles et al . ( 2021b ) , detailed in Appendix A.3 , with the ground truth being a Gaussian centered at the target agent future position . | The paper proposes a multi-modal trajectory prediction pipeline. The idea is that based on past trajectories and a map of the road a model is trained to predict a heat map with the most likely end-points. With those end-points a trajectory can be inferred for each agent in a collision free way by leveraging all heat maps jointly. With those trajectories, multiple, consistent scene level trajectories are predicted. The main contribution of the paper is on the reasoning about the end-points given by the predicted heat map. | SP:f9b8beceeac894bbe86cf0d2c93f844454ef7cd3 |
Coherent and Consistent Relational Transfer Learning with Autoencoders | 1 INTRODUCTION . Humans are capable of learning concepts such that they can be applied to many different scenarios ( Inhelder & Piaget , 1964 ; Piaget , 2005 ; Lake et al. , 2017 ) . An important characteristic is that human-like concepts remain coherent across contexts , whereby their logical consistency in one context is retained in another ( Nye et al. , 2021 ) . As an example , consider the concept of ordinality which permits comparison over ordered sets , e.g . “ A is larger than B ” , and pertains to a multitude of properties : position , size , volume , reach , etc . So long as one of these properties can be attributed to an object , a set of objects can be compared on that basis ; in this sense ordinality generalises between objects . All in all , if the concept of ordinality were to be learned in its most general form , it should be coherent across properties and objects . In this paper , we seek to define the conditions that allow a learned concept to transfer well across properties and objects in the case of sub-symbolic learners ( d ’ Avila Garcez & Lamb , 2020 ; Santoro et al. , 2021 ; Greff et al. , 2020 ) . We define consistency and coherence of sub-symbolic learners borrowing from analogous definitions from symbolic AI . We propose a neural-symbolic autoencoder architecture consisting of a neural encoder for objects coupled with modular relation-decoders ( Serafini & Garcez , 2016 ; Donadello et al. , 2017 ; Badreddine et al. , 2020 ; Wang et al. , 2017 ; Nickel et al. , 2016a ; Dai et al. , 2020 ) , and we show that this architecture is capable of achieving an improved transfer learning performance by being coherent across properties and objects . We therefore claim that retaining consistency across domains dictates concept coherence , offering a more fine-grained measure of transfer learning than accuracy alone . The proposed architecture is a practical instantiation of this formalisation and is evaluated in this paper on a new Partial Relation Transfer ( PRT ) task and data set . We begin by expressing the symbolic application of a set of relations to some domain of interest as a model-theoretic structure , such as A is larger than B , and defining an analogous soft-structure for non-symbolic learners where relations are modelled by relation-decoders that compute beliefs . We then propose formal definitions for consistency and coherence of soft-structures which provide a practical consistency score calculation to the evaluation of autoencoders . Finally , we present a benchmark PRT learning task with the use of a new BlockStacks data set derived from the CLEVR data set rendering agent . We then compare our proposed archi- tecture with several existing relation-decoder models on transfer learning tasks from the MNIST data set to BlockStacks . Our experiments show that relation-decoders which maintain consistency over unobserved regions of representational space retain coherence across domains whilst achieving better transfer learning performance . The contributions of this paper are : • A formal definition of coherence and consistency for sub-symbolic learners with a practical evaluation score . • A neural model and learning task for partial relation transfer including a new data set to evaluate concept coherence . • A comprehensive critical evaluation of results in comparison with multiple state-of-the-art relation-decoder models . In Section 2 we provide the required background , Section 3 introduces soft-structures and formally defines coherence and consistency , Section 4 provides a practical consistency loss and Section 5 then outlines our neural-symbolic architecture . After detailing the PRT task in Section 6 , we present results in Section 7 and complete the paper in Section 8 with a discussion and concluding remarks , including limitations and future work . We provide related work in Appendix A . 2 PRELIMINARIES . Notations : We reserve uppercase calligraphic letters to denote sets and lowercase versions of the same letter to denote their elements , e.g. , S = { s1 , . . . , sn } is a set S of n elements si . We indicate with |S| = n the cardinality of S. We use uppercase roman letters to denote a random variable e.g. , S , and use the uppercase calligraphic version of the same letter ( S ) to denote the set from which the random variable takes values according to some corresponding probability distribution , pS , over the elements of the set , such that ∑|S| i=1 pS ( si ) = 1 for a discrete S. For brevity , we may write pS ( si ) as p ( si ) , where the random variable is implied by the argument . We use bold font lowercase letters to denote vector elements , e.g. , si ∈ Rd is an d-dimensional vector element from the set S = Rd . Logic and model-theoretic background : We assume a formal language L composed of variables , predicates ( i.e . relations ) , logical connectives ¬ ( negation ) , ∨ ( disjunction ) and ∧ ( conjunction ) , → ( implication ) and universal quantification ∀ ( for all ) with their conventional meaning ( see Shapiro & Kouri Kissel ( 2021 ) ) . Relations express relational knowledge over elements of a domain . For instance , r ( s1 , s2 ) states that elements s1 and s2 are related through the binary relation r. The meaning of relations is defined by an interpretation , ISσ over elements of an non-empty domain S. Together a S and ISσ form a structure Sσ . Definition 1 ( Signature , Interpretation , Structure ) The signature of a language L is σ = { r ∈ L : r is a relation } , whose elements have arity given by ar : σ → N , where N is the set of natural numbers . For each r ∈ σ , ar ( r ) denotes the arity of r. Given a signature σ and a non-empty domain S , an interpretation ISσ of σ over elements of S assigns to each relation r ∈ σ a set ISσ ( r ) ⊆ Sar ( r ) . A structure is a tuple Sσ = ( S , ISσ ) . Note that for a fixed domain S and signature σ , different interpretations yield different structures . We construct universally quantified first-order formulae ( called sentences ) using the signature σ of L , whose truth-value is defined with respect to a given structure Sσ . To do so , we first consider ground instances of a formula . These are given by replacing all the variables in the formula with elements from the domain S. For instance , r ( s1 , s2 ) , where s1 and s2 are elements of S , is a ground instance of an atomic formula r ( i , j ) where i and j are variables in L. Given a structure Sσ = ( S , ISσ ) a relation r and a tuple ( s1 , . . . , sar ( r ) ) ∈ Sar ( r ) , a ground instance r ( s1 , . . . , sar ( r ) ) is true in the structure Sσ if and only if ( s1 , . . . , sar ( r ) ) ∈ ISσ ( r ) . The truth value of a sentence in a given structure Sσ depends on the truth value of its respective ground instances . Specifically , a sentence is true in a structure Sσ if and only if all of its ground instances are true in Sσ . When a sentence , τ , is true in a structure , Sσ , we say that the structure satisfies τ , denoted as Sσ |= τ . A set of sentences form a theory , T . A model of T is a structure that satisfies every sentence in T . Definition 2 ( Model of a theory ) Let T be a theory written in a language L and let Sσ = ( S , ISσ ) be a structure , where σ is the signature of L. Sσ is a model of T if and only if Sσ |= τ for every sentence τ ∈ T . Example 1 Suppose we have the structure Sσ = ( S , ISσ ) , where S is a domain of images of handwritten digits and σ the signature of binary relations σ = { isGreater , isEqual , isLess , isSuccessor , isPredecessor } , or for short σ = { G , E , L , S , P } . Let T be the theory that defines ordinality including , for instance , the sentence ∀i , j.G ( i , j ) → ¬E ( i , j ) ( if a digit is greater than another then they are not equal ) . Any structure Sσ = ( S , ISσ ) with interpretations ISσ of σ that captures a total order over the elements of S is a model of T . 3 APPROXIMATING STRUCTURES THAT HAVE REAL-WORLD DOMAINS . In this section we turn our attention to the challenge of learning a model over a real-world domain , given a signature and theory . Here a learner must determine an appropriate interpretation over real-world data , such as images or other perceptions . This can be challenging because , firstly , we may only have a partial description of the interpretation , and secondly data may be noisy and contain information that is not relevant to the theory . For instance MNIST , a relatively simple data set by current standards , consists of stylistic details such as line thickness and digit skew ( Chen & Batmanghelich , 2020 ) , which are irrelevant to the notion of ordinality , which makes obtaining the structure from Example 1 more complicated . Nevertheless , statistical machine learning models are able to discover commonalities in data which help to infer the underlying semantics ( i.e . interpretation ) and disregard the noise . Following the convention in disentanglement literature ( Bengio et al. , 2013 ; Kingma & Welling , 2014 ; Higgins et al. , 2017 ; 2018 ) , we take the assumption that realworld observations S are drawn from some conditional distribution pS|Z , where Z is a latent random variable , itself drawn from prior pZ . It is therefore useful to define a domain encoding of the form , ψS : S → Z , ( 1 ) tasked with approximating the conditioned expectation of the posterior , i.e . ψS ( s ) = E [ pZ|S ( Z|s ) ] . Since obtaining an interpretation from domain encodings , for a given signature , may require dealing with noise , we express the interpretation of relations over real-world data by belief functions over the space Z ( Paris & Vencovská , 2015 ; Paris , 1994 ) , and refer to these as relation-decoders : φr : Zar ( r ) → ( 0 , 1 ) ( 2 ) with φ = { φr : r ∈ σ } . Concretely , for a binary relation r and ordered pair ( si , sj ) ∈ S2 , φr ( ψS ( si ) , ψS ( sj ) ) describes the belief that ( si , sj ) ∈ ISσ ( r ) . A belief φr ( ψS ( si ) , ψS ( sj ) ) ≈ 1 signifies a strong belief that ( si , sj ) ∈ ISσ ( r ) and φr ( ψS ( si ) , ψS ( sj ) ) ≈ 0 signifies a strong belief that ( si , sj ) /∈ ISσ ( r ) . Together , ψS and φ allow us to define a belief-based analogue to a structure . Definition 3 ( Soft-Structure/Soft-Substructure ) Given signature σ , a possibly infinite set Z and relation-decoders φ , a soft-structure is a tuple Z̃σ = ( Z , φ ) . For ( finite ) domain S and encoding ψS : S → Z , S̃σ = ( ψS ( S ) , φ ) is a ( finite ) soft-substructure of Z̃σ , with sub-domain ψS ( S ) = { ψS ( s ) |s ∈ S } ⊆ Z . A soft-structure can be used to learn a structure over a real-world domain through learning ψS and φ . Clearly , a finite soft-substructure is a soft-structure . To determine the degree to which a softstructure supports any given structure we introduce the following measure : p ( Sσ|S̃σ ) = ∏ r∈σ ∏ O∈Sar ( r ) ( φr ( ψS ( O ) ) ) γrO , Sσ ( 1− φr ( ψS ( O ) ) ) 1−γ r O , Sσ ( 3 ) where γrO , Sσ = 1 if O ∈ ISσ ( r ) , and 0 otherwise ; we use φr ( ψS ( O ) ) as shorthand for φr ( ψS ( s1 ) , . . . , ψS ( sn ) ) for n = ar ( r ) . Eqn . 3 expresses the assumption that , given a finite softstructure , the beliefs in what constitutes the ( different ) interpretations of ( different ) relations are independent of one another . It is straightforward to show that ∑ Sσ p ( Sσ|S̃σ ) = 1 ( summed over all possible structures with domain S and signature σ ) and so it can be treated as a probability measure , where p ( Sσ|S̃σ ) ≈ 1 means that there is a high probability that the interpretation sampled from S̃σ will be ISσ . If we have a theory T over σ then it is natural to ask with what weight S̃σ supports any given structure that is a model of T . In the following , we use model weight , ΓS̃σT , to describe the support given by S̃σ to models of T : ΓS̃σT = ∑ Sσ∈MTS p ( Sσ|S̃σ ) ( 4 ) where MTS is the set of all structures with domain S that are models of T . This lets us compare soft-structures , wherein a good soft-structure will be one that has a high model weight : Definition 4 ( -Consistency of Soft-Structure ) Given a finite soft-structure S̃σ , if 1 − ΓS̃σT ≤ then we say that the soft-structure is -consistent with theory T . We propose -consistency as an appropriate quantified measure of the notion of consistency presented in ( Nye et al. , 2021 ) . A consistent soft-structure S̃σ ensures that φ gives high belief only to interpretations that satisfy , i.e. , are consistent with , T . However , this expression is limited to the domain encodings of S̃σ , i.e . ψS ( S ) . Going a step further , for a concept to be learned in a manner comparable to what a human might learn , we would expect that this consistency carries over to new domains and their corresponding soft-structures , as defined in what follows . | The main motivation of this paper is the observation that once we humans have learned the meaning of a relation, we can apply it to different contexts and scenarios effortlessly. For instance, once we understand the meaning of a concept "larger than", we can apply this in many context such as integers, person lengths, building sizes, etc. The authors argue that it is interesting to study how we can build learning systems that similarly are able to transfers learned relations across domains. In order to express relations, a language based on first-order logic is developed, and semantics is given to this language using an interpretation function, mapping relation symbols to the tuples of elements over which these relations hold. In order to facilitate learning these relations, the authors develop a quantitative account of the logic. They develop so-called "soft structures", which can essentially learn a symbolic structure over real-world domains. This learning is done using the Variational Auto-Encoder framework, where an encoder maps the real-world input (for instance, an image), to a latent space Z. The interpretation of relations over the real world data is then implemented by "relation decoders" (one decoder for each relation), which are belief functions assigning a number in (0, 1) to relations. In order to train such models, they define a measure on how well a soft structure support a symbolic structure, and they use this as a term in the loss function. They also provide a formal account of what it means for two soft structures to be "coherent", which intuitively mean that relations that apply to one domain (e.g., ordinality in numbers) also can be applied to another domain (e.g., ordinality in block sizes). They authors then derive an efficient consistency loss, since the original one is intractable because it requires computing beliefs over all interpretations of the relations. They then propose a Dynamic Comparator relation decoder model, and a new learning task called "Partial Relation Transfer learning". They compare against several existing relation decoder models, and empirically show that their relation decoder retains coherence across domains, while achieving better transfer learning performance. | SP:59af34bc83c862cddb84840001c21d0f9f6d8719 |
Coherent and Consistent Relational Transfer Learning with Autoencoders | 1 INTRODUCTION . Humans are capable of learning concepts such that they can be applied to many different scenarios ( Inhelder & Piaget , 1964 ; Piaget , 2005 ; Lake et al. , 2017 ) . An important characteristic is that human-like concepts remain coherent across contexts , whereby their logical consistency in one context is retained in another ( Nye et al. , 2021 ) . As an example , consider the concept of ordinality which permits comparison over ordered sets , e.g . “ A is larger than B ” , and pertains to a multitude of properties : position , size , volume , reach , etc . So long as one of these properties can be attributed to an object , a set of objects can be compared on that basis ; in this sense ordinality generalises between objects . All in all , if the concept of ordinality were to be learned in its most general form , it should be coherent across properties and objects . In this paper , we seek to define the conditions that allow a learned concept to transfer well across properties and objects in the case of sub-symbolic learners ( d ’ Avila Garcez & Lamb , 2020 ; Santoro et al. , 2021 ; Greff et al. , 2020 ) . We define consistency and coherence of sub-symbolic learners borrowing from analogous definitions from symbolic AI . We propose a neural-symbolic autoencoder architecture consisting of a neural encoder for objects coupled with modular relation-decoders ( Serafini & Garcez , 2016 ; Donadello et al. , 2017 ; Badreddine et al. , 2020 ; Wang et al. , 2017 ; Nickel et al. , 2016a ; Dai et al. , 2020 ) , and we show that this architecture is capable of achieving an improved transfer learning performance by being coherent across properties and objects . We therefore claim that retaining consistency across domains dictates concept coherence , offering a more fine-grained measure of transfer learning than accuracy alone . The proposed architecture is a practical instantiation of this formalisation and is evaluated in this paper on a new Partial Relation Transfer ( PRT ) task and data set . We begin by expressing the symbolic application of a set of relations to some domain of interest as a model-theoretic structure , such as A is larger than B , and defining an analogous soft-structure for non-symbolic learners where relations are modelled by relation-decoders that compute beliefs . We then propose formal definitions for consistency and coherence of soft-structures which provide a practical consistency score calculation to the evaluation of autoencoders . Finally , we present a benchmark PRT learning task with the use of a new BlockStacks data set derived from the CLEVR data set rendering agent . We then compare our proposed archi- tecture with several existing relation-decoder models on transfer learning tasks from the MNIST data set to BlockStacks . Our experiments show that relation-decoders which maintain consistency over unobserved regions of representational space retain coherence across domains whilst achieving better transfer learning performance . The contributions of this paper are : • A formal definition of coherence and consistency for sub-symbolic learners with a practical evaluation score . • A neural model and learning task for partial relation transfer including a new data set to evaluate concept coherence . • A comprehensive critical evaluation of results in comparison with multiple state-of-the-art relation-decoder models . In Section 2 we provide the required background , Section 3 introduces soft-structures and formally defines coherence and consistency , Section 4 provides a practical consistency loss and Section 5 then outlines our neural-symbolic architecture . After detailing the PRT task in Section 6 , we present results in Section 7 and complete the paper in Section 8 with a discussion and concluding remarks , including limitations and future work . We provide related work in Appendix A . 2 PRELIMINARIES . Notations : We reserve uppercase calligraphic letters to denote sets and lowercase versions of the same letter to denote their elements , e.g. , S = { s1 , . . . , sn } is a set S of n elements si . We indicate with |S| = n the cardinality of S. We use uppercase roman letters to denote a random variable e.g. , S , and use the uppercase calligraphic version of the same letter ( S ) to denote the set from which the random variable takes values according to some corresponding probability distribution , pS , over the elements of the set , such that ∑|S| i=1 pS ( si ) = 1 for a discrete S. For brevity , we may write pS ( si ) as p ( si ) , where the random variable is implied by the argument . We use bold font lowercase letters to denote vector elements , e.g. , si ∈ Rd is an d-dimensional vector element from the set S = Rd . Logic and model-theoretic background : We assume a formal language L composed of variables , predicates ( i.e . relations ) , logical connectives ¬ ( negation ) , ∨ ( disjunction ) and ∧ ( conjunction ) , → ( implication ) and universal quantification ∀ ( for all ) with their conventional meaning ( see Shapiro & Kouri Kissel ( 2021 ) ) . Relations express relational knowledge over elements of a domain . For instance , r ( s1 , s2 ) states that elements s1 and s2 are related through the binary relation r. The meaning of relations is defined by an interpretation , ISσ over elements of an non-empty domain S. Together a S and ISσ form a structure Sσ . Definition 1 ( Signature , Interpretation , Structure ) The signature of a language L is σ = { r ∈ L : r is a relation } , whose elements have arity given by ar : σ → N , where N is the set of natural numbers . For each r ∈ σ , ar ( r ) denotes the arity of r. Given a signature σ and a non-empty domain S , an interpretation ISσ of σ over elements of S assigns to each relation r ∈ σ a set ISσ ( r ) ⊆ Sar ( r ) . A structure is a tuple Sσ = ( S , ISσ ) . Note that for a fixed domain S and signature σ , different interpretations yield different structures . We construct universally quantified first-order formulae ( called sentences ) using the signature σ of L , whose truth-value is defined with respect to a given structure Sσ . To do so , we first consider ground instances of a formula . These are given by replacing all the variables in the formula with elements from the domain S. For instance , r ( s1 , s2 ) , where s1 and s2 are elements of S , is a ground instance of an atomic formula r ( i , j ) where i and j are variables in L. Given a structure Sσ = ( S , ISσ ) a relation r and a tuple ( s1 , . . . , sar ( r ) ) ∈ Sar ( r ) , a ground instance r ( s1 , . . . , sar ( r ) ) is true in the structure Sσ if and only if ( s1 , . . . , sar ( r ) ) ∈ ISσ ( r ) . The truth value of a sentence in a given structure Sσ depends on the truth value of its respective ground instances . Specifically , a sentence is true in a structure Sσ if and only if all of its ground instances are true in Sσ . When a sentence , τ , is true in a structure , Sσ , we say that the structure satisfies τ , denoted as Sσ |= τ . A set of sentences form a theory , T . A model of T is a structure that satisfies every sentence in T . Definition 2 ( Model of a theory ) Let T be a theory written in a language L and let Sσ = ( S , ISσ ) be a structure , where σ is the signature of L. Sσ is a model of T if and only if Sσ |= τ for every sentence τ ∈ T . Example 1 Suppose we have the structure Sσ = ( S , ISσ ) , where S is a domain of images of handwritten digits and σ the signature of binary relations σ = { isGreater , isEqual , isLess , isSuccessor , isPredecessor } , or for short σ = { G , E , L , S , P } . Let T be the theory that defines ordinality including , for instance , the sentence ∀i , j.G ( i , j ) → ¬E ( i , j ) ( if a digit is greater than another then they are not equal ) . Any structure Sσ = ( S , ISσ ) with interpretations ISσ of σ that captures a total order over the elements of S is a model of T . 3 APPROXIMATING STRUCTURES THAT HAVE REAL-WORLD DOMAINS . In this section we turn our attention to the challenge of learning a model over a real-world domain , given a signature and theory . Here a learner must determine an appropriate interpretation over real-world data , such as images or other perceptions . This can be challenging because , firstly , we may only have a partial description of the interpretation , and secondly data may be noisy and contain information that is not relevant to the theory . For instance MNIST , a relatively simple data set by current standards , consists of stylistic details such as line thickness and digit skew ( Chen & Batmanghelich , 2020 ) , which are irrelevant to the notion of ordinality , which makes obtaining the structure from Example 1 more complicated . Nevertheless , statistical machine learning models are able to discover commonalities in data which help to infer the underlying semantics ( i.e . interpretation ) and disregard the noise . Following the convention in disentanglement literature ( Bengio et al. , 2013 ; Kingma & Welling , 2014 ; Higgins et al. , 2017 ; 2018 ) , we take the assumption that realworld observations S are drawn from some conditional distribution pS|Z , where Z is a latent random variable , itself drawn from prior pZ . It is therefore useful to define a domain encoding of the form , ψS : S → Z , ( 1 ) tasked with approximating the conditioned expectation of the posterior , i.e . ψS ( s ) = E [ pZ|S ( Z|s ) ] . Since obtaining an interpretation from domain encodings , for a given signature , may require dealing with noise , we express the interpretation of relations over real-world data by belief functions over the space Z ( Paris & Vencovská , 2015 ; Paris , 1994 ) , and refer to these as relation-decoders : φr : Zar ( r ) → ( 0 , 1 ) ( 2 ) with φ = { φr : r ∈ σ } . Concretely , for a binary relation r and ordered pair ( si , sj ) ∈ S2 , φr ( ψS ( si ) , ψS ( sj ) ) describes the belief that ( si , sj ) ∈ ISσ ( r ) . A belief φr ( ψS ( si ) , ψS ( sj ) ) ≈ 1 signifies a strong belief that ( si , sj ) ∈ ISσ ( r ) and φr ( ψS ( si ) , ψS ( sj ) ) ≈ 0 signifies a strong belief that ( si , sj ) /∈ ISσ ( r ) . Together , ψS and φ allow us to define a belief-based analogue to a structure . Definition 3 ( Soft-Structure/Soft-Substructure ) Given signature σ , a possibly infinite set Z and relation-decoders φ , a soft-structure is a tuple Z̃σ = ( Z , φ ) . For ( finite ) domain S and encoding ψS : S → Z , S̃σ = ( ψS ( S ) , φ ) is a ( finite ) soft-substructure of Z̃σ , with sub-domain ψS ( S ) = { ψS ( s ) |s ∈ S } ⊆ Z . A soft-structure can be used to learn a structure over a real-world domain through learning ψS and φ . Clearly , a finite soft-substructure is a soft-structure . To determine the degree to which a softstructure supports any given structure we introduce the following measure : p ( Sσ|S̃σ ) = ∏ r∈σ ∏ O∈Sar ( r ) ( φr ( ψS ( O ) ) ) γrO , Sσ ( 1− φr ( ψS ( O ) ) ) 1−γ r O , Sσ ( 3 ) where γrO , Sσ = 1 if O ∈ ISσ ( r ) , and 0 otherwise ; we use φr ( ψS ( O ) ) as shorthand for φr ( ψS ( s1 ) , . . . , ψS ( sn ) ) for n = ar ( r ) . Eqn . 3 expresses the assumption that , given a finite softstructure , the beliefs in what constitutes the ( different ) interpretations of ( different ) relations are independent of one another . It is straightforward to show that ∑ Sσ p ( Sσ|S̃σ ) = 1 ( summed over all possible structures with domain S and signature σ ) and so it can be treated as a probability measure , where p ( Sσ|S̃σ ) ≈ 1 means that there is a high probability that the interpretation sampled from S̃σ will be ISσ . If we have a theory T over σ then it is natural to ask with what weight S̃σ supports any given structure that is a model of T . In the following , we use model weight , ΓS̃σT , to describe the support given by S̃σ to models of T : ΓS̃σT = ∑ Sσ∈MTS p ( Sσ|S̃σ ) ( 4 ) where MTS is the set of all structures with domain S that are models of T . This lets us compare soft-structures , wherein a good soft-structure will be one that has a high model weight : Definition 4 ( -Consistency of Soft-Structure ) Given a finite soft-structure S̃σ , if 1 − ΓS̃σT ≤ then we say that the soft-structure is -consistent with theory T . We propose -consistency as an appropriate quantified measure of the notion of consistency presented in ( Nye et al. , 2021 ) . A consistent soft-structure S̃σ ensures that φ gives high belief only to interpretations that satisfy , i.e. , are consistent with , T . However , this expression is limited to the domain encodings of S̃σ , i.e . ψS ( S ) . Going a step further , for a concept to be learned in a manner comparable to what a human might learn , we would expect that this consistency carries over to new domains and their corresponding soft-structures , as defined in what follows . | This work introduces a more general definition of structures namely soft structures that can be used to learn structure over richer domains in a data-driven manner. Further, they introduce suitable definitions of coherence and consistency for the proposed soft structures. Next, the paper proposes a neural architecture that is essentially, jointly learning relation encodings jointly with representations for set elements. The key argument of the paper is that transfer is to be expected across shared relations despite their overloaded usage across domains — for example, greater than (GE) can have completely different interpretations in the source / target domains. | SP:59af34bc83c862cddb84840001c21d0f9f6d8719 |
Coherent and Consistent Relational Transfer Learning with Autoencoders | 1 INTRODUCTION . Humans are capable of learning concepts such that they can be applied to many different scenarios ( Inhelder & Piaget , 1964 ; Piaget , 2005 ; Lake et al. , 2017 ) . An important characteristic is that human-like concepts remain coherent across contexts , whereby their logical consistency in one context is retained in another ( Nye et al. , 2021 ) . As an example , consider the concept of ordinality which permits comparison over ordered sets , e.g . “ A is larger than B ” , and pertains to a multitude of properties : position , size , volume , reach , etc . So long as one of these properties can be attributed to an object , a set of objects can be compared on that basis ; in this sense ordinality generalises between objects . All in all , if the concept of ordinality were to be learned in its most general form , it should be coherent across properties and objects . In this paper , we seek to define the conditions that allow a learned concept to transfer well across properties and objects in the case of sub-symbolic learners ( d ’ Avila Garcez & Lamb , 2020 ; Santoro et al. , 2021 ; Greff et al. , 2020 ) . We define consistency and coherence of sub-symbolic learners borrowing from analogous definitions from symbolic AI . We propose a neural-symbolic autoencoder architecture consisting of a neural encoder for objects coupled with modular relation-decoders ( Serafini & Garcez , 2016 ; Donadello et al. , 2017 ; Badreddine et al. , 2020 ; Wang et al. , 2017 ; Nickel et al. , 2016a ; Dai et al. , 2020 ) , and we show that this architecture is capable of achieving an improved transfer learning performance by being coherent across properties and objects . We therefore claim that retaining consistency across domains dictates concept coherence , offering a more fine-grained measure of transfer learning than accuracy alone . The proposed architecture is a practical instantiation of this formalisation and is evaluated in this paper on a new Partial Relation Transfer ( PRT ) task and data set . We begin by expressing the symbolic application of a set of relations to some domain of interest as a model-theoretic structure , such as A is larger than B , and defining an analogous soft-structure for non-symbolic learners where relations are modelled by relation-decoders that compute beliefs . We then propose formal definitions for consistency and coherence of soft-structures which provide a practical consistency score calculation to the evaluation of autoencoders . Finally , we present a benchmark PRT learning task with the use of a new BlockStacks data set derived from the CLEVR data set rendering agent . We then compare our proposed archi- tecture with several existing relation-decoder models on transfer learning tasks from the MNIST data set to BlockStacks . Our experiments show that relation-decoders which maintain consistency over unobserved regions of representational space retain coherence across domains whilst achieving better transfer learning performance . The contributions of this paper are : • A formal definition of coherence and consistency for sub-symbolic learners with a practical evaluation score . • A neural model and learning task for partial relation transfer including a new data set to evaluate concept coherence . • A comprehensive critical evaluation of results in comparison with multiple state-of-the-art relation-decoder models . In Section 2 we provide the required background , Section 3 introduces soft-structures and formally defines coherence and consistency , Section 4 provides a practical consistency loss and Section 5 then outlines our neural-symbolic architecture . After detailing the PRT task in Section 6 , we present results in Section 7 and complete the paper in Section 8 with a discussion and concluding remarks , including limitations and future work . We provide related work in Appendix A . 2 PRELIMINARIES . Notations : We reserve uppercase calligraphic letters to denote sets and lowercase versions of the same letter to denote their elements , e.g. , S = { s1 , . . . , sn } is a set S of n elements si . We indicate with |S| = n the cardinality of S. We use uppercase roman letters to denote a random variable e.g. , S , and use the uppercase calligraphic version of the same letter ( S ) to denote the set from which the random variable takes values according to some corresponding probability distribution , pS , over the elements of the set , such that ∑|S| i=1 pS ( si ) = 1 for a discrete S. For brevity , we may write pS ( si ) as p ( si ) , where the random variable is implied by the argument . We use bold font lowercase letters to denote vector elements , e.g. , si ∈ Rd is an d-dimensional vector element from the set S = Rd . Logic and model-theoretic background : We assume a formal language L composed of variables , predicates ( i.e . relations ) , logical connectives ¬ ( negation ) , ∨ ( disjunction ) and ∧ ( conjunction ) , → ( implication ) and universal quantification ∀ ( for all ) with their conventional meaning ( see Shapiro & Kouri Kissel ( 2021 ) ) . Relations express relational knowledge over elements of a domain . For instance , r ( s1 , s2 ) states that elements s1 and s2 are related through the binary relation r. The meaning of relations is defined by an interpretation , ISσ over elements of an non-empty domain S. Together a S and ISσ form a structure Sσ . Definition 1 ( Signature , Interpretation , Structure ) The signature of a language L is σ = { r ∈ L : r is a relation } , whose elements have arity given by ar : σ → N , where N is the set of natural numbers . For each r ∈ σ , ar ( r ) denotes the arity of r. Given a signature σ and a non-empty domain S , an interpretation ISσ of σ over elements of S assigns to each relation r ∈ σ a set ISσ ( r ) ⊆ Sar ( r ) . A structure is a tuple Sσ = ( S , ISσ ) . Note that for a fixed domain S and signature σ , different interpretations yield different structures . We construct universally quantified first-order formulae ( called sentences ) using the signature σ of L , whose truth-value is defined with respect to a given structure Sσ . To do so , we first consider ground instances of a formula . These are given by replacing all the variables in the formula with elements from the domain S. For instance , r ( s1 , s2 ) , where s1 and s2 are elements of S , is a ground instance of an atomic formula r ( i , j ) where i and j are variables in L. Given a structure Sσ = ( S , ISσ ) a relation r and a tuple ( s1 , . . . , sar ( r ) ) ∈ Sar ( r ) , a ground instance r ( s1 , . . . , sar ( r ) ) is true in the structure Sσ if and only if ( s1 , . . . , sar ( r ) ) ∈ ISσ ( r ) . The truth value of a sentence in a given structure Sσ depends on the truth value of its respective ground instances . Specifically , a sentence is true in a structure Sσ if and only if all of its ground instances are true in Sσ . When a sentence , τ , is true in a structure , Sσ , we say that the structure satisfies τ , denoted as Sσ |= τ . A set of sentences form a theory , T . A model of T is a structure that satisfies every sentence in T . Definition 2 ( Model of a theory ) Let T be a theory written in a language L and let Sσ = ( S , ISσ ) be a structure , where σ is the signature of L. Sσ is a model of T if and only if Sσ |= τ for every sentence τ ∈ T . Example 1 Suppose we have the structure Sσ = ( S , ISσ ) , where S is a domain of images of handwritten digits and σ the signature of binary relations σ = { isGreater , isEqual , isLess , isSuccessor , isPredecessor } , or for short σ = { G , E , L , S , P } . Let T be the theory that defines ordinality including , for instance , the sentence ∀i , j.G ( i , j ) → ¬E ( i , j ) ( if a digit is greater than another then they are not equal ) . Any structure Sσ = ( S , ISσ ) with interpretations ISσ of σ that captures a total order over the elements of S is a model of T . 3 APPROXIMATING STRUCTURES THAT HAVE REAL-WORLD DOMAINS . In this section we turn our attention to the challenge of learning a model over a real-world domain , given a signature and theory . Here a learner must determine an appropriate interpretation over real-world data , such as images or other perceptions . This can be challenging because , firstly , we may only have a partial description of the interpretation , and secondly data may be noisy and contain information that is not relevant to the theory . For instance MNIST , a relatively simple data set by current standards , consists of stylistic details such as line thickness and digit skew ( Chen & Batmanghelich , 2020 ) , which are irrelevant to the notion of ordinality , which makes obtaining the structure from Example 1 more complicated . Nevertheless , statistical machine learning models are able to discover commonalities in data which help to infer the underlying semantics ( i.e . interpretation ) and disregard the noise . Following the convention in disentanglement literature ( Bengio et al. , 2013 ; Kingma & Welling , 2014 ; Higgins et al. , 2017 ; 2018 ) , we take the assumption that realworld observations S are drawn from some conditional distribution pS|Z , where Z is a latent random variable , itself drawn from prior pZ . It is therefore useful to define a domain encoding of the form , ψS : S → Z , ( 1 ) tasked with approximating the conditioned expectation of the posterior , i.e . ψS ( s ) = E [ pZ|S ( Z|s ) ] . Since obtaining an interpretation from domain encodings , for a given signature , may require dealing with noise , we express the interpretation of relations over real-world data by belief functions over the space Z ( Paris & Vencovská , 2015 ; Paris , 1994 ) , and refer to these as relation-decoders : φr : Zar ( r ) → ( 0 , 1 ) ( 2 ) with φ = { φr : r ∈ σ } . Concretely , for a binary relation r and ordered pair ( si , sj ) ∈ S2 , φr ( ψS ( si ) , ψS ( sj ) ) describes the belief that ( si , sj ) ∈ ISσ ( r ) . A belief φr ( ψS ( si ) , ψS ( sj ) ) ≈ 1 signifies a strong belief that ( si , sj ) ∈ ISσ ( r ) and φr ( ψS ( si ) , ψS ( sj ) ) ≈ 0 signifies a strong belief that ( si , sj ) /∈ ISσ ( r ) . Together , ψS and φ allow us to define a belief-based analogue to a structure . Definition 3 ( Soft-Structure/Soft-Substructure ) Given signature σ , a possibly infinite set Z and relation-decoders φ , a soft-structure is a tuple Z̃σ = ( Z , φ ) . For ( finite ) domain S and encoding ψS : S → Z , S̃σ = ( ψS ( S ) , φ ) is a ( finite ) soft-substructure of Z̃σ , with sub-domain ψS ( S ) = { ψS ( s ) |s ∈ S } ⊆ Z . A soft-structure can be used to learn a structure over a real-world domain through learning ψS and φ . Clearly , a finite soft-substructure is a soft-structure . To determine the degree to which a softstructure supports any given structure we introduce the following measure : p ( Sσ|S̃σ ) = ∏ r∈σ ∏ O∈Sar ( r ) ( φr ( ψS ( O ) ) ) γrO , Sσ ( 1− φr ( ψS ( O ) ) ) 1−γ r O , Sσ ( 3 ) where γrO , Sσ = 1 if O ∈ ISσ ( r ) , and 0 otherwise ; we use φr ( ψS ( O ) ) as shorthand for φr ( ψS ( s1 ) , . . . , ψS ( sn ) ) for n = ar ( r ) . Eqn . 3 expresses the assumption that , given a finite softstructure , the beliefs in what constitutes the ( different ) interpretations of ( different ) relations are independent of one another . It is straightforward to show that ∑ Sσ p ( Sσ|S̃σ ) = 1 ( summed over all possible structures with domain S and signature σ ) and so it can be treated as a probability measure , where p ( Sσ|S̃σ ) ≈ 1 means that there is a high probability that the interpretation sampled from S̃σ will be ISσ . If we have a theory T over σ then it is natural to ask with what weight S̃σ supports any given structure that is a model of T . In the following , we use model weight , ΓS̃σT , to describe the support given by S̃σ to models of T : ΓS̃σT = ∑ Sσ∈MTS p ( Sσ|S̃σ ) ( 4 ) where MTS is the set of all structures with domain S that are models of T . This lets us compare soft-structures , wherein a good soft-structure will be one that has a high model weight : Definition 4 ( -Consistency of Soft-Structure ) Given a finite soft-structure S̃σ , if 1 − ΓS̃σT ≤ then we say that the soft-structure is -consistent with theory T . We propose -consistency as an appropriate quantified measure of the notion of consistency presented in ( Nye et al. , 2021 ) . A consistent soft-structure S̃σ ensures that φ gives high belief only to interpretations that satisfy , i.e. , are consistent with , T . However , this expression is limited to the domain encodings of S̃σ , i.e . ψS ( S ) . Going a step further , for a concept to be learned in a manner comparable to what a human might learn , we would expect that this consistency carries over to new domains and their corresponding soft-structures , as defined in what follows . | This paper proposes a loss based on theory for the relation prediction problem, using the sentence of the relationship as the supervision. The paper designs an relation transfer learning experiment. By supervising one relation, other relations can be transferred. | SP:59af34bc83c862cddb84840001c21d0f9f6d8719 |
Scaling Laws for Neural Machine Translation | 1 INTRODUCTION Scaling properties of neural networks have long been an intriguing topic of study ( Ahmad & Tesauro , 1988 ; Amari et al. , 1992 ) . Along with the practical success of modern neural networks at scale , theoretical understanding of the factors governing the quality and training dynamics of large neural networks has also being developing ( Advani & Saxe , 2017 ; Rosenfeld et al. , 2019 ; Geiger et al. , 2020 ; Ghorbani et al. , 2020 ; Cho , 2020 ; Hutter , 2021 ; Bahri et al. , 2021 ; Loureiro et al. , 2021 ) . In particular , scaling model sizes , datasets and the total computation budget has been identified as a reliable approach to improve generalization performance on several machine learning tasks . For many of these tasks the scaling behavior of neural networks is highly predictable ; model fit or test loss can be described precisely as a function of its number of parameters ( Hestness et al. , 2017 ; Kaplan et al. , 2020 ; Henighan et al. , 2020 ; Hernandez et al. , 2021 ; Rosenfeld et al. , 2019 ) . Neural machine translation ( NMT ) has long enjoyed the benefits of scaling ( Huang et al. , 2019 ; Arivazhagan et al. , 2019 ; Lepikhin et al. , 2021 ) , but studies investigating the scaling behavior of NMT models are missing . We present a large-scale systematic study of scaling laws for encoder-decoder Transformer models applied to NMT ( Vaswani et al. , 2017 ) . We start with highlighting the major differences between decoder-only language models , where the majority of the previous work has focused , and encoder-decoder ( conditional ) language models applied to NMT . The two differ along a few crucial dimensions . The first difference results from the very nature of the separate architectures being used , i.e . decoder-only vs encoder-decoder . The presence of separate architectural components complicates the study of scaling properties due to the increased degree of freedom . Second , contrary to language modeling , the task of machine translation is conditional : the task is predictive rather than fully generative . Furthermore , this prediction task is ambiguous : there is no one right answer for a given source , and translations can vary substantially depending on the translator ’ s incentives . This manifests itself as different scaling benefits for different test sets . To take an extreme example , a test set translated by someone who writes nearly wordfor-word translations may benefit less from model scaling than one translated by someone who considers each translation a work of art . In this work , these differences in difficulty coincide with the translation direction of the test set ; that is , whether the source was translated into the target ( source-original ) or vice versa ( target-original ) . Source-original data has translated text on the target side , which contains several artifacts of “ translationese ” that distinguish it from text originally written in that language , often lacking the diversity and complexity of “ natural ” text ( Koppel & Ordan , 2011 ) , while target-original data requires the prediction of more complex natural text on the target side . Finally , unlike language models , NMT is evaluated on metrics that quantify generation quality against reference translations ( for eg . BLEU ) ( Papineni et al. , 2002 ) instead of evaluating model fit ( perplexity ) on an evaluation set . In this paper , we aim to provide empirical answers to the following research questions : 1 . Does the encoder-decoder architecture for NMT share the same scaling law function as language models ? Contrary to previous work on LM , we show that a univariate law depending on the total number of parameters in the network does not adequately describe the scaling behavior of NMT models . Our scaling laws parameterize the cross entropy loss as a bivariate function of the number of encoder parameters and the number of decoder parameters as separate variables . Our results indicate that the scaling behavior is largely determined by the total capacity of the model , and the capacity allocation between the encoder and the decoder . 2 . How does the naturalness of source/target side data affect scaling behavior ? We study the effect of naturalness of the source and the target text , both for training and evaluation . When evaluating with target side natural text , scaling the model capacity continues improving model quality throughout our range of measurements . On the other hand , improvements on cross-entropy saturate ( or reaches the irreducible error region ) on source side natural evaluation sets even for moderately-sized models . 3 . Do scaling improvements in cross-entropy translate into corresponding improvements in generation quality ? Finally we study the relationship between generation quality and cross-entropy and how their correlation changes as we : ( i ) Scale different components of the model ( encoder vs decoder ) and ( ii ) Evaluate on source-natural or target-natural test sets . Our results on multiple language pairs and training/test data compositions validate that model scaling predictably improves the cross-entropy on validation data . However , our findings also raise several questions regarding the effect of naturalness of training and evaluation text and how cross-entropy eventually relates with generation quality for auto-regressive generative models . 1 2 EFFECT OF SCALING ON CROSS-ENTROPY 2.1 EXPERIMENTAL SETTING Model Architectures and Training We train a series of pre-layer norm Transformer networks with varying sizes ( Xiong et al. , 2020 ) . Models are trained with per-token cross-entropy loss and Adafactor optimizer ( Shazeer & Stern , 2018 ) . All models are trained with a fixed batch-size of 500k tokens and dropout rate of 0.1 for residuals , feed-forward activations and attention . All models are trained to near convergence for 500k training steps . Details of the model hyper-parameters are described in Appendix C. Model Scaling Transformer architecture consists of Transformer Blocks : a cascade of self-attention , cross-attention and feed-forward layers , each having multiple adjustable hyper-parameters ( e.g . model-dimension , number of attention heads , attention projection dimension etc. ) . Considering the combinatorial expansion of the search space for scaling each one , in this study we choose to vary only 1A few weeks before the publication of this manuscript on Arxiv , Gordon et al . ( 2021 ) appeared on OpenReview . While both papers study scaling laws for NMT , our studies focus on different parameter regimes ( 393K-56M vs 100M-4B ) . See Appendix A for a comparison of the results . the total number of Transformer Blocks , while keeping the internal hyper-parameters intact across different scales . In other words , we scale the depth of the Transformers while keeping width and other variables fixed . 2 We use GPipe pipeline parallelism for our experiments ( Huang et al. , 2019 ) . In an encoder-decoder Transformer architecture for NMT , depth scaling can naturally be implemented by varying encoder-decoder blocks independently or symmetrically . Hence , we examine the change in the cross-entropy loss as the number of parameters increase with three depth scaling approaches : Encoder Scaling : vary encoder depth ( 2 to 64 ) while decoder depth is fixed ( 6 layers ) . Decoder Scaling : vary decoder depth ( 2 to 64 ) while encoder depth is fixed ( 6 layers ) . Symmetric Scaling : increasing decoder and encoder layers together ( from 2 to 64 ) , i.e . the number of Transformer Blocks in the encoder and decoder being equal . For all experiments , configuration of the individual layers is unchanged : the model dimension , width of the feed-forward layer , and number of attention heads are fixed respectively at 1024 , 8192 , and 16 . 3 Each encoder layer adds approximately 20M parameters to the model while each decoder layer adds around 25M parameters . We train 200 such models which scale the model size by approximately a factor of 50 ( from 80M parameters to 4B parameters ) . Following the convention , we do not count the parameters in the embedding and softmax layers towards the model size . We focus our study on large-scale models : our smallest models require 200 TPUv3 days to train to convergence while our largest models require 2700 TPUv3 days of training . Language Pairs We report results on four language pairs : English→German , German→English , English→Chinese , and Chinese→English . We use in-house web-crawled training datasets with around 2.2 billion sentence pairs ( approximately 55 billion tokens ) for English↔German and 781 million sentence pairs for English↔Chinese . Evaluation Sets We use a variety of test sets for evaluation covering different domains : ( i ) WebDomain ( ii ) News-Domain ( iii ) Wikipedia ( iv ) Patents . The news-domain test sets come from the WMT2019 ( Barrault et al. , 2019 ) evaluation campaign ( newstest2019 ) for all language pairs . The other test sets are internal test sets representing the different domains , ranging from 500 to 5000 sentence pairs . For each domain , we randomly sample sentences in the source language and use professional translators to generate a reference translation in the target language . Throughout the paper , we will refer this type of test sets as source-original as the source sentences have been crawled from the web while the reference translations are added later . For most of the domains , we also have a target-original counterpart which is generated in the opposite direction : Sentences are crawled in the target language and human translated into the source language . Earlier work ( Freitag et al. , 2019 ; 2020 ; Graham et al. , 2020 ) showed that it is important to differentiate between the two different kinds of test sets as the style of natural sentences and human ( or machine ) translations ( translationese ) is quite different . Cross-entropy loss is evaluated on the different test sets during training . To reduce the variation caused by the parameter fluctuations at the end of the training , we present the median loss over the last 50k steps of the training as the final loss . 2.2 RESULTS Figure 1 shows the empirical evolution of the test loss on the Web-Domain test sets for encoder and decoder scaling for English→German . To compare the empirical results with the scaling laws present in the literature for decoder only models ( Kaplan et al. , 2020 ; Henighan et al. , 2020 ) , we have fitted a power law of the form L̂ ( N ) = αN−p+L∞ to the data . 4 Here , N is the total number of parameters outside of embedding / softmax layers and { α , p , L∞ } are fitted parameters . As Figure 1 suggests , scaling the encoder has different effects on the test loss compared to scaling the decoder . As such , simple power-laws that only consider the total parameter count , fail to capture the correct scaling behavior of the model . 2In Appendix H , we examine the role of scaling approach in our results . See Section 2 for more details . 3A complete description of the model architecture is provided in Appendix C 4Details of the curve fitting procedure are presented in Appendix I. laws are unable to capture the correct scaling behavior . R-squared ( 100× explained variancetotal variance ) and maximum absolute deviation ( ‖ · ‖∞ ) are reported for each fit . Proposed Scaling Law To tackle this issue , we present a new scaling law that reflects the encoderdecoder nature of the architecture as well as the bilingual format of the data . Let Ne and Nd be the number of non-embedding parameters in the encoder and the decoder respectively . Then , our proposed scaling law has the form L̂ ( Ne , Nd ) = αN −pe e N −pd d + L∞ ( 1 ) where { α , pe , pd , L∞ } are test set specific ( fitted ) parameters . In this formulation , α corresponds to the maximum loss reduction that one can hope from scaling , while pe and pd are the scaling exponents for encoder and decoder respectively . L∞ corresponds to the irreducible loss of the data . In Figure 2 , we fit our scaling law to the empirical loss values achieved by encoder/decoder scaling models on Web-Domain test sets . The results suggest that our scaling law almost fully captures the variation in the data ( R2 > 99 % ) . In particular , in contrast to traditional scaling laws , our proposed formulation is able to capture the different scaling behaviors of encoder and decoder scaled models . While these improved fits are encouraging , to truly show that a candidate scaling law captures the model scaling behavior in NMT , it is necessary to verify the out-of-sample prediction power of the law . We provide two such verifications ; we compare the predictions of the scaling law fitted on encoder / decoder scaling models with the empirical loss values from ( i ) symmetrically scaled models , ( ii ) models randomly sampled from the 2D grid of all potential encoder-decoder configurations.5 Symmetrically scaled models are chosen for this verification as they are widely used by the NLP community ( Raffel et al. , 2020 ) . Randomly sampled models are chosen to ensure that our results are not biased towards any particular model class . Figures 3 & 4 present the out-of-sample results . As the plots suggest , the predictions of the scaling law match the empirical results with remarkable accuracy . These results suggest that Eq . ( 1 ) is able to capture the model scaling behavior across different scaling approaches and model sizes . To ensure that our results generalize across different language pairs , we examine the fit of our scaling law on encoder / decoder scaling models trained on German→English ( De→En ) , Chinese-to-English ( Zh→En ) , and English-to-Chinese ( En→Zh ) translation tasks . Figure 5 presents the results evaluated on in-domain held-out data . Similar to the En→De case , our proposed functional form is able to closely capture the scaling behavior of these models . To keep the scope of our experiments manageable , we focus on depth scaling in our analysis . In Appendix H , we ran large-scale experiments to see if other scaling approaches show significant deviations from the trends identified here . For large-scale models , which are the focus of this study , we observe that different scaling approaches yield very similar results : the final achieved test loss across different scaling approaches shows just 1− 2 % variation . We observed that for small models ( < 100M parameters ) , this variation tends to grow , suggesting that model shape plays a key role in the performance of small models . We leave further analysis of this phenomenon to future research . 5See Appendix F for the full description of the models . 2.3 ANALYSIS The above results suggest that the scaling law formalized in Eq . ( 1 ) captures the scaling behavior of the Transformer NMT models in multiple language pairs . As such , we can study the fitted coefficients to fully understand the scaling properties of these models . Several observations are in order : Decoder vs Encoder Scaling : On all our test sets , the decoder exponents were observed to be larger than the encoder exponents , pd > pe ( see Figure 9 ) . As a result , when improving the test loss is concerned , it is much more effective to scale the decoder rather than the encoder . This is contrary to the usual practice ; due to latency considerations , many practitioners train NMT models with deep encoders and shallow decoders ( Kasai et al. , 2021 ) . Our results suggest this practice could be sub-optimal in terms of loss reduction . Proposition 1 below provides guidance on how to allocate parameters in between the encoder and decoder optimally . The proof is presented in Appendix G. Proposition 1 ( Optimal Scaling ) . Assume the loss performance of the model is described by Eq . ( 1 ) . Let B denote the budget for the total number of parameters . Then , the optimal encoder / decoder sizes ( denoted respectively by N∗e and N ∗ d ) are N ∗ e = pe pe+pd B and N∗d = pd pe+pd B . In addition , when optimally scaling the model , the scaling law reduces to L̂opt ( B ) = α∗B− ( pd+pe ) +L∞ , where α∗ ≡ α ( pe+pdpe ) pe ( pe+pdpd ) pd . Proposition 1 suggests that when NeNd = pe pd , the optimal scaling behavior can be achieved . Inspection of the functional form of Eq . ( 1 ) suggests that as long as Nd/Ne is fixed as the model scales ( i.e . encoder and decoder grow proportionally together ) , the optimal scaling exponent , ( pe + pd ) , can be achieved , albeit with a potentially sub-optimal multiplicative constant , α # . To examine the magnitude of this sub-optimality , in Figure 10 , we compare the multiplicative constants resulting from proportional scaling of the encoder and decoder with different values of Nd/Ne . The results suggest that as long as the parameter allocation is not extremely far from ( N∗e , N ∗ d ) , the scaling scheme is approximately optimal . In particular , symmetrically scaling the encoder and decoder layers , is barely distinguishable from the optimal scaling scheme . 3 EFFECT OF DATASET COMPOSITION BIAS ON SCALING BEHAVIOR Translation deals with the problem of mapping a sequence in one language into another language . A good translation should not only be adequate and fluent , but should ideally also adopt the style of a sentence naturally written in the target language . This necessitates MT models to make sense of natural looking inputs and generate natural looking outputs . As mentioned in Section 2 , the examples used to train or test NMT models carry a critical bias , which we refer to as composition bias . Composition bias is introduced because of the unavailability of source-target examples ( pairs ) that are both natural in the accessible data generating distribution . For any given naturally generated text in a language , the corresponding text in the other language is either translated by humans , introducing translationese bias or translated by other machine translation systems , introducing MT bias . We consider both biases affecting the problem from a similar angle , hence we bundle them and call it composition bias . While machine translation by design has composition bias in the training/test sets employed ( Freitag et al. , 2020 ; Riley et al. , 2020 ) , its effect on model scaling is unknown . In this section we investigate the role of composition bias in scaling and identify critical factors playing a role . We caution the reader to not take the composition bias as a problem specific to NMT . In fact as most training corpora in NMT are web-crawled , they can contain machine translation output on either the source or target side . Considering the growth of generated content in the web by machine learning models 6 7 , it is not improbable that a proportion of the content collected and used by machine learning models is going to be biased by other models that are continuously generating content . The Effect of Test Set Construction : We first examine the impact of composition bias on the test sets . Figure 9 shows the fitted scaling exponents for all our test sets . The plot suggests that the scaling powers for source-original test sets are drastically different from those of target-original test sets . This behavior is in direct contrast with language modeling setting ( Kaplan et al. , 2020 ) where it was observed that the evaluation on different test sets merely acted as a scaling penalty that only changed the multiplicative constants of the scaling law . To elucidate this phenomenon further , in Figure 11 of the appendix , we compare the scaling trends for different source and target original test sets . Several observations are in order : Test sets with a similar composition approach ( source or target original ) have a qualitatively similar scaling behavior . However , scaling behavior is vastly different between the two composition approaches . Reducible loss quickly decays to zero for source original test sets . In fact , we observe that scaling our baseline 6L-6L model by a factor of 2.5 is sufficient for ensuring that reducible loss is below 0.05 for all source original test sets . In contrast , on target original test sets , the loss decays much more slowly with model size . For comparison , to ensure that reducible loss is below 0.05 for all target original test sets , we estimate that the baseline model has to be scaled up by a factor of 11 . Because of this behavior , the value of larger models in NMT is closely tied to their evaluation sets : On source original data , due to larger scaling exponents , even moderate increases in model size are sufficient for pushing the reducible loss close to zero . Hence , beyond a few hundred million parameters , there is no benefit in increasing the model size . In contrast , for target original data , which generally have smaller scaling exponents , large models are needed to push the reducible loss to zero . The Effect of Training Set Construction : In this section , we briefly examine the role of training data construction on the scaling behavior . To do this , we generate two En→De datasets , that were not used in the previous experiments . One fully target original and another completely source original . To generate the target original dataset , we compile a set of German documents from the web . Documents are screened to ensure the data is not machine generated . We use a Hybrid model ( with 380M parameters ) ( Chen et al. , 2018 ) to back-translate ( BT ) these documents to English . Similarly , for the source original data , we collect human generated English documents and ( forward ) translate them to German using a hybrid model ( with approximately 327M parameters ) . Both datasets provide us with approximately 2.2 billion training examples . We mimic the experimental setup of Section 2 . Note that even though these datasets are not human generated , they reflect important aspects of training large NMT models . Many modern NMT datasets are harvested from the web and as a result , are contaminated with machine generated data . Moreover , many popular data augmentation algorithms such as Back Translation ( Sennrich et al. , 2016 ) , sequence level distillation ( Kim & Rush , 2016 ) and self training ( He et al. , 2020 ) purposefully add machine generated data into the training pipeline in order to take advantage of monolingual data . Figure 6 describes the scaling behavior for models trained on target-original data . We observe that even though larger models are successful in reducing the training loss , they are unable to improve the test loss after roughly 400M parameters . Once this size threshold is exceeded , models overfit the training data and the test loss starts to deteriorate across all of our test sets . We hypothesize that this size threshold corresponds to the capacity of the original back-translation model . This assertion suggests that in order for back-translation to be beneficial for training large models , it has to be performed with a models with comparable capacity or higher . Although quite intriguing , we leave the verification of this hypothesis to future work . Figure 7 paints another interesting picture for the models trained on the source-original data only , implying the target side having the composition bias , expected to be simpler , dull and not rich in its content , in short - not natural looking . As experiments suggest , even our smallest models are able to achieve extremely low loss values ( roughly 0.16 ) , with an apparent overfitting pattern . We believe the same phenomenon is also related to the `` data simplification '' effect seeked by non-autoregressive models in NMT ( Zhou et al. , 2021 ) . 6https : //openai.com/blog/gpt-3-apps/ 7https : //blog.google/products/translate/one-billion-installs/ 4 EVOLUTION OF GENERATION QUALITY We examine the effects of scaling on the output quality as measured by BLEU score 8 . For this analysis , we focus on output generated via beam search ( Wu et al. , 2016 ) . For tractability purposes , we do not attempt to tune the ( many ) hyper-parameters of beam-search for each model . Instead , we use the configuration optimized for the baseline model ( listed in Appendix J ) in all the decoding tasks . Figure 8 presents the co-evolution of BLEU score and cross-entropy loss throughout the training for all of our models . Depending on the construction of the test sets , two different empirical behaviors emerge . On target-original test sets , larger models are able to improve ( lower ) the test loss . These improvements in the loss are accompanied with consistent improvements ( increases ) in BLEU score . In fact , we observe that a simple power law can capture the relationship between BLEU score and cross-entropy loss for high-quality models . 9 In contrast , on source-original test sets , this relationship is absent ; larger models consistently achieve better test losses , however , beyond a certain threshold , BLEU scores begin to deteriorate . Figures 27 and 28 exhibit that this phenomenon is not due to over-training ; the BLEU score gap between large and small models is persistent throughout training . To ensure that this observation truly reflects the generation quality of the models ( as opposed to potential biases of BLEU score ) , we repeat our analysis with BLEURT score ( Sellam et al. , 2020a ; b ) . The results are presented in Appendix K. The results show that BLEURT scores closely mirror the behavior of BLEU scores with respect to model scaling . A careful look at the left-subplots of Figures 8 brings up another interesting trend . At similar values of the test loss , encoder-scaled models result in better generation quality compared to decoder-scaled models . This findings agrees with previous work that relied on encoder-scaling when optimizing for BLEU and inference latency ( Kasai et al. , 2021 ) . Whether these differences in the effects of encoder-scaling and decoder-scaling are caused by insufficient search algorithms , or just different model fits from different architectural priors is left to future work . 8We computed the BLEU scores using an internal reimplementation of Moses scorer : mteval-v13a.pl . 9Fitting details are presented in Appendix J . 5 CONCLUSION AND LIMITATION In this work we have attempted to quantify the evolution of model quality as a function of model capacity for encoder-decoder NMT models . While a univariate scaling law describing the crossentropy as a function of the total number of parameters in the model is insufficient , a bivariate law treating the number of encoder and decoder parameters as separate variables adequately describes the scaling behavior of these models under various scaling strategies . We validate this behavior on a variety of language pairs and evaluation sets . Whether this behavior is intrinsic to the encoder-decoder architecture , or arising from the nature of the NMT task , requires further study . Next , we show that this scaling behavior is highly dependent on the composition of the evaluation data , specifically on whether the source or target sentences are “ original ” . Our findings indicate that target-original evaluation sets continue benefiting from model scaling , while the reducible error on source-original evaluation sets quickly saturates to 0 . This could be an artifact of the lack of diversity in translated text ; a simpler target distribution doesn ’ t require much capacity to model while generating fluent or natural-looking text could benefit much more from scale . We also study how the composition of training data affects the scaling behavior of models . When training on target-original ( back-translated ) text , model quality keeps improving until a point after which the trend saturates . In our study the capacity where saturation manifests first is perilously close to the capacity of the model used for back-translation , indicating that the capacity of the generative model used to generate synthetic text might have a role to play , but this requires further investigation . When training on source-original text , even low-capacity models are sufficient to reach the irreducible loss region , painting a gloomy picture for synthetic data . While we have explored these ideas in the context of machine translation , given the proliferation of generative models this problem will likely be a challenge for future practitioners training on web-scraped monolingual datasets as well . For low-resource languages , the proliferation of machine translated text is already a problem given that a significant portion of web text in these languages is machine translated . Finally , we attempt to understand how generation quality evolves with the improvements in crossentropy resulting from model scaling . As with our previous findings , dataset composition plays a major role in determining the trends . For source-original evaluation sets , the correlation between cross-entropy and generation quality breaks down . On target-original evaluation , we observe an inverse correlation between cross-entropy and BLEU , suggesting that improved model fit results in a corresponding improvement in generation quality . The slope of this relationship is different for encoder-scaling and decoder-scaling , with encoder-scaled models performing better on BLEU than decoder-scaled models , at the same level of cross-entropy loss . Whether this is an artifact of our search strategy or the difference in architectural priors is something that requires further investigation . 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Yonghui Wu , Mike Schuster , Zhifeng Chen , Quoc V Le , Mohammad Norouzi , Wolfgang Macherey , Maxim Krikun , Yuan Cao , Qin Gao , Klaus Macherey , et al . Google ’ s neural machine translation system : Bridging the gap between human and machine translation . arXiv preprint arXiv:1609.08144 , 2016 . Ruibin Xiong , Yunchang Yang , Di He , Kai Zheng , Shuxin Zheng , Chen Xing , Huishuai Zhang , Yanyan Lan , Liwei Wang , and Tieyan Liu . On layer normalization in the transformer architecture . In International Conference on Machine Learning , pp . 10524–10533 . PMLR , 2020 . Chunting Zhou , Graham Neubig , and Jiatao Gu . Understanding knowledge distillation in nonautoregressive machine translation , 2021 . A CONNECTIONS TO RECENT LITERATURE In this section , we provide a more detailed overview of the scaling laws literature immediately relevant to our investigation . Hestness et al . ( 2017 ) examine data scaling laws for LSTM-based encoder-decoder NMT models . They show that the model performance exhibits an empirical powerlaw relationship with the size of the training data . Rosenfeld et al . ( 2019 ) study model and data scaling laws for image classification and language modeling tasks . They provide a bivariate scaling law that jointly capture the model and data scaling behaviors for each of these tasks . Crucially , they show that scaling laws can be used to effectively extrapolate beyond model/data configurations used for fitting them . In particular , they show that one can use smaller models to fit the scaling law and then use the law to accurately predict the performance of larger models . Our results ( Figures 3 and 4 ) validate that this behavior also holds for our setting . Kaplan et al . ( 2020 ) study the scaling behavior of decoder-only Transformer models used in language modeling . They identify three factors that principally determine the model performance : the number of ( non-embedding ) model parameters , the size of the training dataset , and the amount of compute used for training . In the scenario where data and compute are abundant , they show that model performance exhibits an empirical power-law relationship with the total number of ( non-embedding ) parameters in the network . Henighan et al . ( 2020 ) verify this power-law behavior for other autoregressive generative modeling tasks such as image modeling , video modeling , and mathematical problem solving . In contrast , in our setting , we observe that the total number of parameters is not a good predictor of the test loss in NMT . Instead , we show that encoder-decoder NMT models scale according to a bivariate function of the number of encoder parameters and the number of decoder parameters . Kaplan et al . ( 2020 ) also examine how scaling affects the out-of-distribution generalization behavior in language models . Their results suggest that different test sets exhibit rather similar scaling trends as model size increases . In comparison , in the NMT setting , we show that different test sets , depending on their naturalness structure , exhibit drastically different scaling behaviors . The most closely related study to ours is the work of Gordon et al . ( 2021 ) . They also examine scaling laws for NMT , albeit at a different parameter regime : Gordon et al . focus on small models ( 393K56M parameters ) while here , we focus on large scale models ( 100M-4B parameters ) . In addition , we examine the general scenario where encoder and decoder can scale asymmetrically ( which is a common practice in NMT ) while Gordon et al . only consider symmetric scaling . Despite the differences in the experimental setup , when focusing on the symmetric scaling setting , the conclusions of both studies mostly align . In particular , when symmetrically scaling the model , our scaling law simplifies to L ( N ) = αN−p + L∞ which agrees with the conclusions of Gordon et al . One difference in our conclusions is the presence of the L∞ term in our scaling law . This term , which captures the irreducible loss in the learning problem , does not appear in Gordon et al ’ s analysis . We suspect that this might be due to the small models used in that analysis which can not statistically observe such asymptotic terms . Gordon et al . also examine the relationship between test cross-entropy loss and BLEU score . They observe that as models scale and cross-entropy loss improves , BLEU score also improves in tandem . In comparison , we unveil a more intricate relationship in our study : We show that when the test data is target-original , improvements in the test cross-entropy loss are correlated with improvements in generation quality ( as measured by BLEU/BLEURT scores ) . However , when the test set is sourceoriginal , the correlation between cross-entropy loss and generation quality breaks down . In this setting , we observe that generation quality improves up until a certain threshold and then deteriorates ( see the right plot in Figure 8 ) . B MISCELLANEOUS FIGURES | The paper provides interesting scaling properties for autoregressive neural machine translation models, which are validated on comprehensive experiments. The authors investigate the scaling properties from three perspectives: 1. Quantify the evolution of model quality as a function of model capacity for encoder-decoder NMT models. 2. How the composition of training/test data affects the scaling behavior of NMT models? 3. How generation quality evolves with the improvements in cross-entropy resulting from model scaling? | SP:7c0300ec9f6fe3242ba9e6d6f7098059fa60be0f |
Scaling Laws for Neural Machine Translation | 1 INTRODUCTION Scaling properties of neural networks have long been an intriguing topic of study ( Ahmad & Tesauro , 1988 ; Amari et al. , 1992 ) . Along with the practical success of modern neural networks at scale , theoretical understanding of the factors governing the quality and training dynamics of large neural networks has also being developing ( Advani & Saxe , 2017 ; Rosenfeld et al. , 2019 ; Geiger et al. , 2020 ; Ghorbani et al. , 2020 ; Cho , 2020 ; Hutter , 2021 ; Bahri et al. , 2021 ; Loureiro et al. , 2021 ) . In particular , scaling model sizes , datasets and the total computation budget has been identified as a reliable approach to improve generalization performance on several machine learning tasks . For many of these tasks the scaling behavior of neural networks is highly predictable ; model fit or test loss can be described precisely as a function of its number of parameters ( Hestness et al. , 2017 ; Kaplan et al. , 2020 ; Henighan et al. , 2020 ; Hernandez et al. , 2021 ; Rosenfeld et al. , 2019 ) . Neural machine translation ( NMT ) has long enjoyed the benefits of scaling ( Huang et al. , 2019 ; Arivazhagan et al. , 2019 ; Lepikhin et al. , 2021 ) , but studies investigating the scaling behavior of NMT models are missing . We present a large-scale systematic study of scaling laws for encoder-decoder Transformer models applied to NMT ( Vaswani et al. , 2017 ) . We start with highlighting the major differences between decoder-only language models , where the majority of the previous work has focused , and encoder-decoder ( conditional ) language models applied to NMT . The two differ along a few crucial dimensions . The first difference results from the very nature of the separate architectures being used , i.e . decoder-only vs encoder-decoder . The presence of separate architectural components complicates the study of scaling properties due to the increased degree of freedom . Second , contrary to language modeling , the task of machine translation is conditional : the task is predictive rather than fully generative . Furthermore , this prediction task is ambiguous : there is no one right answer for a given source , and translations can vary substantially depending on the translator ’ s incentives . This manifests itself as different scaling benefits for different test sets . To take an extreme example , a test set translated by someone who writes nearly wordfor-word translations may benefit less from model scaling than one translated by someone who considers each translation a work of art . In this work , these differences in difficulty coincide with the translation direction of the test set ; that is , whether the source was translated into the target ( source-original ) or vice versa ( target-original ) . Source-original data has translated text on the target side , which contains several artifacts of “ translationese ” that distinguish it from text originally written in that language , often lacking the diversity and complexity of “ natural ” text ( Koppel & Ordan , 2011 ) , while target-original data requires the prediction of more complex natural text on the target side . Finally , unlike language models , NMT is evaluated on metrics that quantify generation quality against reference translations ( for eg . BLEU ) ( Papineni et al. , 2002 ) instead of evaluating model fit ( perplexity ) on an evaluation set . In this paper , we aim to provide empirical answers to the following research questions : 1 . Does the encoder-decoder architecture for NMT share the same scaling law function as language models ? Contrary to previous work on LM , we show that a univariate law depending on the total number of parameters in the network does not adequately describe the scaling behavior of NMT models . Our scaling laws parameterize the cross entropy loss as a bivariate function of the number of encoder parameters and the number of decoder parameters as separate variables . Our results indicate that the scaling behavior is largely determined by the total capacity of the model , and the capacity allocation between the encoder and the decoder . 2 . How does the naturalness of source/target side data affect scaling behavior ? We study the effect of naturalness of the source and the target text , both for training and evaluation . When evaluating with target side natural text , scaling the model capacity continues improving model quality throughout our range of measurements . On the other hand , improvements on cross-entropy saturate ( or reaches the irreducible error region ) on source side natural evaluation sets even for moderately-sized models . 3 . Do scaling improvements in cross-entropy translate into corresponding improvements in generation quality ? Finally we study the relationship between generation quality and cross-entropy and how their correlation changes as we : ( i ) Scale different components of the model ( encoder vs decoder ) and ( ii ) Evaluate on source-natural or target-natural test sets . Our results on multiple language pairs and training/test data compositions validate that model scaling predictably improves the cross-entropy on validation data . However , our findings also raise several questions regarding the effect of naturalness of training and evaluation text and how cross-entropy eventually relates with generation quality for auto-regressive generative models . 1 2 EFFECT OF SCALING ON CROSS-ENTROPY 2.1 EXPERIMENTAL SETTING Model Architectures and Training We train a series of pre-layer norm Transformer networks with varying sizes ( Xiong et al. , 2020 ) . Models are trained with per-token cross-entropy loss and Adafactor optimizer ( Shazeer & Stern , 2018 ) . All models are trained with a fixed batch-size of 500k tokens and dropout rate of 0.1 for residuals , feed-forward activations and attention . All models are trained to near convergence for 500k training steps . Details of the model hyper-parameters are described in Appendix C. Model Scaling Transformer architecture consists of Transformer Blocks : a cascade of self-attention , cross-attention and feed-forward layers , each having multiple adjustable hyper-parameters ( e.g . model-dimension , number of attention heads , attention projection dimension etc. ) . Considering the combinatorial expansion of the search space for scaling each one , in this study we choose to vary only 1A few weeks before the publication of this manuscript on Arxiv , Gordon et al . ( 2021 ) appeared on OpenReview . While both papers study scaling laws for NMT , our studies focus on different parameter regimes ( 393K-56M vs 100M-4B ) . See Appendix A for a comparison of the results . the total number of Transformer Blocks , while keeping the internal hyper-parameters intact across different scales . In other words , we scale the depth of the Transformers while keeping width and other variables fixed . 2 We use GPipe pipeline parallelism for our experiments ( Huang et al. , 2019 ) . In an encoder-decoder Transformer architecture for NMT , depth scaling can naturally be implemented by varying encoder-decoder blocks independently or symmetrically . Hence , we examine the change in the cross-entropy loss as the number of parameters increase with three depth scaling approaches : Encoder Scaling : vary encoder depth ( 2 to 64 ) while decoder depth is fixed ( 6 layers ) . Decoder Scaling : vary decoder depth ( 2 to 64 ) while encoder depth is fixed ( 6 layers ) . Symmetric Scaling : increasing decoder and encoder layers together ( from 2 to 64 ) , i.e . the number of Transformer Blocks in the encoder and decoder being equal . For all experiments , configuration of the individual layers is unchanged : the model dimension , width of the feed-forward layer , and number of attention heads are fixed respectively at 1024 , 8192 , and 16 . 3 Each encoder layer adds approximately 20M parameters to the model while each decoder layer adds around 25M parameters . We train 200 such models which scale the model size by approximately a factor of 50 ( from 80M parameters to 4B parameters ) . Following the convention , we do not count the parameters in the embedding and softmax layers towards the model size . We focus our study on large-scale models : our smallest models require 200 TPUv3 days to train to convergence while our largest models require 2700 TPUv3 days of training . Language Pairs We report results on four language pairs : English→German , German→English , English→Chinese , and Chinese→English . We use in-house web-crawled training datasets with around 2.2 billion sentence pairs ( approximately 55 billion tokens ) for English↔German and 781 million sentence pairs for English↔Chinese . Evaluation Sets We use a variety of test sets for evaluation covering different domains : ( i ) WebDomain ( ii ) News-Domain ( iii ) Wikipedia ( iv ) Patents . The news-domain test sets come from the WMT2019 ( Barrault et al. , 2019 ) evaluation campaign ( newstest2019 ) for all language pairs . The other test sets are internal test sets representing the different domains , ranging from 500 to 5000 sentence pairs . For each domain , we randomly sample sentences in the source language and use professional translators to generate a reference translation in the target language . Throughout the paper , we will refer this type of test sets as source-original as the source sentences have been crawled from the web while the reference translations are added later . For most of the domains , we also have a target-original counterpart which is generated in the opposite direction : Sentences are crawled in the target language and human translated into the source language . Earlier work ( Freitag et al. , 2019 ; 2020 ; Graham et al. , 2020 ) showed that it is important to differentiate between the two different kinds of test sets as the style of natural sentences and human ( or machine ) translations ( translationese ) is quite different . Cross-entropy loss is evaluated on the different test sets during training . To reduce the variation caused by the parameter fluctuations at the end of the training , we present the median loss over the last 50k steps of the training as the final loss . 2.2 RESULTS Figure 1 shows the empirical evolution of the test loss on the Web-Domain test sets for encoder and decoder scaling for English→German . To compare the empirical results with the scaling laws present in the literature for decoder only models ( Kaplan et al. , 2020 ; Henighan et al. , 2020 ) , we have fitted a power law of the form L̂ ( N ) = αN−p+L∞ to the data . 4 Here , N is the total number of parameters outside of embedding / softmax layers and { α , p , L∞ } are fitted parameters . As Figure 1 suggests , scaling the encoder has different effects on the test loss compared to scaling the decoder . As such , simple power-laws that only consider the total parameter count , fail to capture the correct scaling behavior of the model . 2In Appendix H , we examine the role of scaling approach in our results . See Section 2 for more details . 3A complete description of the model architecture is provided in Appendix C 4Details of the curve fitting procedure are presented in Appendix I. laws are unable to capture the correct scaling behavior . R-squared ( 100× explained variancetotal variance ) and maximum absolute deviation ( ‖ · ‖∞ ) are reported for each fit . Proposed Scaling Law To tackle this issue , we present a new scaling law that reflects the encoderdecoder nature of the architecture as well as the bilingual format of the data . Let Ne and Nd be the number of non-embedding parameters in the encoder and the decoder respectively . Then , our proposed scaling law has the form L̂ ( Ne , Nd ) = αN −pe e N −pd d + L∞ ( 1 ) where { α , pe , pd , L∞ } are test set specific ( fitted ) parameters . In this formulation , α corresponds to the maximum loss reduction that one can hope from scaling , while pe and pd are the scaling exponents for encoder and decoder respectively . L∞ corresponds to the irreducible loss of the data . In Figure 2 , we fit our scaling law to the empirical loss values achieved by encoder/decoder scaling models on Web-Domain test sets . The results suggest that our scaling law almost fully captures the variation in the data ( R2 > 99 % ) . In particular , in contrast to traditional scaling laws , our proposed formulation is able to capture the different scaling behaviors of encoder and decoder scaled models . While these improved fits are encouraging , to truly show that a candidate scaling law captures the model scaling behavior in NMT , it is necessary to verify the out-of-sample prediction power of the law . We provide two such verifications ; we compare the predictions of the scaling law fitted on encoder / decoder scaling models with the empirical loss values from ( i ) symmetrically scaled models , ( ii ) models randomly sampled from the 2D grid of all potential encoder-decoder configurations.5 Symmetrically scaled models are chosen for this verification as they are widely used by the NLP community ( Raffel et al. , 2020 ) . Randomly sampled models are chosen to ensure that our results are not biased towards any particular model class . Figures 3 & 4 present the out-of-sample results . As the plots suggest , the predictions of the scaling law match the empirical results with remarkable accuracy . These results suggest that Eq . ( 1 ) is able to capture the model scaling behavior across different scaling approaches and model sizes . To ensure that our results generalize across different language pairs , we examine the fit of our scaling law on encoder / decoder scaling models trained on German→English ( De→En ) , Chinese-to-English ( Zh→En ) , and English-to-Chinese ( En→Zh ) translation tasks . Figure 5 presents the results evaluated on in-domain held-out data . Similar to the En→De case , our proposed functional form is able to closely capture the scaling behavior of these models . To keep the scope of our experiments manageable , we focus on depth scaling in our analysis . In Appendix H , we ran large-scale experiments to see if other scaling approaches show significant deviations from the trends identified here . For large-scale models , which are the focus of this study , we observe that different scaling approaches yield very similar results : the final achieved test loss across different scaling approaches shows just 1− 2 % variation . We observed that for small models ( < 100M parameters ) , this variation tends to grow , suggesting that model shape plays a key role in the performance of small models . We leave further analysis of this phenomenon to future research . 5See Appendix F for the full description of the models . 2.3 ANALYSIS The above results suggest that the scaling law formalized in Eq . ( 1 ) captures the scaling behavior of the Transformer NMT models in multiple language pairs . As such , we can study the fitted coefficients to fully understand the scaling properties of these models . Several observations are in order : Decoder vs Encoder Scaling : On all our test sets , the decoder exponents were observed to be larger than the encoder exponents , pd > pe ( see Figure 9 ) . As a result , when improving the test loss is concerned , it is much more effective to scale the decoder rather than the encoder . This is contrary to the usual practice ; due to latency considerations , many practitioners train NMT models with deep encoders and shallow decoders ( Kasai et al. , 2021 ) . Our results suggest this practice could be sub-optimal in terms of loss reduction . Proposition 1 below provides guidance on how to allocate parameters in between the encoder and decoder optimally . The proof is presented in Appendix G. Proposition 1 ( Optimal Scaling ) . Assume the loss performance of the model is described by Eq . ( 1 ) . Let B denote the budget for the total number of parameters . Then , the optimal encoder / decoder sizes ( denoted respectively by N∗e and N ∗ d ) are N ∗ e = pe pe+pd B and N∗d = pd pe+pd B . In addition , when optimally scaling the model , the scaling law reduces to L̂opt ( B ) = α∗B− ( pd+pe ) +L∞ , where α∗ ≡ α ( pe+pdpe ) pe ( pe+pdpd ) pd . Proposition 1 suggests that when NeNd = pe pd , the optimal scaling behavior can be achieved . Inspection of the functional form of Eq . ( 1 ) suggests that as long as Nd/Ne is fixed as the model scales ( i.e . encoder and decoder grow proportionally together ) , the optimal scaling exponent , ( pe + pd ) , can be achieved , albeit with a potentially sub-optimal multiplicative constant , α # . To examine the magnitude of this sub-optimality , in Figure 10 , we compare the multiplicative constants resulting from proportional scaling of the encoder and decoder with different values of Nd/Ne . The results suggest that as long as the parameter allocation is not extremely far from ( N∗e , N ∗ d ) , the scaling scheme is approximately optimal . In particular , symmetrically scaling the encoder and decoder layers , is barely distinguishable from the optimal scaling scheme . 3 EFFECT OF DATASET COMPOSITION BIAS ON SCALING BEHAVIOR Translation deals with the problem of mapping a sequence in one language into another language . A good translation should not only be adequate and fluent , but should ideally also adopt the style of a sentence naturally written in the target language . This necessitates MT models to make sense of natural looking inputs and generate natural looking outputs . As mentioned in Section 2 , the examples used to train or test NMT models carry a critical bias , which we refer to as composition bias . Composition bias is introduced because of the unavailability of source-target examples ( pairs ) that are both natural in the accessible data generating distribution . For any given naturally generated text in a language , the corresponding text in the other language is either translated by humans , introducing translationese bias or translated by other machine translation systems , introducing MT bias . We consider both biases affecting the problem from a similar angle , hence we bundle them and call it composition bias . While machine translation by design has composition bias in the training/test sets employed ( Freitag et al. , 2020 ; Riley et al. , 2020 ) , its effect on model scaling is unknown . In this section we investigate the role of composition bias in scaling and identify critical factors playing a role . We caution the reader to not take the composition bias as a problem specific to NMT . In fact as most training corpora in NMT are web-crawled , they can contain machine translation output on either the source or target side . Considering the growth of generated content in the web by machine learning models 6 7 , it is not improbable that a proportion of the content collected and used by machine learning models is going to be biased by other models that are continuously generating content . The Effect of Test Set Construction : We first examine the impact of composition bias on the test sets . Figure 9 shows the fitted scaling exponents for all our test sets . The plot suggests that the scaling powers for source-original test sets are drastically different from those of target-original test sets . This behavior is in direct contrast with language modeling setting ( Kaplan et al. , 2020 ) where it was observed that the evaluation on different test sets merely acted as a scaling penalty that only changed the multiplicative constants of the scaling law . To elucidate this phenomenon further , in Figure 11 of the appendix , we compare the scaling trends for different source and target original test sets . Several observations are in order : Test sets with a similar composition approach ( source or target original ) have a qualitatively similar scaling behavior . However , scaling behavior is vastly different between the two composition approaches . Reducible loss quickly decays to zero for source original test sets . In fact , we observe that scaling our baseline 6L-6L model by a factor of 2.5 is sufficient for ensuring that reducible loss is below 0.05 for all source original test sets . In contrast , on target original test sets , the loss decays much more slowly with model size . For comparison , to ensure that reducible loss is below 0.05 for all target original test sets , we estimate that the baseline model has to be scaled up by a factor of 11 . Because of this behavior , the value of larger models in NMT is closely tied to their evaluation sets : On source original data , due to larger scaling exponents , even moderate increases in model size are sufficient for pushing the reducible loss close to zero . Hence , beyond a few hundred million parameters , there is no benefit in increasing the model size . In contrast , for target original data , which generally have smaller scaling exponents , large models are needed to push the reducible loss to zero . The Effect of Training Set Construction : In this section , we briefly examine the role of training data construction on the scaling behavior . To do this , we generate two En→De datasets , that were not used in the previous experiments . One fully target original and another completely source original . To generate the target original dataset , we compile a set of German documents from the web . Documents are screened to ensure the data is not machine generated . We use a Hybrid model ( with 380M parameters ) ( Chen et al. , 2018 ) to back-translate ( BT ) these documents to English . Similarly , for the source original data , we collect human generated English documents and ( forward ) translate them to German using a hybrid model ( with approximately 327M parameters ) . Both datasets provide us with approximately 2.2 billion training examples . We mimic the experimental setup of Section 2 . Note that even though these datasets are not human generated , they reflect important aspects of training large NMT models . Many modern NMT datasets are harvested from the web and as a result , are contaminated with machine generated data . Moreover , many popular data augmentation algorithms such as Back Translation ( Sennrich et al. , 2016 ) , sequence level distillation ( Kim & Rush , 2016 ) and self training ( He et al. , 2020 ) purposefully add machine generated data into the training pipeline in order to take advantage of monolingual data . Figure 6 describes the scaling behavior for models trained on target-original data . We observe that even though larger models are successful in reducing the training loss , they are unable to improve the test loss after roughly 400M parameters . Once this size threshold is exceeded , models overfit the training data and the test loss starts to deteriorate across all of our test sets . We hypothesize that this size threshold corresponds to the capacity of the original back-translation model . This assertion suggests that in order for back-translation to be beneficial for training large models , it has to be performed with a models with comparable capacity or higher . Although quite intriguing , we leave the verification of this hypothesis to future work . Figure 7 paints another interesting picture for the models trained on the source-original data only , implying the target side having the composition bias , expected to be simpler , dull and not rich in its content , in short - not natural looking . As experiments suggest , even our smallest models are able to achieve extremely low loss values ( roughly 0.16 ) , with an apparent overfitting pattern . We believe the same phenomenon is also related to the `` data simplification '' effect seeked by non-autoregressive models in NMT ( Zhou et al. , 2021 ) . 6https : //openai.com/blog/gpt-3-apps/ 7https : //blog.google/products/translate/one-billion-installs/ 4 EVOLUTION OF GENERATION QUALITY We examine the effects of scaling on the output quality as measured by BLEU score 8 . For this analysis , we focus on output generated via beam search ( Wu et al. , 2016 ) . For tractability purposes , we do not attempt to tune the ( many ) hyper-parameters of beam-search for each model . Instead , we use the configuration optimized for the baseline model ( listed in Appendix J ) in all the decoding tasks . Figure 8 presents the co-evolution of BLEU score and cross-entropy loss throughout the training for all of our models . Depending on the construction of the test sets , two different empirical behaviors emerge . On target-original test sets , larger models are able to improve ( lower ) the test loss . These improvements in the loss are accompanied with consistent improvements ( increases ) in BLEU score . In fact , we observe that a simple power law can capture the relationship between BLEU score and cross-entropy loss for high-quality models . 9 In contrast , on source-original test sets , this relationship is absent ; larger models consistently achieve better test losses , however , beyond a certain threshold , BLEU scores begin to deteriorate . Figures 27 and 28 exhibit that this phenomenon is not due to over-training ; the BLEU score gap between large and small models is persistent throughout training . To ensure that this observation truly reflects the generation quality of the models ( as opposed to potential biases of BLEU score ) , we repeat our analysis with BLEURT score ( Sellam et al. , 2020a ; b ) . The results are presented in Appendix K. The results show that BLEURT scores closely mirror the behavior of BLEU scores with respect to model scaling . A careful look at the left-subplots of Figures 8 brings up another interesting trend . At similar values of the test loss , encoder-scaled models result in better generation quality compared to decoder-scaled models . This findings agrees with previous work that relied on encoder-scaling when optimizing for BLEU and inference latency ( Kasai et al. , 2021 ) . Whether these differences in the effects of encoder-scaling and decoder-scaling are caused by insufficient search algorithms , or just different model fits from different architectural priors is left to future work . 8We computed the BLEU scores using an internal reimplementation of Moses scorer : mteval-v13a.pl . 9Fitting details are presented in Appendix J . 5 CONCLUSION AND LIMITATION In this work we have attempted to quantify the evolution of model quality as a function of model capacity for encoder-decoder NMT models . While a univariate scaling law describing the crossentropy as a function of the total number of parameters in the model is insufficient , a bivariate law treating the number of encoder and decoder parameters as separate variables adequately describes the scaling behavior of these models under various scaling strategies . We validate this behavior on a variety of language pairs and evaluation sets . Whether this behavior is intrinsic to the encoder-decoder architecture , or arising from the nature of the NMT task , requires further study . Next , we show that this scaling behavior is highly dependent on the composition of the evaluation data , specifically on whether the source or target sentences are “ original ” . Our findings indicate that target-original evaluation sets continue benefiting from model scaling , while the reducible error on source-original evaluation sets quickly saturates to 0 . This could be an artifact of the lack of diversity in translated text ; a simpler target distribution doesn ’ t require much capacity to model while generating fluent or natural-looking text could benefit much more from scale . We also study how the composition of training data affects the scaling behavior of models . When training on target-original ( back-translated ) text , model quality keeps improving until a point after which the trend saturates . In our study the capacity where saturation manifests first is perilously close to the capacity of the model used for back-translation , indicating that the capacity of the generative model used to generate synthetic text might have a role to play , but this requires further investigation . When training on source-original text , even low-capacity models are sufficient to reach the irreducible loss region , painting a gloomy picture for synthetic data . While we have explored these ideas in the context of machine translation , given the proliferation of generative models this problem will likely be a challenge for future practitioners training on web-scraped monolingual datasets as well . For low-resource languages , the proliferation of machine translated text is already a problem given that a significant portion of web text in these languages is machine translated . Finally , we attempt to understand how generation quality evolves with the improvements in crossentropy resulting from model scaling . As with our previous findings , dataset composition plays a major role in determining the trends . For source-original evaluation sets , the correlation between cross-entropy and generation quality breaks down . On target-original evaluation , we observe an inverse correlation between cross-entropy and BLEU , suggesting that improved model fit results in a corresponding improvement in generation quality . The slope of this relationship is different for encoder-scaling and decoder-scaling , with encoder-scaled models performing better on BLEU than decoder-scaled models , at the same level of cross-entropy loss . Whether this is an artifact of our search strategy or the difference in architectural priors is something that requires further investigation . 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Yonghui Wu , Mike Schuster , Zhifeng Chen , Quoc V Le , Mohammad Norouzi , Wolfgang Macherey , Maxim Krikun , Yuan Cao , Qin Gao , Klaus Macherey , et al . Google ’ s neural machine translation system : Bridging the gap between human and machine translation . arXiv preprint arXiv:1609.08144 , 2016 . Ruibin Xiong , Yunchang Yang , Di He , Kai Zheng , Shuxin Zheng , Chen Xing , Huishuai Zhang , Yanyan Lan , Liwei Wang , and Tieyan Liu . On layer normalization in the transformer architecture . In International Conference on Machine Learning , pp . 10524–10533 . PMLR , 2020 . Chunting Zhou , Graham Neubig , and Jiatao Gu . Understanding knowledge distillation in nonautoregressive machine translation , 2021 . A CONNECTIONS TO RECENT LITERATURE In this section , we provide a more detailed overview of the scaling laws literature immediately relevant to our investigation . Hestness et al . ( 2017 ) examine data scaling laws for LSTM-based encoder-decoder NMT models . They show that the model performance exhibits an empirical powerlaw relationship with the size of the training data . Rosenfeld et al . ( 2019 ) study model and data scaling laws for image classification and language modeling tasks . They provide a bivariate scaling law that jointly capture the model and data scaling behaviors for each of these tasks . Crucially , they show that scaling laws can be used to effectively extrapolate beyond model/data configurations used for fitting them . In particular , they show that one can use smaller models to fit the scaling law and then use the law to accurately predict the performance of larger models . Our results ( Figures 3 and 4 ) validate that this behavior also holds for our setting . Kaplan et al . ( 2020 ) study the scaling behavior of decoder-only Transformer models used in language modeling . They identify three factors that principally determine the model performance : the number of ( non-embedding ) model parameters , the size of the training dataset , and the amount of compute used for training . In the scenario where data and compute are abundant , they show that model performance exhibits an empirical power-law relationship with the total number of ( non-embedding ) parameters in the network . Henighan et al . ( 2020 ) verify this power-law behavior for other autoregressive generative modeling tasks such as image modeling , video modeling , and mathematical problem solving . In contrast , in our setting , we observe that the total number of parameters is not a good predictor of the test loss in NMT . Instead , we show that encoder-decoder NMT models scale according to a bivariate function of the number of encoder parameters and the number of decoder parameters . Kaplan et al . ( 2020 ) also examine how scaling affects the out-of-distribution generalization behavior in language models . Their results suggest that different test sets exhibit rather similar scaling trends as model size increases . In comparison , in the NMT setting , we show that different test sets , depending on their naturalness structure , exhibit drastically different scaling behaviors . The most closely related study to ours is the work of Gordon et al . ( 2021 ) . They also examine scaling laws for NMT , albeit at a different parameter regime : Gordon et al . focus on small models ( 393K56M parameters ) while here , we focus on large scale models ( 100M-4B parameters ) . In addition , we examine the general scenario where encoder and decoder can scale asymmetrically ( which is a common practice in NMT ) while Gordon et al . only consider symmetric scaling . Despite the differences in the experimental setup , when focusing on the symmetric scaling setting , the conclusions of both studies mostly align . In particular , when symmetrically scaling the model , our scaling law simplifies to L ( N ) = αN−p + L∞ which agrees with the conclusions of Gordon et al . One difference in our conclusions is the presence of the L∞ term in our scaling law . This term , which captures the irreducible loss in the learning problem , does not appear in Gordon et al ’ s analysis . We suspect that this might be due to the small models used in that analysis which can not statistically observe such asymptotic terms . Gordon et al . also examine the relationship between test cross-entropy loss and BLEU score . They observe that as models scale and cross-entropy loss improves , BLEU score also improves in tandem . In comparison , we unveil a more intricate relationship in our study : We show that when the test data is target-original , improvements in the test cross-entropy loss are correlated with improvements in generation quality ( as measured by BLEU/BLEURT scores ) . However , when the test set is sourceoriginal , the correlation between cross-entropy loss and generation quality breaks down . In this setting , we observe that generation quality improves up until a certain threshold and then deteriorates ( see the right plot in Figure 8 ) . B MISCELLANEOUS FIGURES | The paper provides scaling laws for machine translation models. They experiment on different language pairs and domains and focus on large-scale Transformer-based NMT models. They find that the scaling behavior of cross-entropy loss can be described as a bivariate function of encoder and decoder size and make recommendations for optimal allocation of encoder/decoder capacity. They also find that “translationese” can affect the model scaling behavior. Finally, they study the correlations between cross-entropy loss and BLEU/BLEURT scores. | SP:7c0300ec9f6fe3242ba9e6d6f7098059fa60be0f |
Scaling Laws for Neural Machine Translation | 1 INTRODUCTION Scaling properties of neural networks have long been an intriguing topic of study ( Ahmad & Tesauro , 1988 ; Amari et al. , 1992 ) . Along with the practical success of modern neural networks at scale , theoretical understanding of the factors governing the quality and training dynamics of large neural networks has also being developing ( Advani & Saxe , 2017 ; Rosenfeld et al. , 2019 ; Geiger et al. , 2020 ; Ghorbani et al. , 2020 ; Cho , 2020 ; Hutter , 2021 ; Bahri et al. , 2021 ; Loureiro et al. , 2021 ) . In particular , scaling model sizes , datasets and the total computation budget has been identified as a reliable approach to improve generalization performance on several machine learning tasks . For many of these tasks the scaling behavior of neural networks is highly predictable ; model fit or test loss can be described precisely as a function of its number of parameters ( Hestness et al. , 2017 ; Kaplan et al. , 2020 ; Henighan et al. , 2020 ; Hernandez et al. , 2021 ; Rosenfeld et al. , 2019 ) . Neural machine translation ( NMT ) has long enjoyed the benefits of scaling ( Huang et al. , 2019 ; Arivazhagan et al. , 2019 ; Lepikhin et al. , 2021 ) , but studies investigating the scaling behavior of NMT models are missing . We present a large-scale systematic study of scaling laws for encoder-decoder Transformer models applied to NMT ( Vaswani et al. , 2017 ) . We start with highlighting the major differences between decoder-only language models , where the majority of the previous work has focused , and encoder-decoder ( conditional ) language models applied to NMT . The two differ along a few crucial dimensions . The first difference results from the very nature of the separate architectures being used , i.e . decoder-only vs encoder-decoder . The presence of separate architectural components complicates the study of scaling properties due to the increased degree of freedom . Second , contrary to language modeling , the task of machine translation is conditional : the task is predictive rather than fully generative . Furthermore , this prediction task is ambiguous : there is no one right answer for a given source , and translations can vary substantially depending on the translator ’ s incentives . This manifests itself as different scaling benefits for different test sets . To take an extreme example , a test set translated by someone who writes nearly wordfor-word translations may benefit less from model scaling than one translated by someone who considers each translation a work of art . In this work , these differences in difficulty coincide with the translation direction of the test set ; that is , whether the source was translated into the target ( source-original ) or vice versa ( target-original ) . Source-original data has translated text on the target side , which contains several artifacts of “ translationese ” that distinguish it from text originally written in that language , often lacking the diversity and complexity of “ natural ” text ( Koppel & Ordan , 2011 ) , while target-original data requires the prediction of more complex natural text on the target side . Finally , unlike language models , NMT is evaluated on metrics that quantify generation quality against reference translations ( for eg . BLEU ) ( Papineni et al. , 2002 ) instead of evaluating model fit ( perplexity ) on an evaluation set . In this paper , we aim to provide empirical answers to the following research questions : 1 . Does the encoder-decoder architecture for NMT share the same scaling law function as language models ? Contrary to previous work on LM , we show that a univariate law depending on the total number of parameters in the network does not adequately describe the scaling behavior of NMT models . Our scaling laws parameterize the cross entropy loss as a bivariate function of the number of encoder parameters and the number of decoder parameters as separate variables . Our results indicate that the scaling behavior is largely determined by the total capacity of the model , and the capacity allocation between the encoder and the decoder . 2 . How does the naturalness of source/target side data affect scaling behavior ? We study the effect of naturalness of the source and the target text , both for training and evaluation . When evaluating with target side natural text , scaling the model capacity continues improving model quality throughout our range of measurements . On the other hand , improvements on cross-entropy saturate ( or reaches the irreducible error region ) on source side natural evaluation sets even for moderately-sized models . 3 . Do scaling improvements in cross-entropy translate into corresponding improvements in generation quality ? Finally we study the relationship between generation quality and cross-entropy and how their correlation changes as we : ( i ) Scale different components of the model ( encoder vs decoder ) and ( ii ) Evaluate on source-natural or target-natural test sets . Our results on multiple language pairs and training/test data compositions validate that model scaling predictably improves the cross-entropy on validation data . However , our findings also raise several questions regarding the effect of naturalness of training and evaluation text and how cross-entropy eventually relates with generation quality for auto-regressive generative models . 1 2 EFFECT OF SCALING ON CROSS-ENTROPY 2.1 EXPERIMENTAL SETTING Model Architectures and Training We train a series of pre-layer norm Transformer networks with varying sizes ( Xiong et al. , 2020 ) . Models are trained with per-token cross-entropy loss and Adafactor optimizer ( Shazeer & Stern , 2018 ) . All models are trained with a fixed batch-size of 500k tokens and dropout rate of 0.1 for residuals , feed-forward activations and attention . All models are trained to near convergence for 500k training steps . Details of the model hyper-parameters are described in Appendix C. Model Scaling Transformer architecture consists of Transformer Blocks : a cascade of self-attention , cross-attention and feed-forward layers , each having multiple adjustable hyper-parameters ( e.g . model-dimension , number of attention heads , attention projection dimension etc. ) . Considering the combinatorial expansion of the search space for scaling each one , in this study we choose to vary only 1A few weeks before the publication of this manuscript on Arxiv , Gordon et al . ( 2021 ) appeared on OpenReview . While both papers study scaling laws for NMT , our studies focus on different parameter regimes ( 393K-56M vs 100M-4B ) . See Appendix A for a comparison of the results . the total number of Transformer Blocks , while keeping the internal hyper-parameters intact across different scales . In other words , we scale the depth of the Transformers while keeping width and other variables fixed . 2 We use GPipe pipeline parallelism for our experiments ( Huang et al. , 2019 ) . In an encoder-decoder Transformer architecture for NMT , depth scaling can naturally be implemented by varying encoder-decoder blocks independently or symmetrically . Hence , we examine the change in the cross-entropy loss as the number of parameters increase with three depth scaling approaches : Encoder Scaling : vary encoder depth ( 2 to 64 ) while decoder depth is fixed ( 6 layers ) . Decoder Scaling : vary decoder depth ( 2 to 64 ) while encoder depth is fixed ( 6 layers ) . Symmetric Scaling : increasing decoder and encoder layers together ( from 2 to 64 ) , i.e . the number of Transformer Blocks in the encoder and decoder being equal . For all experiments , configuration of the individual layers is unchanged : the model dimension , width of the feed-forward layer , and number of attention heads are fixed respectively at 1024 , 8192 , and 16 . 3 Each encoder layer adds approximately 20M parameters to the model while each decoder layer adds around 25M parameters . We train 200 such models which scale the model size by approximately a factor of 50 ( from 80M parameters to 4B parameters ) . Following the convention , we do not count the parameters in the embedding and softmax layers towards the model size . We focus our study on large-scale models : our smallest models require 200 TPUv3 days to train to convergence while our largest models require 2700 TPUv3 days of training . Language Pairs We report results on four language pairs : English→German , German→English , English→Chinese , and Chinese→English . We use in-house web-crawled training datasets with around 2.2 billion sentence pairs ( approximately 55 billion tokens ) for English↔German and 781 million sentence pairs for English↔Chinese . Evaluation Sets We use a variety of test sets for evaluation covering different domains : ( i ) WebDomain ( ii ) News-Domain ( iii ) Wikipedia ( iv ) Patents . The news-domain test sets come from the WMT2019 ( Barrault et al. , 2019 ) evaluation campaign ( newstest2019 ) for all language pairs . The other test sets are internal test sets representing the different domains , ranging from 500 to 5000 sentence pairs . For each domain , we randomly sample sentences in the source language and use professional translators to generate a reference translation in the target language . Throughout the paper , we will refer this type of test sets as source-original as the source sentences have been crawled from the web while the reference translations are added later . For most of the domains , we also have a target-original counterpart which is generated in the opposite direction : Sentences are crawled in the target language and human translated into the source language . Earlier work ( Freitag et al. , 2019 ; 2020 ; Graham et al. , 2020 ) showed that it is important to differentiate between the two different kinds of test sets as the style of natural sentences and human ( or machine ) translations ( translationese ) is quite different . Cross-entropy loss is evaluated on the different test sets during training . To reduce the variation caused by the parameter fluctuations at the end of the training , we present the median loss over the last 50k steps of the training as the final loss . 2.2 RESULTS Figure 1 shows the empirical evolution of the test loss on the Web-Domain test sets for encoder and decoder scaling for English→German . To compare the empirical results with the scaling laws present in the literature for decoder only models ( Kaplan et al. , 2020 ; Henighan et al. , 2020 ) , we have fitted a power law of the form L̂ ( N ) = αN−p+L∞ to the data . 4 Here , N is the total number of parameters outside of embedding / softmax layers and { α , p , L∞ } are fitted parameters . As Figure 1 suggests , scaling the encoder has different effects on the test loss compared to scaling the decoder . As such , simple power-laws that only consider the total parameter count , fail to capture the correct scaling behavior of the model . 2In Appendix H , we examine the role of scaling approach in our results . See Section 2 for more details . 3A complete description of the model architecture is provided in Appendix C 4Details of the curve fitting procedure are presented in Appendix I. laws are unable to capture the correct scaling behavior . R-squared ( 100× explained variancetotal variance ) and maximum absolute deviation ( ‖ · ‖∞ ) are reported for each fit . Proposed Scaling Law To tackle this issue , we present a new scaling law that reflects the encoderdecoder nature of the architecture as well as the bilingual format of the data . Let Ne and Nd be the number of non-embedding parameters in the encoder and the decoder respectively . Then , our proposed scaling law has the form L̂ ( Ne , Nd ) = αN −pe e N −pd d + L∞ ( 1 ) where { α , pe , pd , L∞ } are test set specific ( fitted ) parameters . In this formulation , α corresponds to the maximum loss reduction that one can hope from scaling , while pe and pd are the scaling exponents for encoder and decoder respectively . L∞ corresponds to the irreducible loss of the data . In Figure 2 , we fit our scaling law to the empirical loss values achieved by encoder/decoder scaling models on Web-Domain test sets . The results suggest that our scaling law almost fully captures the variation in the data ( R2 > 99 % ) . In particular , in contrast to traditional scaling laws , our proposed formulation is able to capture the different scaling behaviors of encoder and decoder scaled models . While these improved fits are encouraging , to truly show that a candidate scaling law captures the model scaling behavior in NMT , it is necessary to verify the out-of-sample prediction power of the law . We provide two such verifications ; we compare the predictions of the scaling law fitted on encoder / decoder scaling models with the empirical loss values from ( i ) symmetrically scaled models , ( ii ) models randomly sampled from the 2D grid of all potential encoder-decoder configurations.5 Symmetrically scaled models are chosen for this verification as they are widely used by the NLP community ( Raffel et al. , 2020 ) . Randomly sampled models are chosen to ensure that our results are not biased towards any particular model class . Figures 3 & 4 present the out-of-sample results . As the plots suggest , the predictions of the scaling law match the empirical results with remarkable accuracy . These results suggest that Eq . ( 1 ) is able to capture the model scaling behavior across different scaling approaches and model sizes . To ensure that our results generalize across different language pairs , we examine the fit of our scaling law on encoder / decoder scaling models trained on German→English ( De→En ) , Chinese-to-English ( Zh→En ) , and English-to-Chinese ( En→Zh ) translation tasks . Figure 5 presents the results evaluated on in-domain held-out data . Similar to the En→De case , our proposed functional form is able to closely capture the scaling behavior of these models . To keep the scope of our experiments manageable , we focus on depth scaling in our analysis . In Appendix H , we ran large-scale experiments to see if other scaling approaches show significant deviations from the trends identified here . For large-scale models , which are the focus of this study , we observe that different scaling approaches yield very similar results : the final achieved test loss across different scaling approaches shows just 1− 2 % variation . We observed that for small models ( < 100M parameters ) , this variation tends to grow , suggesting that model shape plays a key role in the performance of small models . We leave further analysis of this phenomenon to future research . 5See Appendix F for the full description of the models . 2.3 ANALYSIS The above results suggest that the scaling law formalized in Eq . ( 1 ) captures the scaling behavior of the Transformer NMT models in multiple language pairs . As such , we can study the fitted coefficients to fully understand the scaling properties of these models . Several observations are in order : Decoder vs Encoder Scaling : On all our test sets , the decoder exponents were observed to be larger than the encoder exponents , pd > pe ( see Figure 9 ) . As a result , when improving the test loss is concerned , it is much more effective to scale the decoder rather than the encoder . This is contrary to the usual practice ; due to latency considerations , many practitioners train NMT models with deep encoders and shallow decoders ( Kasai et al. , 2021 ) . Our results suggest this practice could be sub-optimal in terms of loss reduction . Proposition 1 below provides guidance on how to allocate parameters in between the encoder and decoder optimally . The proof is presented in Appendix G. Proposition 1 ( Optimal Scaling ) . Assume the loss performance of the model is described by Eq . ( 1 ) . Let B denote the budget for the total number of parameters . Then , the optimal encoder / decoder sizes ( denoted respectively by N∗e and N ∗ d ) are N ∗ e = pe pe+pd B and N∗d = pd pe+pd B . In addition , when optimally scaling the model , the scaling law reduces to L̂opt ( B ) = α∗B− ( pd+pe ) +L∞ , where α∗ ≡ α ( pe+pdpe ) pe ( pe+pdpd ) pd . Proposition 1 suggests that when NeNd = pe pd , the optimal scaling behavior can be achieved . Inspection of the functional form of Eq . ( 1 ) suggests that as long as Nd/Ne is fixed as the model scales ( i.e . encoder and decoder grow proportionally together ) , the optimal scaling exponent , ( pe + pd ) , can be achieved , albeit with a potentially sub-optimal multiplicative constant , α # . To examine the magnitude of this sub-optimality , in Figure 10 , we compare the multiplicative constants resulting from proportional scaling of the encoder and decoder with different values of Nd/Ne . The results suggest that as long as the parameter allocation is not extremely far from ( N∗e , N ∗ d ) , the scaling scheme is approximately optimal . In particular , symmetrically scaling the encoder and decoder layers , is barely distinguishable from the optimal scaling scheme . 3 EFFECT OF DATASET COMPOSITION BIAS ON SCALING BEHAVIOR Translation deals with the problem of mapping a sequence in one language into another language . A good translation should not only be adequate and fluent , but should ideally also adopt the style of a sentence naturally written in the target language . This necessitates MT models to make sense of natural looking inputs and generate natural looking outputs . As mentioned in Section 2 , the examples used to train or test NMT models carry a critical bias , which we refer to as composition bias . Composition bias is introduced because of the unavailability of source-target examples ( pairs ) that are both natural in the accessible data generating distribution . For any given naturally generated text in a language , the corresponding text in the other language is either translated by humans , introducing translationese bias or translated by other machine translation systems , introducing MT bias . We consider both biases affecting the problem from a similar angle , hence we bundle them and call it composition bias . While machine translation by design has composition bias in the training/test sets employed ( Freitag et al. , 2020 ; Riley et al. , 2020 ) , its effect on model scaling is unknown . In this section we investigate the role of composition bias in scaling and identify critical factors playing a role . We caution the reader to not take the composition bias as a problem specific to NMT . In fact as most training corpora in NMT are web-crawled , they can contain machine translation output on either the source or target side . Considering the growth of generated content in the web by machine learning models 6 7 , it is not improbable that a proportion of the content collected and used by machine learning models is going to be biased by other models that are continuously generating content . The Effect of Test Set Construction : We first examine the impact of composition bias on the test sets . Figure 9 shows the fitted scaling exponents for all our test sets . The plot suggests that the scaling powers for source-original test sets are drastically different from those of target-original test sets . This behavior is in direct contrast with language modeling setting ( Kaplan et al. , 2020 ) where it was observed that the evaluation on different test sets merely acted as a scaling penalty that only changed the multiplicative constants of the scaling law . To elucidate this phenomenon further , in Figure 11 of the appendix , we compare the scaling trends for different source and target original test sets . Several observations are in order : Test sets with a similar composition approach ( source or target original ) have a qualitatively similar scaling behavior . However , scaling behavior is vastly different between the two composition approaches . Reducible loss quickly decays to zero for source original test sets . In fact , we observe that scaling our baseline 6L-6L model by a factor of 2.5 is sufficient for ensuring that reducible loss is below 0.05 for all source original test sets . In contrast , on target original test sets , the loss decays much more slowly with model size . For comparison , to ensure that reducible loss is below 0.05 for all target original test sets , we estimate that the baseline model has to be scaled up by a factor of 11 . Because of this behavior , the value of larger models in NMT is closely tied to their evaluation sets : On source original data , due to larger scaling exponents , even moderate increases in model size are sufficient for pushing the reducible loss close to zero . Hence , beyond a few hundred million parameters , there is no benefit in increasing the model size . In contrast , for target original data , which generally have smaller scaling exponents , large models are needed to push the reducible loss to zero . The Effect of Training Set Construction : In this section , we briefly examine the role of training data construction on the scaling behavior . To do this , we generate two En→De datasets , that were not used in the previous experiments . One fully target original and another completely source original . To generate the target original dataset , we compile a set of German documents from the web . Documents are screened to ensure the data is not machine generated . We use a Hybrid model ( with 380M parameters ) ( Chen et al. , 2018 ) to back-translate ( BT ) these documents to English . Similarly , for the source original data , we collect human generated English documents and ( forward ) translate them to German using a hybrid model ( with approximately 327M parameters ) . Both datasets provide us with approximately 2.2 billion training examples . We mimic the experimental setup of Section 2 . Note that even though these datasets are not human generated , they reflect important aspects of training large NMT models . Many modern NMT datasets are harvested from the web and as a result , are contaminated with machine generated data . Moreover , many popular data augmentation algorithms such as Back Translation ( Sennrich et al. , 2016 ) , sequence level distillation ( Kim & Rush , 2016 ) and self training ( He et al. , 2020 ) purposefully add machine generated data into the training pipeline in order to take advantage of monolingual data . Figure 6 describes the scaling behavior for models trained on target-original data . We observe that even though larger models are successful in reducing the training loss , they are unable to improve the test loss after roughly 400M parameters . Once this size threshold is exceeded , models overfit the training data and the test loss starts to deteriorate across all of our test sets . We hypothesize that this size threshold corresponds to the capacity of the original back-translation model . This assertion suggests that in order for back-translation to be beneficial for training large models , it has to be performed with a models with comparable capacity or higher . Although quite intriguing , we leave the verification of this hypothesis to future work . Figure 7 paints another interesting picture for the models trained on the source-original data only , implying the target side having the composition bias , expected to be simpler , dull and not rich in its content , in short - not natural looking . As experiments suggest , even our smallest models are able to achieve extremely low loss values ( roughly 0.16 ) , with an apparent overfitting pattern . We believe the same phenomenon is also related to the `` data simplification '' effect seeked by non-autoregressive models in NMT ( Zhou et al. , 2021 ) . 6https : //openai.com/blog/gpt-3-apps/ 7https : //blog.google/products/translate/one-billion-installs/ 4 EVOLUTION OF GENERATION QUALITY We examine the effects of scaling on the output quality as measured by BLEU score 8 . For this analysis , we focus on output generated via beam search ( Wu et al. , 2016 ) . For tractability purposes , we do not attempt to tune the ( many ) hyper-parameters of beam-search for each model . Instead , we use the configuration optimized for the baseline model ( listed in Appendix J ) in all the decoding tasks . Figure 8 presents the co-evolution of BLEU score and cross-entropy loss throughout the training for all of our models . Depending on the construction of the test sets , two different empirical behaviors emerge . On target-original test sets , larger models are able to improve ( lower ) the test loss . These improvements in the loss are accompanied with consistent improvements ( increases ) in BLEU score . In fact , we observe that a simple power law can capture the relationship between BLEU score and cross-entropy loss for high-quality models . 9 In contrast , on source-original test sets , this relationship is absent ; larger models consistently achieve better test losses , however , beyond a certain threshold , BLEU scores begin to deteriorate . Figures 27 and 28 exhibit that this phenomenon is not due to over-training ; the BLEU score gap between large and small models is persistent throughout training . To ensure that this observation truly reflects the generation quality of the models ( as opposed to potential biases of BLEU score ) , we repeat our analysis with BLEURT score ( Sellam et al. , 2020a ; b ) . The results are presented in Appendix K. The results show that BLEURT scores closely mirror the behavior of BLEU scores with respect to model scaling . A careful look at the left-subplots of Figures 8 brings up another interesting trend . At similar values of the test loss , encoder-scaled models result in better generation quality compared to decoder-scaled models . This findings agrees with previous work that relied on encoder-scaling when optimizing for BLEU and inference latency ( Kasai et al. , 2021 ) . Whether these differences in the effects of encoder-scaling and decoder-scaling are caused by insufficient search algorithms , or just different model fits from different architectural priors is left to future work . 8We computed the BLEU scores using an internal reimplementation of Moses scorer : mteval-v13a.pl . 9Fitting details are presented in Appendix J . 5 CONCLUSION AND LIMITATION In this work we have attempted to quantify the evolution of model quality as a function of model capacity for encoder-decoder NMT models . While a univariate scaling law describing the crossentropy as a function of the total number of parameters in the model is insufficient , a bivariate law treating the number of encoder and decoder parameters as separate variables adequately describes the scaling behavior of these models under various scaling strategies . We validate this behavior on a variety of language pairs and evaluation sets . Whether this behavior is intrinsic to the encoder-decoder architecture , or arising from the nature of the NMT task , requires further study . Next , we show that this scaling behavior is highly dependent on the composition of the evaluation data , specifically on whether the source or target sentences are “ original ” . Our findings indicate that target-original evaluation sets continue benefiting from model scaling , while the reducible error on source-original evaluation sets quickly saturates to 0 . This could be an artifact of the lack of diversity in translated text ; a simpler target distribution doesn ’ t require much capacity to model while generating fluent or natural-looking text could benefit much more from scale . We also study how the composition of training data affects the scaling behavior of models . When training on target-original ( back-translated ) text , model quality keeps improving until a point after which the trend saturates . In our study the capacity where saturation manifests first is perilously close to the capacity of the model used for back-translation , indicating that the capacity of the generative model used to generate synthetic text might have a role to play , but this requires further investigation . When training on source-original text , even low-capacity models are sufficient to reach the irreducible loss region , painting a gloomy picture for synthetic data . While we have explored these ideas in the context of machine translation , given the proliferation of generative models this problem will likely be a challenge for future practitioners training on web-scraped monolingual datasets as well . For low-resource languages , the proliferation of machine translated text is already a problem given that a significant portion of web text in these languages is machine translated . Finally , we attempt to understand how generation quality evolves with the improvements in crossentropy resulting from model scaling . As with our previous findings , dataset composition plays a major role in determining the trends . For source-original evaluation sets , the correlation between cross-entropy and generation quality breaks down . On target-original evaluation , we observe an inverse correlation between cross-entropy and BLEU , suggesting that improved model fit results in a corresponding improvement in generation quality . The slope of this relationship is different for encoder-scaling and decoder-scaling , with encoder-scaled models performing better on BLEU than decoder-scaled models , at the same level of cross-entropy loss . Whether this is an artifact of our search strategy or the difference in architectural priors is something that requires further investigation . 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Yonghui Wu , Mike Schuster , Zhifeng Chen , Quoc V Le , Mohammad Norouzi , Wolfgang Macherey , Maxim Krikun , Yuan Cao , Qin Gao , Klaus Macherey , et al . Google ’ s neural machine translation system : Bridging the gap between human and machine translation . arXiv preprint arXiv:1609.08144 , 2016 . Ruibin Xiong , Yunchang Yang , Di He , Kai Zheng , Shuxin Zheng , Chen Xing , Huishuai Zhang , Yanyan Lan , Liwei Wang , and Tieyan Liu . On layer normalization in the transformer architecture . In International Conference on Machine Learning , pp . 10524–10533 . PMLR , 2020 . Chunting Zhou , Graham Neubig , and Jiatao Gu . Understanding knowledge distillation in nonautoregressive machine translation , 2021 . A CONNECTIONS TO RECENT LITERATURE In this section , we provide a more detailed overview of the scaling laws literature immediately relevant to our investigation . Hestness et al . ( 2017 ) examine data scaling laws for LSTM-based encoder-decoder NMT models . They show that the model performance exhibits an empirical powerlaw relationship with the size of the training data . Rosenfeld et al . ( 2019 ) study model and data scaling laws for image classification and language modeling tasks . They provide a bivariate scaling law that jointly capture the model and data scaling behaviors for each of these tasks . Crucially , they show that scaling laws can be used to effectively extrapolate beyond model/data configurations used for fitting them . In particular , they show that one can use smaller models to fit the scaling law and then use the law to accurately predict the performance of larger models . Our results ( Figures 3 and 4 ) validate that this behavior also holds for our setting . Kaplan et al . ( 2020 ) study the scaling behavior of decoder-only Transformer models used in language modeling . They identify three factors that principally determine the model performance : the number of ( non-embedding ) model parameters , the size of the training dataset , and the amount of compute used for training . In the scenario where data and compute are abundant , they show that model performance exhibits an empirical power-law relationship with the total number of ( non-embedding ) parameters in the network . Henighan et al . ( 2020 ) verify this power-law behavior for other autoregressive generative modeling tasks such as image modeling , video modeling , and mathematical problem solving . In contrast , in our setting , we observe that the total number of parameters is not a good predictor of the test loss in NMT . Instead , we show that encoder-decoder NMT models scale according to a bivariate function of the number of encoder parameters and the number of decoder parameters . Kaplan et al . ( 2020 ) also examine how scaling affects the out-of-distribution generalization behavior in language models . Their results suggest that different test sets exhibit rather similar scaling trends as model size increases . In comparison , in the NMT setting , we show that different test sets , depending on their naturalness structure , exhibit drastically different scaling behaviors . The most closely related study to ours is the work of Gordon et al . ( 2021 ) . They also examine scaling laws for NMT , albeit at a different parameter regime : Gordon et al . focus on small models ( 393K56M parameters ) while here , we focus on large scale models ( 100M-4B parameters ) . In addition , we examine the general scenario where encoder and decoder can scale asymmetrically ( which is a common practice in NMT ) while Gordon et al . only consider symmetric scaling . Despite the differences in the experimental setup , when focusing on the symmetric scaling setting , the conclusions of both studies mostly align . In particular , when symmetrically scaling the model , our scaling law simplifies to L ( N ) = αN−p + L∞ which agrees with the conclusions of Gordon et al . One difference in our conclusions is the presence of the L∞ term in our scaling law . This term , which captures the irreducible loss in the learning problem , does not appear in Gordon et al ’ s analysis . We suspect that this might be due to the small models used in that analysis which can not statistically observe such asymptotic terms . Gordon et al . also examine the relationship between test cross-entropy loss and BLEU score . They observe that as models scale and cross-entropy loss improves , BLEU score also improves in tandem . In comparison , we unveil a more intricate relationship in our study : We show that when the test data is target-original , improvements in the test cross-entropy loss are correlated with improvements in generation quality ( as measured by BLEU/BLEURT scores ) . However , when the test set is sourceoriginal , the correlation between cross-entropy loss and generation quality breaks down . In this setting , we observe that generation quality improves up until a certain threshold and then deteriorates ( see the right plot in Figure 8 ) . B MISCELLANEOUS FIGURES | This paper studies scaling laws for NMT. It confirms and extends some of the existing work on scaling laws, and as such it is a valuable contribution to the field. In particular, new questions addressed include: does the encoder-decoder architecture of NMT share the same scaling law as decoder-only LMs? How does training data (backtranslated or not) affect scaling? | SP:7c0300ec9f6fe3242ba9e6d6f7098059fa60be0f |
Deep Active Learning with Noise Stability | 1 INTRODUCTION . The success of training a deep neural network highly depends on a huge amount of labeled data . Nevertheless , in the era of big data , much data is unlabeled . This imposes a challenge on exploiting appropriate learning fashions . For instance , given a pool of unlabeled data , one can employ unsupervised manner to learn a model . However , the performance of unsupervised or semi-supervised learning is still bound to that of supervised learning ( Yoo & Kweon , 2019 ) . An ideal option could be annotating all the given data , so as to use the supervised way to learn a model . Nevertheless , the cost for data annotation could be extremely high , such as in medical area ( Gorriz et al. , 2017 ; Konyushkova et al. , 2017 ) . Due to the limited labeling budget , it may be more feasible to annotate a portion of the given data . This is also aligned with the prerequisite of semi-supervised learning . Then one can ask : which data is deserved for being annotated ? Active learning was proposed to solve this question . One of the main ideas , namely uncertaintybased method , aims to select a portion of the most uncertain or informative data from an unlabeled pool for annotation . The newly annotated data are then used to train a task model . Note that uncertainty estimation with deep neural networks ( DNNs ) remains challenging , due to the potential over-confidence of deep models . That is , the primitive softmax output1 as a score does not necessarily reflect a reliable uncertainty or confidence for the prediction . Several methods have been proposed to address this challenge . For example , MC-Dropout ( Gal & Ghahramani , 2016 ) implements a Single-Training Multi-Inference algorithm by aggregating multiple confidences sampled by Monte Carlo dropout . The algorithm has been shown to be equivalent to the approximate Bayesian inference . Another typical idea is query-by-committee ( Freund et al. , 1997 ; Gorriz et al. , 2017 ) , or QBC in short , which requires training multiple auxiliary models as a committee . Besides , many other efforts suggest to use specially designed auxiliary modules or training fashions , such as variational auto-encoder ( Sinha et al. , 2019 ) , adversarial learning ( Ducoffe & Precioso , 2018 ; Mayer & Timofte , 2020 ) and Graph Convolutional Network ( Caramalau et al. , 2021 ) . In this work , we study the problem of deep active learning from a different viewpoint , inspired by a relevant topic of noise stability . Noise stability , in terms of the output stability w.r.t . the input 1Considering classification problems noise , is well studied in literature . For example , the authors in ( Bishop , 1995 ) propose to train a model with perturbed input samples as a regularizer to improve the model performance . A recent theoretical work ( Arora et al. , 2018 ) shows that , the stability of each layer ’ s computation to noise injected at lower layers acts as a good indicator of the generalization bounds for DNNs . Behind these achievements is an intuitive implication that , a model ( or more exactly , its parameters ) robust to perturbed input examples tends to be easy to recognize unseen examples . For the active learning problem , we give an analogous intuition by exchanging the role of the input and parameter that , an example robust to the perturbed current parameters tends to be easy to be recognized by a future model . In other words , examples less robust to the parameter perturbation would be regarded as having higher uncertainty . Specifically , we introduce a simple algorithm of uncertainty estimation by measuring how far does the output deviate from the original value , when imposing a small noise on the parameters . The induced distance is used as a measurement of uncertainty used to select unlabeled examples in active learning . We provide a theoretical analysis under the condition that the noise magnitude is small . By imposing standard multivariate Gaussian noise to the parameter , we prove that the proposed noise stability is equivalent to the parameter-output Jacobian norm under the first order Taylor assumption . Furthermore , we show that , our algorithm has a close connection with variance reduction ( Cohn et al. , 1996 ) . That is , selecting unlabeled samples with low noise stability would yield the same effect of reducing the prediction variance of existing training samples . Our method is easy to implement and free of customized auxiliary models . Therefore , it can be exploited in various tasks , such as image classification and semantic segmentation , leading to its task-agnostic nature . We conduct extensive experiments to evaluate our method on various datasets including Cifar10 ( Krizhevsky et al. , 2009 ) , Cifar100 ( Krizhevsky et al. , 2009 ) , SVHN ( Netzer et al. , 2011 ) , Caltech101 ( Fei-Fei et al. , 2006 ) , Cityscapes ( Cordts et al. , 2016 ) , and cryo-ET ( Chen et al. , 2017 ) . The performance of our method significantly exceeds that of the state-of-the-art . The contributions of this work are summarized as follows . 1 . We propose a novel effective method of noise stability to select unlabeled data for active learning . The proposed method is free of any auxiliary models or special training fashions . 2 . We provide a theoretical analysis to show that , selecting unlabeled data with higher noise stability is equivalent to selecting that with higher Jacobian norm w.r.t . the parameter . We also establish a connection between noise stability and variance reduction for the existing training set . 3 . We conduct extensive experiments on benchmark datasets of image classification , semantic segmentation , and 3D cryo-ET subtomogram classification . The results demonstrate the effectiveness of the proposed method . 2 ACTIVE LEARNING WITH NOISE STABILITY . 2.1 PROBLEM DEFINITION . Here we formulate the active learning problem . Given a pool of unlabeled data { XU } , and a labeled pool { XL , YL } which is initially empty . Active learning aims to select a portion of data from { XU } depending on labeling budget ( e.g . select 2500 samples out of 50000 at a time ) . The selected data XN is then annotated by human oracles ( or equivalent ) , and added to the labeled pool . That is , { XL , YL } ← { XL , YL } + { XN , YN } , where YN is the new annotation for XN . The unlabeled pool is then updated by removing the selected data : { XU } ← { XU } − { XN } . { XL , YL } is then used to train a task model f ( . ; θ ) in supervised fashion by minimizing an empirical loss L ( e.g . crossentropy loss for classification ) , where f ( . ; θ ) denotes a neural network f ( . ) parameterized with θ . Note that in this paper , 2.2 UNCERTAINTY ESTIMATION WITH NOISE STABILITY . We use the noise stability to estimate uncertainty for unlabeled data . Specifically , for an input sample x , we explore to what degree will the model ’ s output deviate from the original observation f ( x ; θ ) , when adding random noise to the model parameter θ . Let ξ∆θ denote the added noise , Algorithm 1 : Active learning with noise stability as uncertainty estimation . Input : T : random initialized task model , U : unlabeled pool of training data , L : labeled pool of training data , C : number of cycles in active learning ; Output : θ : Final learned parameter of T ; 1 begin 2 for i← 1 to C do 3 train T with L , obtaining the current parameter θ ; 4 for k ← 1 to K do 5 sample ∆θ ∼ N ( 0 , σ2I ) ; 6 create a perturbed model T ′ parameterized with θ + ξ∆θ ; 7 for every x in U do 8 calculate R ( k ) ( x ) = ‖T ( x ) − T ′ ( x ) ‖2 ; 9 compute every R ( x ) by averaging all R ( k ) ( x ) ; 10 select samples in U with top N largest R ( x ) as { XN } , and obtain their labels { YN } ; 11 update L with L = L ⋃ { XN , YN } ; 12 update U with U = U \ { XN } ; 13 return θ ; where ξ controls the magnitude of the noise . If ∆θ conforms to the standard multivariate Gaussian distribution2 N ( 0 , σ2I ) , the criterion to quantify the degree of noise instability can be formulated as R ( x ) = E ∆θ∼N ( 0 , σ2I ) ‖f ( x ; θ ) − f ( x ; θ + ξ∆θ ) ‖2 , ( 1 ) where ‖.‖2 is the L2 norm of a vector . In active learning , we select samples with largestR ( x ) for annotation . To approximate the expectation , we perform a Monto-carlo sampling from the multivariate Guassian distribution to generate multiple ∆θ ( i ) . Then we get the approximated noise stability by R ( x ) = 1 K K∑ i=1 ‖f ( x ; θ ) − f ( x ; θ + ξ ( i ) ∆θ ( i ) ) ‖2 . ( 2 ) The noise magnitude should be small relative to that of the original parameter ( e.g . ξ ( i ) = 10−3 ‖θ‖‖∆θ ( i ) ‖ ) in order to avoid catastrophic perturbation to the clean model . Note that , we use a single ξ ( i ) for the entire ∆θ ( i ) , i.e . each element in ∆θ ( i ) is equally re-scaled by ξ ( i ) . As for the sampling number , we find the setting K = 5 or 10 works well in practice . A complete procedure of our method is presented in Algorithm 1 . 3 THEORETICAL UNDERSTANDINGS . In this section , we provide theoretical understandings about our simple method for uncertainty estimation . Our main conclusions are summarized as follows . • When the noise magnitude ξ is sufficiently small , selecting data according toR ( x ) in Eq ( 1 ) is equivalent to selecting data by the Frobenius norm of parameter-output Jacobian w.r.t . f . • Based on the above conclusion , selecting new samples with higher R ( x ) are expected to contribute more in reducing the prediction variance of existing training samples . 3.1 NOISE STABILITY AS JACOBIAN NORM . Let f ( x ; θ ) be differentiable ( w.r.t . θ ) at the point θ given x as the input . When the imposed noise ξ∆θ in Eq ( 1 ) has a sufficiently small magnitude , we use the first-order Taylor expansion to estimate f ( x ; θ + ξ∆θ ) as f ( x ; θ + ξ∆θ ) ≈ f ( x ; θ ) + Jθ ( x ; θ ) ξ∆θ , ( 3 ) 2See Appendix A.1 for preliminaries about the standard multivariate Gaussian distribution where Jθ ( x ; θ ) is the Jacobian matrix of f with respect to the parameter θ as Jθ ( x ; θ ) ( i , j ) = ∂f ( x ; θ ) ( i ) /∂θ ( j ) . By substituting the Taylor approximation into Eq ( 1 ) , the uncertainty estimation can be reformulated as R ( x ) = E ∆θ∼N ( 0 , σ2I ) ‖Jθ ( x ; θ ) ξ∆θ‖2 . ( 4 ) Since the elements of ∆θ are independent with each other , it ’ s easy to derive that the expected uncertainty in Eq ( 4 ) is in proportion with the Frobenius norm of Jacobian as R ( x ) = ξ2σ2‖Jθ ( x ; θ ) ‖2F . ( 5 ) In fact , Eq ( 5 ) stands for any appropriate noise distribution with the above mentioned independence prerequisite and zero mean . We put a detailed proof of Eq ( 5 ) in Appendix A.2 . | A method for batch active learning for DNNs is proposed in the paper. The main idea is to consider robustness to parameter perturbation as an uncertainty measure. In other words, prediction deviation when adding a small noise to the parameters is used as a score for selecting data to be labeled in the active learning loop. The authors provide some theoretical analysis for the method to connect it to variance reduction. Experiments on several image classification and semantic segmentation datasets show that the method performs better than several SotA baselines. | SP:3bbafe5ef2c7298cdaf249456d42db2e84d9ec1f |
Deep Active Learning with Noise Stability | 1 INTRODUCTION . The success of training a deep neural network highly depends on a huge amount of labeled data . Nevertheless , in the era of big data , much data is unlabeled . This imposes a challenge on exploiting appropriate learning fashions . For instance , given a pool of unlabeled data , one can employ unsupervised manner to learn a model . However , the performance of unsupervised or semi-supervised learning is still bound to that of supervised learning ( Yoo & Kweon , 2019 ) . An ideal option could be annotating all the given data , so as to use the supervised way to learn a model . Nevertheless , the cost for data annotation could be extremely high , such as in medical area ( Gorriz et al. , 2017 ; Konyushkova et al. , 2017 ) . Due to the limited labeling budget , it may be more feasible to annotate a portion of the given data . This is also aligned with the prerequisite of semi-supervised learning . Then one can ask : which data is deserved for being annotated ? Active learning was proposed to solve this question . One of the main ideas , namely uncertaintybased method , aims to select a portion of the most uncertain or informative data from an unlabeled pool for annotation . The newly annotated data are then used to train a task model . Note that uncertainty estimation with deep neural networks ( DNNs ) remains challenging , due to the potential over-confidence of deep models . That is , the primitive softmax output1 as a score does not necessarily reflect a reliable uncertainty or confidence for the prediction . Several methods have been proposed to address this challenge . For example , MC-Dropout ( Gal & Ghahramani , 2016 ) implements a Single-Training Multi-Inference algorithm by aggregating multiple confidences sampled by Monte Carlo dropout . The algorithm has been shown to be equivalent to the approximate Bayesian inference . Another typical idea is query-by-committee ( Freund et al. , 1997 ; Gorriz et al. , 2017 ) , or QBC in short , which requires training multiple auxiliary models as a committee . Besides , many other efforts suggest to use specially designed auxiliary modules or training fashions , such as variational auto-encoder ( Sinha et al. , 2019 ) , adversarial learning ( Ducoffe & Precioso , 2018 ; Mayer & Timofte , 2020 ) and Graph Convolutional Network ( Caramalau et al. , 2021 ) . In this work , we study the problem of deep active learning from a different viewpoint , inspired by a relevant topic of noise stability . Noise stability , in terms of the output stability w.r.t . the input 1Considering classification problems noise , is well studied in literature . For example , the authors in ( Bishop , 1995 ) propose to train a model with perturbed input samples as a regularizer to improve the model performance . A recent theoretical work ( Arora et al. , 2018 ) shows that , the stability of each layer ’ s computation to noise injected at lower layers acts as a good indicator of the generalization bounds for DNNs . Behind these achievements is an intuitive implication that , a model ( or more exactly , its parameters ) robust to perturbed input examples tends to be easy to recognize unseen examples . For the active learning problem , we give an analogous intuition by exchanging the role of the input and parameter that , an example robust to the perturbed current parameters tends to be easy to be recognized by a future model . In other words , examples less robust to the parameter perturbation would be regarded as having higher uncertainty . Specifically , we introduce a simple algorithm of uncertainty estimation by measuring how far does the output deviate from the original value , when imposing a small noise on the parameters . The induced distance is used as a measurement of uncertainty used to select unlabeled examples in active learning . We provide a theoretical analysis under the condition that the noise magnitude is small . By imposing standard multivariate Gaussian noise to the parameter , we prove that the proposed noise stability is equivalent to the parameter-output Jacobian norm under the first order Taylor assumption . Furthermore , we show that , our algorithm has a close connection with variance reduction ( Cohn et al. , 1996 ) . That is , selecting unlabeled samples with low noise stability would yield the same effect of reducing the prediction variance of existing training samples . Our method is easy to implement and free of customized auxiliary models . Therefore , it can be exploited in various tasks , such as image classification and semantic segmentation , leading to its task-agnostic nature . We conduct extensive experiments to evaluate our method on various datasets including Cifar10 ( Krizhevsky et al. , 2009 ) , Cifar100 ( Krizhevsky et al. , 2009 ) , SVHN ( Netzer et al. , 2011 ) , Caltech101 ( Fei-Fei et al. , 2006 ) , Cityscapes ( Cordts et al. , 2016 ) , and cryo-ET ( Chen et al. , 2017 ) . The performance of our method significantly exceeds that of the state-of-the-art . The contributions of this work are summarized as follows . 1 . We propose a novel effective method of noise stability to select unlabeled data for active learning . The proposed method is free of any auxiliary models or special training fashions . 2 . We provide a theoretical analysis to show that , selecting unlabeled data with higher noise stability is equivalent to selecting that with higher Jacobian norm w.r.t . the parameter . We also establish a connection between noise stability and variance reduction for the existing training set . 3 . We conduct extensive experiments on benchmark datasets of image classification , semantic segmentation , and 3D cryo-ET subtomogram classification . The results demonstrate the effectiveness of the proposed method . 2 ACTIVE LEARNING WITH NOISE STABILITY . 2.1 PROBLEM DEFINITION . Here we formulate the active learning problem . Given a pool of unlabeled data { XU } , and a labeled pool { XL , YL } which is initially empty . Active learning aims to select a portion of data from { XU } depending on labeling budget ( e.g . select 2500 samples out of 50000 at a time ) . The selected data XN is then annotated by human oracles ( or equivalent ) , and added to the labeled pool . That is , { XL , YL } ← { XL , YL } + { XN , YN } , where YN is the new annotation for XN . The unlabeled pool is then updated by removing the selected data : { XU } ← { XU } − { XN } . { XL , YL } is then used to train a task model f ( . ; θ ) in supervised fashion by minimizing an empirical loss L ( e.g . crossentropy loss for classification ) , where f ( . ; θ ) denotes a neural network f ( . ) parameterized with θ . Note that in this paper , 2.2 UNCERTAINTY ESTIMATION WITH NOISE STABILITY . We use the noise stability to estimate uncertainty for unlabeled data . Specifically , for an input sample x , we explore to what degree will the model ’ s output deviate from the original observation f ( x ; θ ) , when adding random noise to the model parameter θ . Let ξ∆θ denote the added noise , Algorithm 1 : Active learning with noise stability as uncertainty estimation . Input : T : random initialized task model , U : unlabeled pool of training data , L : labeled pool of training data , C : number of cycles in active learning ; Output : θ : Final learned parameter of T ; 1 begin 2 for i← 1 to C do 3 train T with L , obtaining the current parameter θ ; 4 for k ← 1 to K do 5 sample ∆θ ∼ N ( 0 , σ2I ) ; 6 create a perturbed model T ′ parameterized with θ + ξ∆θ ; 7 for every x in U do 8 calculate R ( k ) ( x ) = ‖T ( x ) − T ′ ( x ) ‖2 ; 9 compute every R ( x ) by averaging all R ( k ) ( x ) ; 10 select samples in U with top N largest R ( x ) as { XN } , and obtain their labels { YN } ; 11 update L with L = L ⋃ { XN , YN } ; 12 update U with U = U \ { XN } ; 13 return θ ; where ξ controls the magnitude of the noise . If ∆θ conforms to the standard multivariate Gaussian distribution2 N ( 0 , σ2I ) , the criterion to quantify the degree of noise instability can be formulated as R ( x ) = E ∆θ∼N ( 0 , σ2I ) ‖f ( x ; θ ) − f ( x ; θ + ξ∆θ ) ‖2 , ( 1 ) where ‖.‖2 is the L2 norm of a vector . In active learning , we select samples with largestR ( x ) for annotation . To approximate the expectation , we perform a Monto-carlo sampling from the multivariate Guassian distribution to generate multiple ∆θ ( i ) . Then we get the approximated noise stability by R ( x ) = 1 K K∑ i=1 ‖f ( x ; θ ) − f ( x ; θ + ξ ( i ) ∆θ ( i ) ) ‖2 . ( 2 ) The noise magnitude should be small relative to that of the original parameter ( e.g . ξ ( i ) = 10−3 ‖θ‖‖∆θ ( i ) ‖ ) in order to avoid catastrophic perturbation to the clean model . Note that , we use a single ξ ( i ) for the entire ∆θ ( i ) , i.e . each element in ∆θ ( i ) is equally re-scaled by ξ ( i ) . As for the sampling number , we find the setting K = 5 or 10 works well in practice . A complete procedure of our method is presented in Algorithm 1 . 3 THEORETICAL UNDERSTANDINGS . In this section , we provide theoretical understandings about our simple method for uncertainty estimation . Our main conclusions are summarized as follows . • When the noise magnitude ξ is sufficiently small , selecting data according toR ( x ) in Eq ( 1 ) is equivalent to selecting data by the Frobenius norm of parameter-output Jacobian w.r.t . f . • Based on the above conclusion , selecting new samples with higher R ( x ) are expected to contribute more in reducing the prediction variance of existing training samples . 3.1 NOISE STABILITY AS JACOBIAN NORM . Let f ( x ; θ ) be differentiable ( w.r.t . θ ) at the point θ given x as the input . When the imposed noise ξ∆θ in Eq ( 1 ) has a sufficiently small magnitude , we use the first-order Taylor expansion to estimate f ( x ; θ + ξ∆θ ) as f ( x ; θ + ξ∆θ ) ≈ f ( x ; θ ) + Jθ ( x ; θ ) ξ∆θ , ( 3 ) 2See Appendix A.1 for preliminaries about the standard multivariate Gaussian distribution where Jθ ( x ; θ ) is the Jacobian matrix of f with respect to the parameter θ as Jθ ( x ; θ ) ( i , j ) = ∂f ( x ; θ ) ( i ) /∂θ ( j ) . By substituting the Taylor approximation into Eq ( 1 ) , the uncertainty estimation can be reformulated as R ( x ) = E ∆θ∼N ( 0 , σ2I ) ‖Jθ ( x ; θ ) ξ∆θ‖2 . ( 4 ) Since the elements of ∆θ are independent with each other , it ’ s easy to derive that the expected uncertainty in Eq ( 4 ) is in proportion with the Frobenius norm of Jacobian as R ( x ) = ξ2σ2‖Jθ ( x ; θ ) ‖2F . ( 5 ) In fact , Eq ( 5 ) stands for any appropriate noise distribution with the above mentioned independence prerequisite and zero mean . We put a detailed proof of Eq ( 5 ) in Appendix A.2 . | This paper provides a new method for solving the active learning problem, i.e. choosing from a set of unlabeled examples that should be labelled and added to the training set. The proposed method makes use of 'noise stability' of inputs–specifically if the model output corresponding to an input changes to a large extent when small amounts of noise are added to the model parameters, then this input should be labelled. The authors compare their method to a range of existing AL methods and draw theoretical connections to predictive variance reduction. | SP:3bbafe5ef2c7298cdaf249456d42db2e84d9ec1f |
Deep Active Learning with Noise Stability | 1 INTRODUCTION . The success of training a deep neural network highly depends on a huge amount of labeled data . Nevertheless , in the era of big data , much data is unlabeled . This imposes a challenge on exploiting appropriate learning fashions . For instance , given a pool of unlabeled data , one can employ unsupervised manner to learn a model . However , the performance of unsupervised or semi-supervised learning is still bound to that of supervised learning ( Yoo & Kweon , 2019 ) . An ideal option could be annotating all the given data , so as to use the supervised way to learn a model . Nevertheless , the cost for data annotation could be extremely high , such as in medical area ( Gorriz et al. , 2017 ; Konyushkova et al. , 2017 ) . Due to the limited labeling budget , it may be more feasible to annotate a portion of the given data . This is also aligned with the prerequisite of semi-supervised learning . Then one can ask : which data is deserved for being annotated ? Active learning was proposed to solve this question . One of the main ideas , namely uncertaintybased method , aims to select a portion of the most uncertain or informative data from an unlabeled pool for annotation . The newly annotated data are then used to train a task model . Note that uncertainty estimation with deep neural networks ( DNNs ) remains challenging , due to the potential over-confidence of deep models . That is , the primitive softmax output1 as a score does not necessarily reflect a reliable uncertainty or confidence for the prediction . Several methods have been proposed to address this challenge . For example , MC-Dropout ( Gal & Ghahramani , 2016 ) implements a Single-Training Multi-Inference algorithm by aggregating multiple confidences sampled by Monte Carlo dropout . The algorithm has been shown to be equivalent to the approximate Bayesian inference . Another typical idea is query-by-committee ( Freund et al. , 1997 ; Gorriz et al. , 2017 ) , or QBC in short , which requires training multiple auxiliary models as a committee . Besides , many other efforts suggest to use specially designed auxiliary modules or training fashions , such as variational auto-encoder ( Sinha et al. , 2019 ) , adversarial learning ( Ducoffe & Precioso , 2018 ; Mayer & Timofte , 2020 ) and Graph Convolutional Network ( Caramalau et al. , 2021 ) . In this work , we study the problem of deep active learning from a different viewpoint , inspired by a relevant topic of noise stability . Noise stability , in terms of the output stability w.r.t . the input 1Considering classification problems noise , is well studied in literature . For example , the authors in ( Bishop , 1995 ) propose to train a model with perturbed input samples as a regularizer to improve the model performance . A recent theoretical work ( Arora et al. , 2018 ) shows that , the stability of each layer ’ s computation to noise injected at lower layers acts as a good indicator of the generalization bounds for DNNs . Behind these achievements is an intuitive implication that , a model ( or more exactly , its parameters ) robust to perturbed input examples tends to be easy to recognize unseen examples . For the active learning problem , we give an analogous intuition by exchanging the role of the input and parameter that , an example robust to the perturbed current parameters tends to be easy to be recognized by a future model . In other words , examples less robust to the parameter perturbation would be regarded as having higher uncertainty . Specifically , we introduce a simple algorithm of uncertainty estimation by measuring how far does the output deviate from the original value , when imposing a small noise on the parameters . The induced distance is used as a measurement of uncertainty used to select unlabeled examples in active learning . We provide a theoretical analysis under the condition that the noise magnitude is small . By imposing standard multivariate Gaussian noise to the parameter , we prove that the proposed noise stability is equivalent to the parameter-output Jacobian norm under the first order Taylor assumption . Furthermore , we show that , our algorithm has a close connection with variance reduction ( Cohn et al. , 1996 ) . That is , selecting unlabeled samples with low noise stability would yield the same effect of reducing the prediction variance of existing training samples . Our method is easy to implement and free of customized auxiliary models . Therefore , it can be exploited in various tasks , such as image classification and semantic segmentation , leading to its task-agnostic nature . We conduct extensive experiments to evaluate our method on various datasets including Cifar10 ( Krizhevsky et al. , 2009 ) , Cifar100 ( Krizhevsky et al. , 2009 ) , SVHN ( Netzer et al. , 2011 ) , Caltech101 ( Fei-Fei et al. , 2006 ) , Cityscapes ( Cordts et al. , 2016 ) , and cryo-ET ( Chen et al. , 2017 ) . The performance of our method significantly exceeds that of the state-of-the-art . The contributions of this work are summarized as follows . 1 . We propose a novel effective method of noise stability to select unlabeled data for active learning . The proposed method is free of any auxiliary models or special training fashions . 2 . We provide a theoretical analysis to show that , selecting unlabeled data with higher noise stability is equivalent to selecting that with higher Jacobian norm w.r.t . the parameter . We also establish a connection between noise stability and variance reduction for the existing training set . 3 . We conduct extensive experiments on benchmark datasets of image classification , semantic segmentation , and 3D cryo-ET subtomogram classification . The results demonstrate the effectiveness of the proposed method . 2 ACTIVE LEARNING WITH NOISE STABILITY . 2.1 PROBLEM DEFINITION . Here we formulate the active learning problem . Given a pool of unlabeled data { XU } , and a labeled pool { XL , YL } which is initially empty . Active learning aims to select a portion of data from { XU } depending on labeling budget ( e.g . select 2500 samples out of 50000 at a time ) . The selected data XN is then annotated by human oracles ( or equivalent ) , and added to the labeled pool . That is , { XL , YL } ← { XL , YL } + { XN , YN } , where YN is the new annotation for XN . The unlabeled pool is then updated by removing the selected data : { XU } ← { XU } − { XN } . { XL , YL } is then used to train a task model f ( . ; θ ) in supervised fashion by minimizing an empirical loss L ( e.g . crossentropy loss for classification ) , where f ( . ; θ ) denotes a neural network f ( . ) parameterized with θ . Note that in this paper , 2.2 UNCERTAINTY ESTIMATION WITH NOISE STABILITY . We use the noise stability to estimate uncertainty for unlabeled data . Specifically , for an input sample x , we explore to what degree will the model ’ s output deviate from the original observation f ( x ; θ ) , when adding random noise to the model parameter θ . Let ξ∆θ denote the added noise , Algorithm 1 : Active learning with noise stability as uncertainty estimation . Input : T : random initialized task model , U : unlabeled pool of training data , L : labeled pool of training data , C : number of cycles in active learning ; Output : θ : Final learned parameter of T ; 1 begin 2 for i← 1 to C do 3 train T with L , obtaining the current parameter θ ; 4 for k ← 1 to K do 5 sample ∆θ ∼ N ( 0 , σ2I ) ; 6 create a perturbed model T ′ parameterized with θ + ξ∆θ ; 7 for every x in U do 8 calculate R ( k ) ( x ) = ‖T ( x ) − T ′ ( x ) ‖2 ; 9 compute every R ( x ) by averaging all R ( k ) ( x ) ; 10 select samples in U with top N largest R ( x ) as { XN } , and obtain their labels { YN } ; 11 update L with L = L ⋃ { XN , YN } ; 12 update U with U = U \ { XN } ; 13 return θ ; where ξ controls the magnitude of the noise . If ∆θ conforms to the standard multivariate Gaussian distribution2 N ( 0 , σ2I ) , the criterion to quantify the degree of noise instability can be formulated as R ( x ) = E ∆θ∼N ( 0 , σ2I ) ‖f ( x ; θ ) − f ( x ; θ + ξ∆θ ) ‖2 , ( 1 ) where ‖.‖2 is the L2 norm of a vector . In active learning , we select samples with largestR ( x ) for annotation . To approximate the expectation , we perform a Monto-carlo sampling from the multivariate Guassian distribution to generate multiple ∆θ ( i ) . Then we get the approximated noise stability by R ( x ) = 1 K K∑ i=1 ‖f ( x ; θ ) − f ( x ; θ + ξ ( i ) ∆θ ( i ) ) ‖2 . ( 2 ) The noise magnitude should be small relative to that of the original parameter ( e.g . ξ ( i ) = 10−3 ‖θ‖‖∆θ ( i ) ‖ ) in order to avoid catastrophic perturbation to the clean model . Note that , we use a single ξ ( i ) for the entire ∆θ ( i ) , i.e . each element in ∆θ ( i ) is equally re-scaled by ξ ( i ) . As for the sampling number , we find the setting K = 5 or 10 works well in practice . A complete procedure of our method is presented in Algorithm 1 . 3 THEORETICAL UNDERSTANDINGS . In this section , we provide theoretical understandings about our simple method for uncertainty estimation . Our main conclusions are summarized as follows . • When the noise magnitude ξ is sufficiently small , selecting data according toR ( x ) in Eq ( 1 ) is equivalent to selecting data by the Frobenius norm of parameter-output Jacobian w.r.t . f . • Based on the above conclusion , selecting new samples with higher R ( x ) are expected to contribute more in reducing the prediction variance of existing training samples . 3.1 NOISE STABILITY AS JACOBIAN NORM . Let f ( x ; θ ) be differentiable ( w.r.t . θ ) at the point θ given x as the input . When the imposed noise ξ∆θ in Eq ( 1 ) has a sufficiently small magnitude , we use the first-order Taylor expansion to estimate f ( x ; θ + ξ∆θ ) as f ( x ; θ + ξ∆θ ) ≈ f ( x ; θ ) + Jθ ( x ; θ ) ξ∆θ , ( 3 ) 2See Appendix A.1 for preliminaries about the standard multivariate Gaussian distribution where Jθ ( x ; θ ) is the Jacobian matrix of f with respect to the parameter θ as Jθ ( x ; θ ) ( i , j ) = ∂f ( x ; θ ) ( i ) /∂θ ( j ) . By substituting the Taylor approximation into Eq ( 1 ) , the uncertainty estimation can be reformulated as R ( x ) = E ∆θ∼N ( 0 , σ2I ) ‖Jθ ( x ; θ ) ξ∆θ‖2 . ( 4 ) Since the elements of ∆θ are independent with each other , it ’ s easy to derive that the expected uncertainty in Eq ( 4 ) is in proportion with the Frobenius norm of Jacobian as R ( x ) = ξ2σ2‖Jθ ( x ; θ ) ‖2F . ( 5 ) In fact , Eq ( 5 ) stands for any appropriate noise distribution with the above mentioned independence prerequisite and zero mean . We put a detailed proof of Eq ( 5 ) in Appendix A.2 . | This paper proposes a single-model active learning method for classification and segmentation. The key idea is to add noise to the parameters in order to generate multiple outputs and be able to estimate the uncertainty of a sample. The paper presents results for classification and segmentation in different datasets. | SP:3bbafe5ef2c7298cdaf249456d42db2e84d9ec1f |
Generalization of Overparametrized Deep Neural Network Under Noisy Observations | 1 INTRODUCTION Over the past few years , Neural Tangent Kernel ( NTK ) [ Arora et al. , 2019b ; Jacot et al. , 2018 ; Lee et al. , 2018 ; Chizat & Bach , 2018 ] has been one of the most seminal discoveries in the theory of neural network . The underpinning idea of the NTK-type theory comes from the observation that in a wide-enough neural net , model parameters updated by gradient descent ( GD ) stay close to their initializations during the training , so that the dynamics of the networks can be approximated by the first-order Taylor expansion with respect to its parameters at initialization . The linearization of learning dynamics on neural networks has been helpful in showing the linear convergence of the training error on both overparametrized shallow [ Li & Liang , 2018 ; Du et al. , 2018 ] and deep neural networks [ Allen-Zhu et al. , 2018 ; Zou et al. , 2018 ; 2020 ] , as well as the characterizations of generalization error on both models [ Arora et al. , 2019a ; Cao & Gu , 2019 ] . These findings clearly lead to the equivalence between learning dynamics of neural networks and the kernel methods in an reproducing kernel Hilbert spaces ( RKHS ) associated with NTK . 1 Specifically , Arora et al . [ 2019a ] provided the O ( n−1/2 ) generalization bound of shallow neural network , where n denotes the training sample size . Recently , in the context of nonparametric regression , two papers Nitanda & Suzuki [ 2020 ] and Hu et al . [ 2021 ] showed that neural network can obtain the convergence rate faster than O ( n−1/2 ) by specifying the complexities of target function and hypothesis space . Specifically , Nitanda & Suzuki [ 2020 ] showed that the shallow neural network with smoothly approximated ReLU ( swish , see Ramachandran et al . [ 2017 ] ) activation trained via ` 2-regularized averaged stochastic gradient descent ( SGD ) can recover the target function from RKHSs induced from NTK with swish activation . Similarly , Hu et al . [ 2021 ] showed that a shallow neural network with ReLU activation trained via ` 2-regularized GD can generalize well , when the target function ( i.e. , f ? ρ ) is fromHNTK1 . Notably , the rate that the papers Nitanda & Suzuki [ 2020 ] and Hu et al . [ 2021 ] obtained is minimax optimal , meaning that no estimators perform substantially better than the ` 2-regularized GD or averaged SGD algorithms for recovering functions from respective function spaces . Nevertheless , 1Henceforth , we denoteHNTK1 andHNTKL as RKHSs induced from NTK of shallow L = 1 and deep neural networks L ≥ 2 with ReLU activations , respecitvely . these results are restricted to shallow neural networks , and can not explain the generalization abilities of deep neural network ( DNN ) . Similarly with Arora et al . [ 2019a ] , Cao & Gu [ 2019 ] obtained the O ( n−1/2 ) generalization bound , showing that the SGD generalize well for f ? ρ ∈ HNTKL , when f ? ρ has a bounded RKHS norm . However , the rate they obtained is slower than the minimax rate we can actually achieve . Furthermore , their results become vacuous under the presence of additive noises on the data set . Motivated from these observations , the fundamental question in this study is as follows : When the noisy dataset is generated from a function fromHNTKL , does the overparametrized DNN obtained via ( ` 2-regularized ) GD provably generalize well the unseen data ? We consider a neural network that has L ≥ 2 hidden layers with width m n . ( i.e. , overparametrized deep neural network . ) We focus on the least-squares loss and assume that the activation function is ReLU . A positivity assumption of NTK from ReLU DNN is imposed , meaning that λ0 > 0 , where λ0 denotes the minimum eigenvalue of the NTK . We give a more formal mathematical definition of ReLU DNN in the following Subsection 2.2 . Under these settings , we provide an affirmative answer to the above question by investigating the behavior of L2-prediction error of the obtained neural network with respect to GD iterations . 1.1 CONTRIBUTIONS Our derivations of algorithm-dependent prediction risk bound require the analysis on training dynamics of the estimated neural network through ( regularized ) GD algorithm . We include these results as the contributions of our paper , which can be of independent interests as well . • In an unregulaized case , under the assumption λ0 > 0 , we show randomly initialized GD converges to 0 training loss at a linear rate . As will be detailed in subsection 3.3 , this is the different result from the seminal work of Allen-Zhu et al . [ 2018 ] , where they also prove a linear convergence of training loss of ReLU DNN , but under different data distribution assumption . • We show that the DNN updated via vanilla GD does not recover the ground truth function f ? ρ ∈ HNTKL under noisy observations , if the DNN is trained for either too short or too long : that is , the prediction error is bounded away from 0 by some constant as n goes to infinity . • In regularized case , we prove the mean-squared error ( MSE ) of DNN is upper bounded by some positive constant . Additionally , we proved the dynamics of the estimated neural network get close to the solution of kernel ridge regression associated with NTK from ReLU DNN . • We show that the ` 2-regularization can be helpful in achieving the minimax optimal rate of the prediction risk for recovering f ? ρ ∈ HNTKL under the noisy data . Specifically , it is shown that after some iterations of ` 2-regularized GD , the minimax optimal rate ( which is O ( n− d 2d−1 ) , where d is a feature dimension . ) can be achieved . Note that our paper is an extension of Hu et al . [ 2021 ] to DNN model , showing that the ` 2-regularized DNN can achieve a minimax optimal rate of prediction error for recovering f ? ρ ∈ HNTKL . However , we would like to emphasize that our work is not a trivial application of their work from at least two technical aspects . These aspects are more detailed in the following subsection . 1.2 TECHNICAL COMPARISONS WITH HU ET AL . [ 2021 ] Firstly , in the analysis of training loss of regularized shallow neural-net , Hu et al . [ 2021 ] begin the proof by decomposing the difference between two individual predictions into two terms : one that is related with the gram matrix evaluated at each iteration of the algorithm and the perturbation term . Henceforth , we name this decompostion as “ Gram+Pert ” decomposition . This decomposition can be checked with the equality ( E.2 ) in the supplementary PDF of Hu et al . [ 2021 ] . The key ingredients for the decomposition are ( i ) the simple gradient structure of the shallow neural net , and ( ii ) the partitioning of the nodes in the hidden-layer into two sets : a set of nodes whose activation patterns change from their initializations during training , and the complement of the set . This construction of the sets peels off the ReLU activation in the difference so that the GD algorithm can be involved in the analysis . However , because of the compositional structure of the network , the same nodes partitioning technique can not be applied for obtaining the decomposition in the DNN setting with ReLU activation . To avoid this difficulty , we employ a specially designed diagonal matrix Σ̃ and this matrix can peel off the ReLU function for each layer of the network . ( See the definition of Σ̃ in the proof of Theorem 3.5 in the Appendix . ) Recursive applications of this diagonal matrix across the entire hidden layers enable the Gram+Pert decomposition in our setting . It should be noted that the diagnoal matrix Σ̃ had been employed in Zou et al . [ 2020 ] , which analyzed the behavior of training loss of classification problem via ReLU DNN under logistic loss . However , since their result is dependent on different data distribution assumption under the different loss function from ours , they didn ’ t employ the Gram+Pert decomposition . Thus their technical approaches are different from ours . Secondly , Hu et al . [ 2021 ] directly penalized the weight parameter W by adding ‖W‖2F to the objective function . The ` 2-regularization solely on the W has an effect of pushing the weight towards the origin . This makes ‖W ( k ) −W ( 0 ) ‖2 ≤ O ( 1 ) 2 , allowing most activation patterns of the nodes in the hidden layer can change during the training , even in overparametrized setting . Here , W ( k ) denotes the updated weight parameter at kth itertaion of algorithm , and ‖ · ‖2 denotes the spectral norm of the matrix . Nonetheless , this doesn ’ t affect the analysis on obtaining the upper-bound of MSE in shallow neural net , since the network has only a single hidden layer . In contrast , in the DNN setting , we allow the non-convex interactions of parameters across the hidden layers . To the best of our knowledge , a technique for controlling the size of ` 2-norm of network gradient has not been developed under this setting , yet . We circumvent this difficulty by regularizing the distance between the updated and the initialized parameter , instead by directly regularizing the updated parameter . This ensures that the updated parameter by ` 2-regularized GD stays in a close neighborhood to its initialization , so that with heavy over-parametrization , the dynamics of network becomes linearized in parameter and we can ignore the non-convex interactions of parameters across the hidden layers . Specifically , under suitable model parameter setting , we prove that ‖W ( k ) ` −W ( 0 ) ` ‖2 ≤ ÕP ( 1√ m ) 3 over all ` ∈ { 1 , . . . , L } . Here , ÕP ( · ) hides the dependencies on the model parameters ; L , ω , and n. This result allows us to adopt the so-called “ Forward Stability ” argument developed by Allen-Zhu et al . [ 2018 ] , and eventually leads to the control of network gradient under ` 2 sense . 1.3 ADDITIONAL RELATED WORKS There has been another line of works trying to characterize the generalizabilities of DNN under noisy observation settings . Specifically , it has been shown that the neural network model can achieve minimax style optimal convergence rates of L2-prediction risk both in regression [ Bauer & Kohler , 2019 ; Liu et al. , 2019 ; Schmidt-Hieber , 2020 ] and classification [ Kim et al. , 2021 ] problems . Nonetheless , a limitation of the aforementioned papers is that they assume an adequate minimizer of the empirical risk can be obtained . In other words , the mathematical proofs of their theorems do not correspond to implementable algortihms . Recently , several papers , which study the generalization properties of neural network with algorithmic guarantees , appear online . Specifically , Kohler & Krzyzak [ 2019 ] showed that the data interpolants obtained through DNN by vanilla GD is inconsistent . This result is consistent with our result , but they consider the overparametrized DNN that is a linear combination of Ω ( n10d 2 ) smaller neural network , and the activation function they consider is sigmoid function , which is smooth and differentiable . Along this line of research , Kuzborskij & Szepesvári [ 2021 ] ( regression ) and Ji et al . [ 2021 ] ( classification ) showed that when training overparametrized shallow neural network , early stopping of vanilla GD enables us to obtain consistent estimators . Notation . We use the following notation for asymptotics : For sufficiently large n , we write f ( n ) = O ( g ( n ) ) , if there exists a constant K > 0 such that f ( n ) ≤ Kg ( n ) , and f ( n ) = Ω ( g ( n ) ) if f ( n ) ≥ K ′g ( n ) for some constant K ′ > 0 . The notation f ( n ) = Θ ( g ( n ) ) means that f ( n ) = O ( g ( n ) ) and 2This was empirically shown to be true in paper Wei et al . [ 2019 ] . See Figure 3 in their paper . We provide a brief mathematical explanation on why this result is hard to be shown in Appendix C. 3Readers can find the proof of this result in Appendix G. f ( n ) = Ω ( g ( n ) ) . Let 〈A , B〉Tr : = Tr ( A > B ) for the two matrices A , B ∈ Rd1×d2 . We adopt the shorthand notation denoting [ n ] : = { 1 , 2 , . . . , n } for n ∈ N. 2 PROBLEM FORMULATION 2.1 NON-PARAMETRIC REGRESSION Let X ⊂ Rd and Y ⊂ R be the measureable feature space and output space . We denote ρ as a joint probability measure on the product space X × Y , and let ρX be the marginal distribution of the feature space X . We assume that the noisy data-set D : = { ( xi , yi ) } ni=1 are generated from the non-parametric regression model yi = f ? ρ ( xi ) + εi , where εi i.i.d.∼ N ( 0 , 12 ) for i = 1 , . . . , n. Let f̂W ( k ) ( · ) be the value of neural network evaluated with the parameters W at the k-th iterations of GD update rule . At k = 0 , we randomly initialize the weight parameters in the model following He initialization [ He et al. , 2015 ] with a slight modification . Then , the L2 prediction risk is defined as the difference between two expected risks ( i.e. , excess risk ) R ( f̂W ( k ) ) : = Eρ∼ ( x , y ) [ ( y − f̂W ( k ) ( x ) ) 2 ] and R ( f ? ρ ) : = Eρ∼ ( x , y ) [ ( y − f ? ρ ( x ) ) 2 ] , where f ? ρ ( x ) : = E [ y|x ] . Then , we can easily show the prediction risk has a following form : R ( f̂k , f ? ρ ) : = R ( f̂W ( k ) ) −R ( f ? ) = Eρx , ε [ ( f̂W ( k ) ( x ) − f ? ρ ( x ) ) 2 ] . ( 1 ) Note that the expectation is taken over the marginal probability measure of feature space , ρx , and the noise of the data , ε . However , the ( 1 ) is still a random quantity due to the randomness of the initialized parameters ( W ( 0 ) ` ) ` =1 , ... , L . 2.2 DEEP NEURAL NETWORK WITH RELU ACTIVATION Following the setting introduced in Allen-Zhu et al . [ 2018 ] , we consider a fully-connected deep neural networks with L hidden layers and m network width . For L ≥ 2 , the output of the network fW ( · ) ∈ R with input data x ∈ X can be formally written as follows : fW ( x ) = √ m · vTσ ( WLσ ( WL−1 · · ·σ ( W1x ) · · · ) ) , ( 2 ) where Sd−1 is a unit sphere in d-dimensional euclidean space , σ ( · ) is an entry-wise activation function , W1 ∈ Rm×d , W2 , . . . , WL ∈ Rm×m denote the weight matrices for hidden layers and v ∈ Rm×1 denote the weight vector for the output layer . Following the existing literature , we will consider ReLU activation function σ ( x ) = max ( x , 0 ) , which is the most commonly used activation function by practitioners . Random Initialization . Each entries of weight matrices in hidden layers are assumed to be generated from ( Wi , j ) ` =1 , ... , L ∼ N ( 0 , 2m ) , and entries of the output layer are drawn from vj ∼ N ( 0 , ωm ) . This initialization scheme helps the forward propagation neither explode nor vanish at the initialization , seeing Allen-Zhu et al . [ 2018 ] ; Zou et al . [ 2018 ; 2020 ] . Note that we initialize the parameters in the last layer with variance ωm , where ω ≤ 1 is a model parameter to be chosen later for technical convenience . Unregularized GD update rule . We solve a following ` 2-loss function with the given dataset D : LS ( W ) = 1 2 n∑ i=1 ( yi − fW ( xi ) ) 2 . ( 3 ) Let W ( 0 ) 1 , . . . , W ( 0 ) L be the initialized weight matrices introduced above , and we consider a following gradient descent update rule : W ( k ) ` = W ( k−1 ) ` − η∇W ` ( LS ( W ( k−1 ) ` ) ) , ` ∈ [ L ] , k ≥ 1 , ( 4 ) where ∇W ` ( LS ( · ) ) is a partial gradient of the loss function LS ( · ) with respect to the ` -th layer parameters W ` , and η > 0 is the learning rate of the gradient descent . ` 2-regularized GD update rule . The estimator is obtained by minimizing a ` 2-regularized function ; ΦD ( W ) : = LS ( WD ) + µ 2 L∑ ` =1 ∥∥∥WD , ` −W ( 0 ) D , ` ∥∥∥2 F . ( 5 ) Naturally , we update the model parameters { WD , ` } ` =1 , ... , L via modified GD update rule : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] + η2µW ( 0 ) D , ` , ∀ ` ∈ [ L ] , ∀k ≥ 1 . ( 6 ) The notations η1 , η2 are step sizes , and µ > 0 is a tuning parameter on regularization . We adopt the different step sizes for the partial gradient and regularized term for the theoretical conveniences . Furthermore , we add the additional subscript D to the update rule ( 6 ) to denote the variables are under the regularized GD update rule . Recall that the W ( 0 ) D , ` are initialized parameters same with the unregularized case . For simplicity , we fix the output layer , and train L hidden layers for both unregularized and regularized cases . 3 MAIN THEORY First , we describe the neural tangent kernel ( NTK ) matrix of ( 2 ) , which is first proposed by Jacot et al . [ 2018 ] and further studied by Arora et al . [ 2019b ] ; Du et al . [ 2019 ] ; Lee et al . [ 2018 ] ; Yang [ 2019 ] . NTK matrix of DNN is a L-times recursively defined n × n kernel matrix , whose entries are the infinite-width limit of the gram matrix . Let ∇W ` [ fW ( 0 ) ( · ) ] be the gradient of the ReLU DNN ( 2 ) with respect to the weight matrix in ` th hidden layer at random initialization . Note that when ` = 1 , ∇W ` [ fW ( 0 ) ( · ) ] ∈ Rm×d and when ` ∈ { 2 , . . . , L } , ∇W ` [ fW ( 0 ) ( · ) ] ∈ Rm×m . Then , as m→∞ , H ( 0 ) : = ( 1 m L∑ ` =1 〈 ∇W ` [ fW ( 0 ) ( xi ) ] , ∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ) n×n → H∞L , ( 7 ) where H∞L : = { Ker ( xi , xj ) } n i , j=1 . Here , Ker ( · , · ) denotes a NTK function of ( 2 ) to be defined as follows : Definition 3.1 . ( NTK function of ( 2 ) ) . For any x , x′ ∈ X and ` ∈ [ L ] , define Φ ( 0 ) ( x , x′ ) = 〈x , x′〉 , Θ ( ` ) ( x , x′ ) = ( Φ ( ` −1 ) ( x , x ) Φ ( ` −1 ) ( x , x′ ) Φ ( ` −1 ) ( x′ , x ) Φ ( ` −1 ) ( x′ , x′ ) ) ∈ R2×2 , Φ ( ` ) ( x , x′ ) = 2 · E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ ( u ) · σ ( v ) ] , and Φ̇ ( ` ) ( x , x′ ) = 2 · E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ̇ ( u ) · σ̇ ( v ) ] , where σ̇ ( u ) = 1 ( u ≥ 0 ) . Then , we can derive the final expression of NTK function of ( 2 ) as follows : Ker ( x , x′ ) = ω 2 · L∑ ` =1 ( Φ ( ` −1 ) ( x , x′ ) · L∏ ` ′= ` Φ̇ ( ` ′ ) ( x , x′ ) ) . ( 8 ) The expression in ( 8 ) is adapted from Cao & Gu [ 2019 ] . As remarked in Cao & Gu [ 2019 ] , a coefficient 2 in Φ ( ` ) and Φ̇ ( ` ) remove the exponential dependence on the network depth L in the NTK function . However , when compared with the NTK formula in Cao & Gu [ 2019 ] , ( 8 ) is different from two aspects : ( i ) An additional factor ω in ( 8 ) ) comes from the difference in initialization settings of the output layer , in which Cao & Gu [ 2019 ] considers vj ∼ N ( 0 , 1m ) , whereas we consider vj ∼ N ( 0 , ωm ) . ( ii ) Φ ( L ) is not added in the final expression of ( 8 ) ) , whereas it is added in the definition provided in Cao & Gu [ 2019 ] . This is because we only train the L hidden layers but fix the output layer , while Cao & Gu [ 2019 ] train the entire layers of the network including the output layer . As already been pointed by several papers Cho & Saul [ 2009 ] ; Jacot et al . [ 2018 ] , it can be proved that the NTK function ( 8 ) is a positive semi-definite kernel function . Furthermore , Cho & Saul [ 2009 ] prove that the expectations in Φ and Φ̇ have closed form solutions , when the covariance matrices have the form ( 1 tt 1 ) with |t| ≤ 1 : E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ ( u ) · σ ( v ) ] = 1 2π ( t · ( π − arccos ( t ) ) + √ 1− t2 ) , E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ̇ ( u ) · σ̇ ( v ) ] = 1 2π ( π − arccos ( t ) ) . ( 9 ) Clearly , ( 8 ) is symmetric and continuous on the product space X ×X , from which it can be implied that Ker ( · , · ) is a Mercer kernel inducing an unique RKHS . Following Ghorbani et al . [ 2020 ] , we define the RKHS induced by ( 8 ) as : Definition 3.2 . ( NTK induced RKHS ) . For some integer p ∈ N , set of points { x̃j } pj=1 ⊂ X , and weight vector α : = { α1 , . . . , αp } ∈ Rp , define a complete vector space of functions , f : X → R , HNTKL : = cl ( { f ( · ) = p∑ j=1 αjKer ( · , x̃j ) } ) , ( 10 ) where cl ( · ) denotes closure . In the remaining of our work , we assume the regression function f ? ρ ( x ) : = E [ y|x ] belongs toHNTKL . 3.1 ASSUMPTIONS . In this subsection , we state the assumptions imposed on the data distribution with some remarks . ( A1 ) ρX is an uniform distribution on Sd−1 : = { x ∈ Rd | ‖x‖2 = 1 } , and noisy observations are assumed to be bounded . ( i.e. , ρx ∼ Unif ( Sd−1 ) , yi = O ( 1 ) , ∀i ∈ [ n ] . ) ( A2 ) Draw n i.i.d . samples { xi , f ? ρ ( xi ) } ni=1 from the joint measure ρ . Then , with probability at least 1− δ , we have λmin ( H∞L ) = λ0 > 0 . Remark 3.3 . • When the feature space is restricted on the unit sphere , the NTK function in ( 8 ) becomes rotationally invariant zonal kernel . This setting allows to adopt the results of spectral decay of ( 8 ) in the basis of spherical harmonic polynomials for measuring the complexity of hypothesis space , HNTKL . See the Appendix 3.2 and references therein . • Assumption ( A2 ) is commonly employed in NTK related literature for proving global convergence of training error and generalization error of both deep and shallow neural network , Du et al . [ 2018 ; 2019 ] ; Arora et al . [ 2019a ] . Note that the ( A2 ) holds as long as no two xi and xj are parallel to each other , which is true for most of the real-world distributions . See the proof of this claim in Du et al . [ 2019 ] . 3.2 MINIMAX RATE FOR RECOVERING f ? ρ ∈ HNTKL The obtainable minimax rate of L2-prediction error is directly related with the complexity of function space of interest . In our setting , the complexity of RKHS HNTKL can be characterized by the eigen-decay rate of the NTK function . Since Ker ( x , x′ ) is defined on the sphere , the decomposition can be given in the basis of spherical harmonics as follows : Ker ( x , x′ ) = ∞∑ k=0 µk N ( d , k ) ∑ j=1 Yk , j ( x ) Yk , j ( x ′ ) , where Yk , j , j = 1 , . . . , N ( d , k ) are spherical harmonic polynomials of degree k and { µk } ∞k=0 are non-negative eigenvalues . Recently , several researchers , both empirically [ Basri et al. , 2020 ] and theoretically [ Chen & Xu , 2020 ; Geifman et al. , 2020 ; Bietti & Bach , 2021 ] , showed that , for large enough harmonic function frequency k , the decay rate of the eigenvalues µk is in the order of Θ ( k−d ) 4 . Given this result and the fact N ( d , k ) = 2k+d−3k ( k+d−3 d−2 ) grows as kd−2 for large k , it can be easily shown λj = Θ ( j− d d−1 ) , when Ker ( x , x′ ) = ∑∞ j=1 λjφj ( x ) φj ( x ′ ) , for eigen-values λ1 ≥ λ2 ≥ · · · ≥ 0 and orthonormal basis { φj } ∞j=1 . Furthermore , it is a well known fact that if the eigenvalues decay at the rate λj = Θ ( j−2ν ) , then the corresponding minimax rate for estimating function in RKHS is O ( n− 2ν 2ν+1 ) , [ Raskutti et al. , 2014 ; Yuan & Zhou , 2016 ; Hu et al. , 2021 ] . By setting 2ν = dd−1 , we can see the minimax rate for recovering f ? ρ ∈ HNTKL isO ( n− d 2d−1 ) . Remark 3.4 . We defer all the technical proofs of the Theorems in subsections 3.3 and 3.4 in the Appendix for conciseness of the paper . At the beginning of each proof , a high-level proof idea is also provided . We also provide numerical experiments which can corroborate our theoretical findings in the Appendix A . 3.3 ANALYSIS OF UNREGULARIZED DNN In this subsection , we provide the results on the training loss of DNN estimator obtained via minimizing unregularized ` 2-loss ( 3 ) and on the corresponding estimator ’ s L2-prediction riskR ( f̂k , f ? ρ ) . Theorem 3.5 . ( Optimization ) For some δ ∈ [ 0 , 1 ] , if we set the width of the network as m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) , and we set the step-size of gradient descent as η = O ( λ0 n2L2m ) . Then , with probability at least 1−δ over the randomness of initialized parametersW ( 0 ) : = { W ( 0 ) ` } L+1 ` =1 with W ( 0 ) L+1 = v , we have for k = 0 , 1 , 2 , . . . , LS ( W ( k ) ) ≤ ( 1− ηmλ0 2 ) k LS ( W ( 0 ) ) . ( 11 ) In other words , the training loss drops to 0 at a linear rate . We acknowledge a series of past works Allen-Zhu et al . [ 2018 ] ; Du et al . [ 2019 ] have similar spirits with those in Theorem 3.5 . However , it is worth noting that their results are not applicable in our problem settings and data assumptions . Specifically , the result of Du et al . [ 2019 ] is based on the smooth and differentiable activation function , whereas the Theorem 3.5 is about the training error of ReLU activation function , which is not differentiable at 0 . Furthermore , the result of Allen-Zhu et al . [ 2018 ] relies on φ-separateness assumption stating that the every pair of feature vectors { xi , xj } n i 6=j is apart from each other by some constant φ > 0 in a Euclidean norm . In our work , the positivity assumption on the minimum eigenvalue of the NTK is imposed ( i.e. , λ0 > 0 ) . Remark 3.6 . Reducing the order of network width is definitely another line of interesting research direction . We are aware of some works in literature , but we chose not to adopt the techniques since this can make the analysis overly complicated . To the best of our knowledge , the paper that most neatly summarizes this line of literature is Zou & Gu [ 2019 ] . See the table in page 3 in their paper . The order of width they obtained is Ω ( n8L12 φ8 ) , where they impose φ-separateness assumption . Remark 3.7 . There has been an attempt to make a connection between the positivity and φseparateness assumptions . Recently , Zou & Gu [ 2019 ] proved the relation λ0 = Ω ( φn−2 ) 5 in a shallow-neural net setting . See Proposition 3.6. of their work . However , it is still an open question on whether this relation holds in DNN setting as well . The results in Theorem 3.5 suggest a positive conjecture on this question . Indeed , plugging the relation λ0 = Ω ( φn−2 ) in ( 11 ) and in the η = O ( λ0 n2L2m ) yield the discount factor ( 1− Ω ( ηmφ n2 ) ) k and step-size η = O ( φ n4L2m ) , which are exactly the same orders as presented in Allen-Zhu et al . [ 2018 ] . See Theorem 1 of their ArXiv version paper for the clear comparison . We leave the proof of this conjecture as a future work . 4In shallow neural network with ReLU activation without bias terms , it is shown that µk satisfy µ0 , µ1 > 0 , µk = 0 if k = 2j + 1 with j ≥ 1 , and otherwise µk = Θ ( k−d ) . See Bietti & Mairal [ 2019 ] . However , in ReLU DNN , it is shown that these parity constraints can be removed even without bias terms and µk achieves Θ ( k−d ) decay rate for large enough k. Readers can refer Bietti & Bach [ 2021 ] for this result . 5We conjecture that this is not the tightest lower bound on λ0 . Recently , Bartlett et al . [ 2021 ] proves that λ0 & d/n in shallow neural net setting . See Lemma 5.3 in their paper . Theorem 3.8 . ( Generalization ) Let f ? ρ ∈ HNTKL . Fix a failure probability δ ∈ [ 0 , 1 ] . Set the width of the network as m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) , the step-size of gradient descent as η = O ( λ0 n2L2m ) , and the variance parameter ω ≤ O ( ( λ0δ n ) 2/3 ) . Then , if the GD iteration k ≥ Ω ( log ( n ) ηmλ0 ) or k ≤ O ( 1 ηmωL ) , with probability at least 1− δ over the randomness of initialized parametersW ( 0 ) , we have R ( f̂k , f ? ρ ) = Ω ( 1 ) . This theorem states that if the network is trained for too long or too short , the L2-prediction error of f̂W ( k ) is bounded away from 0 by some constant factor . Specifically , the former scenario indicates that the overfitting can be harmful for recovering f ? ρ ∈ HNTKL given the noisy observations . Remark 3.9 . Readers should note that the Theorem 3.8 does not consider if the GD algorithm can achieve low prediction risk R ( f̂k , f ? ρ ) over the range of iterations ( ηmωL ) −1 . k . ( ηmλ0 ) −1 log ( n ) . In the numerical experiment to be followed in Section A , we observe that for some algorithm iterations k∗ , the risk indeed decreases to the same minimum as low as the ` 2- regularized algorithm can achieve , and increases again . This observation implies that the unregularized algorithm can achieve the minimax rate of prediction risk . However , analytically deriving a data-dependent stopping time k∗ in our scenario requires further studies , since we need a sharp characterization of eigen-distribution of NTK matrix of ReLU DNN , denoted as H∞L in this paper . Readers can refer the Theorem 4.2. of Hu et al . [ 2021 ] in shallow-neural network and equation ( 6 ) in Raskutti et al . [ 2014 ] in kernel regression context on how to compute k ? with the given eigenvalues of the associated kernel matrices . Remark 3.10 . From practitioner ’ s point of view , we can simply stop GD before it hits the nearlyzero empirical risk , but monitor the performance on a held-out validation sample , and stop training early when a minimum on the validation sample has been reached . 3.4 ANALYSIS OF ` 2-REGULARIZED DNN In this subsection , we study the training dynamics of ` 2-regularized DNN and the effects of the regularization for obtaining the minimax optimal convergence rate of L2-prediction risk . In the results to be followed , we set the orders of model parameters µ , η1 , η2 in ( 6 ) , and a variance parameter of output layer , ω as follows : µ = Θ ( n d−1 2d−1 ) , η1 = Θ ( 1 m n− 3d−2 2d−1 ) , η2 = Θ ( 1 L n− 3d−2 2d−1 ) , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) . ( 12 ) Theorem 3.11 . ( Optimization ) Suppose we minimize ` 2-regularized objective function ( 5 ) via modified GD ( 6 ) . Set the network width m log3 ( m ) ≥ Ω ( L20n24 δ2 ) and model parameters as in ( 12 ) . Then , with probability at least 1− δ , the mean-squared error follows LS ( W ( k ) D ) /n ≤ ( 1− η2µL ) k · LS ( W ( 0 ) D ) /n+OP ( 1 ) , ( 13 ) for k ≥ 0 . Additionally , after k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) iterations of ( 6 ) , for some constant C > 0 , we have ∥∥∥∥∥uD ( k ) −H∞L ( Cµ · I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ OP ( 1 n ) , ( 14 ) where we denote uD ( k ) : = [ f̂W ( k ) D ( x1 ) , . . . , f̂W ( k ) D ( xn ) ] > . Several comments are in sequel . Theorem 3.11 is , to our knowledge , the first result that rigorously shows the training dynamics of ` 2-regularized ReLU DNN in overparametrized setting . Observe that the first term on the right-hand side of the inequality ( 13 ) converges linearly to 0 , and the second term is some positive constant that is bounded away from 0 . This implies that the MSE of regularized DNN is upper-bounded by some positive constant . Note that we only provide the upper bound , but the results of our numerical experiments indicate that the MSE is lower-bounded by OP ( 1 ) as well . We leave the proof of this conjecture for the future work . The inequality ( 14 ) states that the trained dynamics of the regularized neural network can approximate the optimal solution ( denoted as g ? µ ) of the following kernel ridge regression problem : min f∈HNTK { 1 2 n∑ i=1 ( yi − f ( xi ) ) 2 + Cµ 2 ‖f‖2HNTKL } , ( 15 ) where ‖ · ‖HNTKL denotes a NTK-induced RKHS norm . ( Note that the optimization problem in ( 15 ) is not normalized by sample size n. ) The inequality ( 14 ) states that after approximately ( η2µL ) −1 iterations of ( 6 ) , the error rate becomes OP ( 1 n ) . The approximation error is computed at the training data points under ` 2 norm . This should be compared with the Theorem 5.1 of Hu et al . [ 2021 ] , where they showed that the similar approximation holds “ within ” a certain range of algorithm in shallow neural network setting . In contrast , we show that the approximation holds “ after ” k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) in deep neural network . It should be noted that the difference of results comes from the regularization scheme , where we penalize the ∑L ` =1 ‖W ` −W ( 0 ) ` ‖2F , whereas Hu et al . [ 2021 ] regularized the term ‖W1‖2F . As another important comparison , Hu et al . [ 2019 ] showed the equivalence of a solution of kernel ridge regression associated with NTK and first order Taylor expansion of the regularized neural network dynamics ; note , however , that the uD ( k ) in ( 14 ) is a full neural network dynamics . LetR ( f̂ W ( k ) D , f ? ρ ) be the L2-prediction risk of the regularized estimator f̂W ( k ) D via modified GD ( 6 ) . Next theorem states the result of generalization ability of f̂ W ( k ) D . Theorem 3.12 . ( Generalization ) Let f ? ρ ∈ HNTKL . Suppose the network width mlog3 ( m ) ≥ Ω ( L20n24 δ2 ) and model parameters are set as suggested in ( 12 ) . Then , with probability tending to 1 , we have R ( f̂ W ( k ) D , f ? ρ ) = OP ( n− d 2d−1 ) . The resulting convergence rate is O ( n− d 2d−1 ) with respect to the training sample size n. Note that the rate is always faster than O ( n−1/2 ) and turns out to be the minimax optimal [ Caponnetto & De Vito , 2007 ; Blanchard & Mücke , 2018 ] for recovering f ? ρ ∈ HNTKL in the following sense : lim r→0 lim inf n→∞ inf f̂ sup ρ P [ R ( f̂ , f ? ρ ) > rn− d 2d−1 ] = 1 , ( 16 ) where ρ is a data distribution class satisfying the Assumptions ( A1 ) , ( A2 ) and f ? ρ ∈ HNTKL , and infimum is taken over all estimators D → f̂ . It is worth noting that the minimax rate in ( 16 ) is same with the minimax rate for recovering f ? ρ ∈ HNTK1 . ( i.e. , Hu et al . [ 2021 ] ) This result can be derived from the recent discovery of the equivalence between two function spaces , HNTK1 = HNTKL . See Geifman et al . [ 2020 ] and Chen & Xu [ 2020 ] . Remark 3.13 . A particular choice of µ = Θ ( n d−1 2d−1 ) in ( 12 ) is for obtaining an optimal minimax rate for prediction error in Theorem 3.12 . Specifically , the order of µ determines the L2 distance between the f ? ρ and the kernel regressor g ? µ . That is , ‖f ? ρ − g ? µ‖22 = OP ( µ n ) . With the resultH NTK 1 = HNTKL , the same proof of Lemma D.2 . in Hu et al . [ 2021 ] can be applied for proving this result . 4 CONCLUSION We analyze the convergence rate of L2-prediction error of both the unregularized and the regularized gradient descent for overparameterized DNN with ReLU activation for a regression problem . Under a positivity assumption of NTK , we show that without the adoption of early stopping , the L2-prediction error of the estimated DNN via vanilla GD is bounded away from 0 ( Theorem 3.5 ) , whereas the prediction error of the DNN via ` 2-regularized GD achieves the optimal minimax rate ( Theorem 3.12 ) . The minimax rate O ( n− d 2d−1 ) is a faster than the O ( n−1/2 ) by specifying the complexities of target function and hypothesis space . 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Stochastic gradient descent optimizes over-parameterized deep ReLU networks . arxiv e-prints , art . arXiv preprint arXiv:1811.08888 , 2018 . Difan Zou , Yuan Cao , Dongruo Zhou , and Quanquan Gu . Gradient descent optimizes overparameterized deep ReLU networks . Machine Learning , 109 ( 3 ) :467–492 , 2020 . A NUMERICAL ILLUSTRATIONS In this section , we use synthetic data to corroborate our theoretical findings . We use the He initialization [ He et al. , 2015 ] and employ ( ` 2-regularized ) GD as introduced in subsection 2.2 . For the experiments , we run 1000 epochs of GD and use a fixed step size , setting η1 = η2 = 0.001 . We uniformly generate n feature data xitrain from Sd−1 with d = 2 and generate yi from f ? ρ ( xtraini ) with εi ∼ N ( 0 , 1 ) . To create a function f ? ρ ∈ HNTKL , we use the definition in ( 10 ) with α ∈ Unif ( Sp−1 ) and with p fixed points { x̃j } pj=1 ⊂ Unif ( Sd−1 ) , where p is simply set as 1 . Note that Ker ( · , · ) in ( 10 ) can be calculated via the formulas ( 8 ) and ( 9 ) with specified network depth L. We consider a scenario where we have a network with depth L = 8 and width m = 2000 . The variance parameter of the output layer ( ω ) is set as 1 for unregularized and 0.001 for regularized cases . In Fig 1 . ( a ) , we record the training errors of regularized networks over the GD epochs k ≤ 1000 , where we have n ∈ { 100 , 300 , 500 , 1000 , 5000 } training samples . This aims to verify the inequality ( 13 ) that the MSE of regularized network is bounded away from 0 by some constant . In Fig 1 . ( b ) , the prediction risks of both unregularized and regularized networks are displayed . We approximate the risk with 1500 ∑500 j=1 ( f̂k ( x test j ) − f ? ρ ( xtestj ) ) 2 with a new test data set { xtestj , f ? ρ ( xtestj ) } 500j=1 over k ≤ 1000 for both unregularized and regularized cases . In both cases , they reach the same minimal risks , but the risk of unregularized network increase after it hits the minimal point , whereas the risk of regularized network stays stable . Theorem 3.6 tells us that for the iteration less than the order O ( 1 ηmωL ) , the prediction error is bounded away from 0 . In the experiment for unregularized case , we set η = 0.01 , m = 2000 , L = 8 , and ω = 1 . Plugging in these parameters in the bound says that the minimum can be achieved within a very few iterations . Note that the optimal risk is non-zero as long as we have finite sample sizes n , but converges to 0 at the rate O ( n− d 2d−1 ) . In Fig 1 . ( c ) , we verify that the more training sample sizes we have , the closer the risks of the regularized networks get to 0 . The risk is evaluated at the sample sizes n = { 100 , 300 , 500 , 1000 , 5000 } . We have to acknowledge that there is a discrepancy between our experiment setting and theory . Specifically , due to the limited computing power , we could not run the experiment under the regime of width m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) . But the prediction risk behaves similarly as expected by our theorem , which can be a partial evidence that the statement in Theorem 3.6 still holds in the narrower width of the network . B PRELIMINARY NOTATIONS Before presenting the formal proofs of Lemmas and main results , we introduce several notations used frequently throughout the proofs . First , we denote x ` , i the output of the ` th hidden layer with the input data xi after applying entry-wise ReLU activation function . x ` , i = σ ( W ` σ ( W ` −1 · · ·σ ( W1xi ) · · · ) ) . Denote fW ( k ) ( x ) a value of neural network ( 2 ) evaluated at the collection of network parameters W ( k ) : = { W ( k ) ` } ` =1 , ... , L and W ( k ) ` denotes the ` th hidden layer parameter updated by kth GD iterations . Partial gradient of fW ( k ) ( x ) . We employ the following matrix product notation which was used in several other papers [ Zou et al. , 2018 ; Cao & Gu , 2019 ] : ` 2∏ r= ` 1 Ar : = { A ` 2A ` 2−1 · · ·A ` 1 if ` 1 ≤ ` 2 , I otherwise . ( 17 ) Then , the partial gradient of fW ( k ) ( x ) with respect to W ( k ) ` for 1 ≤ ` ≤ L has a following form : for i ∈ { 1 , . . . , n } , ∇W ` [ fW ( k ) ( xi ) ] = √ m · [ x ( k ) ` −1 , iv T ( L∏ r= ` +1 Σ ( k ) r , i W ( k ) r ) Σ ( k ) ` , i ] > , ` ∈ [ L ] , where Σ ( k ) ` , i : = Diag ( 1 ( 〈w ( k ) ` ,1 , x ( k ) ` −1 , i〉 ≥ 0 ) , . . . ,1 ( 〈w ( k ) ` , m , x ( k ) ` −1 , i〉 ≥ 0 ) ) ∈ Rm×m and w ( k ) ` , j denotes jth column of the matrix W ( k ) ` . Gram matrix H ( k ) . Each entries of empirical gram matrix evaluated at the kth GD update are defined as follows : Hi , j ( k ) = 1 m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr . Note that H ( 0 ) → H∞L as m → ∞ which is proved in Jacot et al . [ 2018 ] ; Yang [ 2019 ] ; Lee et al . [ 2018 ] ; Arora et al . [ 2019b ] . Perturbation region of weight matrices . Consider a collection of weight matrices W̃ = { W̃ ` } ` =1 , ... , L such that W̃ ∈ B ( W ( 0 ) , τ ) : = { W̃ ` : ‖W̃ ` −W ( 0 ) ` ‖2 ≤ τ , ∀ ` ∈ [ L ] } . ( 18 ) For all i ∈ { 1 , . . . , n } and ` = 1 , . . . , L , we denote x ` , i and x̃ ` , i as the outputs of the ` -th layer of the neural network with weight matrices W ( 0 ) and W̃ , and Σ ` , i and Σ̃ ` , i are diagonal matrices with ( Σ ` , i ) jj = 1 ( 〈w ( 0 ) ` , j , x ` −1 , i〉 ≥ 0 ) and ( Σ̃ ` , i ) jj = 1 ( 〈w̃ ` , j , x̃ ` −1 , i〉 ≥ 0 ) , respectively . C WHY IS IT HARD TO PROVE ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) ? In this subsection , we provide a heuristic argument on why it is hard to prove ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) , where W ( k ) D , ` is the model parameter of ` th layer in kth iteration of algorithm . Here , we regularize solely on the model parameter , instead on the relative to the initialization . In this case , we can write the update rule as follows : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . ( 19 ) By recursively applying above equation ( 4.3 ) , we can write W ( k ) D , ` with respect to W ( 0 ) D , ` as follows : W ( k ) D , ` = ( 1− η2µ ) kW ( 0 ) D , ` − η1 k−1∑ ` =0 ( 1− η2µ ) ` ∇W ` [ LS ( W ( k− ` −1 ) D ) ] . Then , we can control the bound as follows : ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ ( 1− ( 1− η2µ ) k ) ∥∥∥W ( 0 ) D , ` ∥∥∥ 2 + η1 η2µ max ` =0 , ... , k−1 ∥∥∥∇W ` [ LS ( W ( k− ` −1 ) D ) ] ∥∥∥ 2 . We know under the initialization setting in our paper , ‖W ( k ) D , ` ‖2 ≤ O ( 1 ) with high-probability ( see Vershynin [ 2018 ] ) , and as long as we can prove the ` 2-norm of gradient is bounded , then we can conclude ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) . However , we are not aware of works in which they control the size of ‖∇W ` [ LS ( W ( k− ` −1 ) D ) ] ‖2 where the non-convex interactions between model parameters across the hidden layers are allowed . To the best of our knowledge , we know the work Allen-Zhu et al . [ 2019 ] deals with the three layer case under this setting . But we need further investigations on whether the techniques employed in their paper can be generalized to arbitrary L-hidden layer setting . D USEFUL LEMMAS A simple fact . Suppose vj i.i.d∼ N ( 0 , ωm ) for j ∈ [ m ] . Then , with probability at least 1 − exp [ −Ω ( m ) ] , ‖v‖22 ≤ O ( ω ) . Proof . Since ∥∥v2j∥∥Ψ1 ≤ O ( ωm ) for j ∈ [ m ] , where ‖ · ‖Ψ1 denotes a sub-exponential norm , Bernstein ’ s inequality for i.i.d . centered sub-exponential random variables can be employed : For any t ≥ 0 , P ( ∣∣∣∣∣∣ m∑ j=1 ( v2j − ω m ) ∣∣∣∣∣∣ ≥ t ) ≤ 2 exp ( − cmin ( t2∑m j=1 ∥∥v2j∥∥2Ψ1 , t maxj ∥∥v2j∥∥Ψ1 ) ) , ( 20 ) where c > 0 is an absolute constant . Note that we used the fact centering does not hurt the subexponentiality of random variable . Choosing t = O ( ω ) concludes the proof . Lemma 4.1 ( Lemma 7.1 . Allen-Zhu et al . [ 2018 ] ) . With probability at least 1 − O ( nL ) · exp [ −Ω ( m/L ) ] , 3/4 ≤ ‖x ( 0 ) ` , i ‖2 ≤ 5/4 for all i ∈ { 1 , . . . , n } and ` ∈ { 1 , . . . , L } . Lemma 4.2 ( Lemma B.1 . Cao & Gu [ 2019 ] ) . If τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , then with probability at least 1 − O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , 1/2 ≤ ‖x̃ ` , i‖2 ≤ 3/2 for all W̃ ∈ B ( W ( 0 ) , τ ) , i ∈ { 1 , . . . , n } and ` ∈ { 1 , . . . , L } . Lemma 4.3 ( Allen-Zhu et al . [ 2018 ] ) . Uniformly over i ∈ { 1 , . . . , n } and 1 ≤ ` 1 ≤ ` 2 ≤ L , the following results hold : 1 . ( Lemma.7.3 , Allen-Zhu et al . [ 2018 ] ) Suppose m ≥ Ω ( nL log ( nL ) ) , then with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥∥∥ ` 2∏ r= ` 1 Σ ( 0 ) r , iW ( 0 ) r ∥∥∥∥∥ 2 ≤ O ( √ L ) . 2 . ( Lemma.7.4 , Allen-Zhu et al . [ 2018 ] ) Suppose m ≥ Ω ( nL log ( nL ) ) , then with probability at least , 1−O ( nL ) · exp [ −Ω ( m/L ) ] , ∥∥∥∥∥v > ( L∏ r= ` 1 Σ ( 0 ) r , iW ( 0 ) r ) ∥∥∥∥∥ 2 ≤ O ( √ w ) . 3 . ( Lemma.8.2 , Allen-Zhu et al . [ 2018 ] ) Suppose τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) . For all W̃ ∈ B ( W ( 0 ) , τ ) , with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥x̃ ` 1 , i − x ( 0 ) ` 1 , i∥∥∥2 ≤ O ( τL5/2√log ( m ) ) . 4 . ( Corollary.8.4 , Allen-Zhu et al . [ 2018 ] ) Suppose τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , then with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥Σ̃ ` 1 , i −Σ ( 0 ) ` 1 , i∥∥∥0 ≤ O ( mτ2/3L ) . 5 . ( Lemma.8.7 , Allen-Zhu et al . [ 2018 ] ) For all ` ∈ [ L ] , let Σ′′ ` , i ∈ [ −3 , 3 ] m×m be the diagonal matrices with at most s = O ( mτ2/3L ) non-zero entries . For all W̃ ∈ B ( W ( 0 ) , τ ) , where τ = O ( 1 L1.5 ) , with probability at least 1−O ( nL ) · exp [ −Ω ( s log ( m ) ) ] , ∥∥∥∥∥vT ( L∏ r= ` 1+1 ( Σ′′r , i + Σ ( 0 ) r , i ) W̃r , i ) ( Σ′′ ` 1 , i + Σ ( 0 ) r , i ) − vT ( L∏ r= ` 1+1 Σ ( 0 ) r , iW ( 0 ) r , i ) Σ ( 0 ) ` 1 , i ∥∥∥∥∥ 2 ≤ O ( τ1/3L2 √ ω log ( m ) ) . Lemma 4.4 ( Lemma B.3 . Cao & Gu [ 2019 ] ) . There exists an absolute constant κ such that , with probability at least 1 − O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , i ∈ 1 , . . . , n and ` ∈ 1 , . . . , L and for all W̃ ∈ B ( W ( 0 ) , τ ) , with τ ≤ κL−6 [ log ( m ) ] −3 , it holds uniformly that∥∥∇W ` [ fW̃ ( xi ) ] ∥∥2 ≤ O ( √ωm ) . Lemma 4.5 . Suppose W̃ ∈ B ( W ( 0 ) , τ ) and τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) . For all u ∈ Rm with a cardinality ‖u‖0 ≤ s , for any 1 ≤ ` ≤ L and i ∈ { 1 , . . . , n } , with probability at least 1−O ( nL ) · exp ( − Ω ( s log ( m ) ) ) −O ( nL ) · exp ( − Ω ( mτ2/3L ) ) , ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≤ √ ωs log ( m ) m · O ( ‖u‖2 ) . Proof . Recall Lemma 4.2 . For any fixed vector u ∈ Rm , with probability at least 1 − O ( nL ) · exp [ −Ω ( mτ2/3L ) ] for τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , for any 1 ≤ ` ≤ L and i ∈ { 1 , . . . , n } , we have the event T , ∥∥∥∥∥ ( L∏ r= ` Σ̃r , iW̃r , i ) u ∥∥∥∥∥ 2 ≤ 3 ‖u‖2 . ( 21 ) Conditioned on this event happens , it is easy to see the random variable v > ( ∏L r=a Σ̃r , iW̃r , i ) u ∼ SG ( 9ω m ‖u‖ 2 2 ) . Based on this observation , we have the probability , P ( ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≥ √ ωs log ( m ) m · O ( ‖u‖2 ) ) ≤ P ( ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≥ √ ωs log ( m ) m · O ( ‖u‖2 ) ∣∣∣ T ) + P ( T c ) ≤ O ( nL ) · exp ( − Ω ( s log ( m ) ) ) +O ( nL ) · exp ( − Ω ( mτ2/3L ) ) , where in the last inequality , union bounds over the indices ` and i , and over the vector u ∈ Rm with ‖u‖0 ≤ s are taken . Lemma 4.6 . Suppose τ ≤ 1 CL9/2 [ log ( m ) ] 3 for some constant C > 0 . Then , for all i ∈ [ n ] and ` ∈ [ L ] , with probability at least 1−O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , we have ∥∥∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥2 ≤ O ( τ1/3L2√ωm log ( m ) ) . Proof . By using the results from Lemma 4.3 , we can control the term : ‖∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ‖2 = √ m · ∥∥∥∥∥x ( k ) ` −1vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` − x ( 0 ) ` −1v T ( L∏ r= ` +1 Σ ( 0 ) r W ( 0 ) r ) Σ ( 0 ) ` ∥∥∥∥∥ 2 ≤ √ m · ‖x ( k ) ` −1 − x ( 0 ) ` −1 ‖2︸ ︷︷ ︸ ≤O ( τL5/2 √ log ( m ) ) · ∥∥∥∥∥vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` ∥∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ ω ) + √ m · ∥∥∥x ( 0 ) ` −1∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) · ∥∥∥∥∥vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` − v T ( L∏ r= ` +1 Σ ( 0 ) r W ( 0 ) r ) Σ ( 0 ) ` ∥∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ω log ( m ) ) ≤ O ( τ1/3L2 √ ωm log ( m ) ) , where , in the last inequality , we used the condition on τ ≤ 1 CL9/2 [ log ( m ) ] 3 < 1 . Remark 4.7 . Note that the results in Lemmas 6.3 ( second and fifth items ) , 6.4 , 6.5 , 6.6 are in the setting of vj ∼ N ( 0 , ωm ) for j ∈ [ m ] . For the notational convenience , in following Lemmas we denote fW ( k ) ( xi ) as ui ( k ) and let u ( k ) : = [ u1 ( k ) , . . . , un ( k ) ] > for k ≥ 0 . Lemma 4.8 . For some δ ∈ [ 0 , 1 ] , if m ≥ Ω ( L log ( nL/δ ) ) , then with probability at least 1 − δ , ‖u ( k ) ‖2 ≤ O ( √nω δ ) for any k ≥ 0 . Proof . Recall the Lemma 4.2 stating that ∥∥∥x ( k ) L , i∥∥∥ 2 = O ( 1 ) for any input data xi for i ∈ [ n ] . Also recall that vj ∼ N ( 0 , ωm ) for j ∈ [ m ] , xL , i ∈ R m and ui ( k ) = √ mv > xL , i ∼ N ( 0 , O ( ω ) ) . Then , we have a following via simple Markov inequality : for any t ≥ 0 , P ( ‖u ( k ) ‖2 ≥ t ) ≤ E [ ‖u ( k ) ‖2 ] t ≤ √ E [ ‖u ( k ) ‖22 ] t ≤ O ( √ nω ) t . Lemma 4.9 . For some δ ∈ [ 0 , 1 ] , if m ≥ Ω ( L log ( nL/δ ) ) , then with probability at least 1− δ , we have ‖u ( 0 ) − y‖2 ≤ O ( √ n δ ) . Proof . By Markov ’ s inequality , for any t ≥ 0 , P ( ‖u ( 0 ) − y‖2 ≥ t ) ≤ Eε , W ( 0 ) , v [ ‖u ( 0 ) − y‖22 ] t2 . ( 22 ) Note that the expectation in the nominator of ( 22 ) is taken over the random noise ε and initialized parameter W ( 0 ) , v. We can expand the nominator as follows : Eε , W ( 0 ) , v [ ‖u ( 0 ) − y‖22 ] = EW ( 0 ) , v‖u ( 0 ) ‖22 + Eε‖y‖22 − 2Eε , W ( 0 ) , v [ y > u ( 0 ) ] . ( 23 ) For the convenience of notation , let y∗ : = [ f ? ρ ( x1 ) , . . . , f ? ρ ( xn ) ] > and ε : = [ ε1 , . . . , εn ] > . Recall that we have y = y∗ + ε , and ‖y∗‖22 = O ( n ) . Also note that by Lemma 4.1 , with probability at least 1−O ( nL ) · exp [ −Ω ( m/L ) ] , for any i = 1 , . . . , n , ‖x ( 0 ) L , i‖22 = O ( 1 ) . Then , we have a random variable ui ( 0 ) = √ mv > xL , i ∼ N ( 0 , O ( ω ) ) . Now , we are ready to derive the orders of three terms on the RHS of ( 23 ) . EW ( 0 ) , v‖u ( 0 ) ‖22 = O ( n ) , Eε‖y‖22 = Eε [ ‖y∗‖22 + ‖ε‖22 − 2y > ε ] = O ( n ) , Eε , W ( 0 ) , v [ y > u ( 0 ) ] = Eε , W ( 0 ) , v [ ( y∗ + ε ) > u ( 0 ) ] = 0 . Combining the above three equalities , we conclude the proof . Lemma 4.10 . Suppose τ = O ( n√ω√ mδλ0 ) . For some δ ∈ [ 0 , 1 ] such that δ ≥ O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , then with probability at least 1− δ , we have ‖H ( k ) −H ( 0 ) ‖2 ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) . Proof . By the definition of gram matrix Hi , j ( k ) for any k ≥ 0 , we have |Hi , j ( k ) −Hi , j ( 0 ) | = ∣∣∣∣∣ 1m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr − 〈 ∇W ` [ fW ( 0 ) ( xi ) ] , ∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ∣∣∣∣∣ ≤ 1 m L∑ ` =1 { ∣∣∣∣〈∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] −∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ∣∣∣∣ + ∣∣∣∣〈∇W ` [ fW ( 0 ) ( xj ) ] , ∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] 〉 Tr ∣∣∣∣ } ≤ 1 m L∑ ` =1 { ∥∥∇W ` [ fW ( k ) ( xi ) ] ∥∥2︸ ︷︷ ︸ ≤O ( √ ωm ) · ∥∥∇W ` [ fW ( k ) ( xj ) ] −∇W ` [ fW ( 0 ) ( xj ) ] ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) + ∥∥∇W ` [ fW ( 0 ) ( xj ) ] ∥∥2︸ ︷︷ ︸ ≤O ( √ ωm ) · ∥∥∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) } ≤ O ( ω7/6n1/3L3 6 √ log3 ( m ) mδλ20 ) . In the second inequality , Lemmas 4.4 and 4.6 are used , and in the last inequality , τ = O ( n√ω√ mδλ0 ) is plugged in . With this , using the fact that Frobenius norm of a matrix is bigger than the operator norm , we bound the term ‖H ( k ) −H ( 0 ) ‖2 as follows : ‖H ( k ) −H ( 0 ) ‖2 ≤ ‖H ( k ) −H ( 0 ) ‖F ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) . Lemma 4.11 . For some δ ∈ [ 0 , 1 ] , with probability at least 1− δ , ‖H∞L −H ( 0 ) ‖2 ≤ O ( ωnL5/2 4 √ log ( nL/δ ) m ) Proof . For some δ ′ ∈ [ 0 , 1 ] , set ε = L3/2 4 √ log ( L/δ ′ ) m from Theorem 3.1. of Arora et al . [ 2019b ] . For any fixed i , j ∈ [ n ] , we have P [ ∣∣H∞i , j −Hi , j ( 0 ) ∣∣ ≤ O ( ωL5/2 4√ log ( L/δ′ ) m ) ] ≥ 1− δ′ . After applying the union bound over all i , j ∈ [ n ] , setting δ = δ ′ n2 , and using the fact that Frobenius norm of a matrix is bigger than the operator norm , we conclude the proof . For two positive semi-definite matrices A and B , if we write A B , then it means A − B is positive semi-definite matrix . Similarly , if we write A B , then it means A−B is positive definite matrix . With these notations , we introduce a following Lemma . Lemma 4.12 ( Lemma D.6 . Hu et al . [ 2021 ] ) . For two positive semi-definite matrices A and B , 1 . Suppose A is non-singular , then A B ⇐⇒ λmax ( BA−1 ) ≤ 1 and A B ⇐⇒ λmax ( BA −1 ) < 1 , where λmax ( · ) denotes the maximum eigenvalue of the input matrix . 2 . Suppose A , B and Q are positive definite matrices , A and B are exchangeable , then A B =⇒ AQA BQB . E PROOF OF THEOREM 3.5 For the convenience of notation , denote ui ( k ) = fW ( k ) ( xi ) and let u ( k ) = [ u1 ( k ) , u2 ( k ) , . . . , un ( k ) ] > . In order to achieve linear convergence rate of the training error , ‖u ( k ) − y‖22 , we decompose the term as follows : ‖u ( k + 1 ) − y‖22 = ∥∥u ( k ) − y + ( u ( k + 1 ) − u ( k ) ) ∥∥2 2 = ‖u ( k ) − y‖22 − 2 ( u ( k ) − y ) > ( u ( k + 1 ) − u ( k ) ) + ‖u ( k + 1 ) − u ( k ) ‖22 . Equipped with this decomposition , the proof consists of the following steps : 1 . Similarly with Du et al . [ 2019 ] , a term ( u ( k + 1 ) − u ( k ) ) is decomposed into two terms , where we denote them as I ( k ) 1 and I ( k ) 2 , respectively . We note that the first term I ( k ) 1 is related with a gram matrix H ( k ) and a second term I ( k ) 2 can be controlled small enough in ` 2 sense with proper choices of the step size and the width of network . 2 . A term ‖u ( k + 1 ) − u ( k ) ‖22 needs to be controlled small enough to ensure 2 ( u ( k ) − y ) > ( u ( k + 1 ) − u ( k ) ) > ‖u ( k + 1 ) − u ( k ) ‖22 so that the loss decreases . 3 . It is shown that the distance between the gram matrix H ( k ) and the NTK matrix H∞L is close enough in terms of operator norm . 4 . Lastly , we inductively show that the weights generated from gradient descent stay within a perturbation region B ( W ( 0 ) , τ ) , irrespective with the number of iterations of algorithm , We start the proof by analyzing the term u ( k + 1 ) − u ( k ) . Step 1 . Control on u ( k + 1 ) − u ( k ) . Recall ( Σ ( k ) ` , i ) jj = 1 ( 〈w ( k ) ` , j , x ( k ) ` −1 , i〉 ≥ 0 ) and we introduce a diagonal matrix Σ̃ ( k ) ` , i , whose jth entry is defined as follows : ( Σ̃ ( k ) ` , i ) jj = ( Σ ( k+1 ) ` , i −Σ ( k ) ` , i ) jj · 〈w ( k+1 ) ` , j , x ( k+1 ) ` −1 , i 〉 〈w ( k+1 ) ` , j , x ( k+1 ) ` −1 , i 〉 − 〈w ( k ) ` , j , x ( k ) ` −1 , i〉 . With this notation , the difference x ( k+1 ) L , i − x ( k ) L , i can be rewritten via the recursive applications of Σ̃ ( k ) ` , i : x ( k+1 ) L , i − x ( k ) L , i = ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) ( W ( k+1 ) L x ( k+1 ) L−1 , i −W ( k ) L x ( k ) L−1 , i ) = ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) W ( k+1 ) L ( x ( k+1 ) L−1 , i − x ( k ) L−1 , i ) + ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) ( W ( k+1 ) L −W ( k ) L ) x ( k ) L−1 , i = L∑ ` =1 ( L∏ r= ` +1 ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) W ( k+1 ) r ) ( Σ ( k ) ` , i + Σ̃ ( k ) ` , i ) ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i ( 24 ) Then , we introduce following notations : D ( k ) ` , i = ( L∏ r= ` +1 Σ ( k ) r , i W ( k ) r ) Σ ( k ) ` , i , D̃ ( k ) ` , i = ( L∏ r= ` +1 ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) W ( k+1 ) r ) ( Σ ( k ) ` , i + Σ̃ ( k ) ` , i ) . Now , we can write ui ( k + 1 ) − ui ( k ) by noting that ui ( k ) = √ m · vTx ( k ) L , i : ui ( k + 1 ) − ui ( k ) = √ m · vT ( x ( k+1 ) L , i − x ( k ) L , i ) = √ m · vT L∑ ` =1 D̃ ( k ) ` , i ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i ( 25 ) = −η √ m · vT L∑ ` =1 D ( k ) ` , i ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i︸ ︷︷ ︸ I ( k ) 1 , i −η √ m · vT L∑ ` =1 ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i︸ ︷︷ ︸ I ( k ) 2 , i Here , I ( k ) 1 , i can be rewritten as follows : I ( k ) 1 , i = −η √ m · vT L∑ ` =1 D ( k ) ` , i n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] x ( k ) ` −1 , i = −η · n∑ j=1 ( uj ( k ) − yj ) · ( √ m L∑ ` =1 vTD ( k ) ` , i ∇W ` [ fW ( k ) ( xj ) ] x ( k ) ` −1 , i ) = −mη · n∑ j=1 ( uj ( k ) − yj ) · 1 m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr = −mη · n∑ j=1 ( uj ( k ) − yj ) ·Hi , j ( k ) . For I ( k ) 2 , i , we need a more careful control . First , we pay our attention on bounding the term ‖v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ‖2 as follows : By triangle inequality , we have∥∥∥∥v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∥∥∥∥ 2 ≤ ∥∥∥∥v > ( D ( k ) ` , i −D ( 0 ) ` , i ) ∥∥∥∥ 2 + ∥∥∥∥v > ( D̃ ( k ) ` , i −D ( 0 ) ` , i ) ∥∥∥∥ 2 . ( 26 ) We control the first term of the right-hand side ( R.H.S ) in ( 26 ) . By the fourth item of the Lemma 4.3 , we know ‖Σ ( k ) r , i −Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) and | ( Σ ( k ) r , i −Σ ( 0 ) r , i ) j , j | ≤ 1 for j ∈ [ m ] . Then , we can plug Σ′′r , i = Σ ( k ) r , i −Σ ( 0 ) r , i in the inequality of the fifth item of Lemma 4.3 . So , the first term of the R.H.S in ( 26 ) can be bounded by O ( τ1/3L2 √ ω log ( m ) ) . The second term of the R.H.S in ( 26 ) can be similarly controlled as the first term . Observe that | ( Σ ( k ) r , i +Σ̃ ( k ) r , i ) jj | ≤ 1 , then we have | ( Σ ( k ) r , i +Σ̃ ( k ) r , i −Σ ( 0 ) r , i ) j , j | ≤ 2 for all j ∈ [ m ] . Note that by the definition of Σ̃ ( k ) r , i , we have ‖Σ̃ ( k ) r , i ‖0 = ‖Σ ( k+1 ) r , i −Σ ( k ) r , i ‖0 ≤ ‖Σ ( k+1 ) r , i −Σ ( 0 ) r , i ‖0+‖Σ ( k ) r , i −Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) . Thus , by the triangle inequality , we have ‖Σ ( k ) r , i + Σ̃ ( k ) r , i − Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) . These observations enable us to plug Σ′′r , i = Σ ( k ) r , i + Σ̃ ( k ) r , i −Σ ( 0 ) r , i in the inequality of the fifth item of Lemma 4.3 , and give the bound on the second term as O ( τ1/3L2 √ ω log ( m ) ) . We have ‖v > ( D̃ ( k ) ` , i − D ( k ) ` , i ) ‖2 ≤ O ( τ1/3L2 √ ω log ( m ) ) . Now , we control the ` 2-norm of the I ( k ) 2 as follows : ∥∥∥I ( k ) 2 ∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 2 , i ∣∣∣ ≤ η √ m · n∑ i=1 [ L∑ ` =1 ∥∥∥∥v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( L2τ1/3 √ ω log ( m ) ) · ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 · ∥∥∥x ( k ) ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) : Lemma 4.2 ] ≤ O ( ηnL2τ1/3 √ ωm log ( m ) ) L∑ ` =1 ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 ≤ O ( ηnL5/2τ1/3 √ ωm log ( m ) ) √√√√ L∑ ` =1 ∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥2F ≤ O ( ηnL5/2τ1/3 √ ωm log ( m ) ) √√√√ n∑ j=1 ( uj ( k ) − yj ) 2 L∑ ` =1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥2F ≤ O ( ηnL3τ1/3ωm √ log ( m ) ) ‖u ( k ) − y‖2 . ( 27 ) Step 2 . Control on ‖u ( k + 1 ) −u ( k ) ‖22 . Recall that by ( 25 ) , x ( k+1 ) L , i −x ( k ) L , i can be written as follows : x ( k+1 ) L , i − x ( k ) L , i = L∑ ` =1 D̃ ( k ) ` , i ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i = −η · L∑ ` =1 D̃ ( k ) ` , i ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i . It is worth noting that , ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥2 2 = ∥∥∥∥∥∥ n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] ∥∥∥∥∥∥ 2 2 ≤ n∑ j=1 ( uj ( k ) − yj ) 2 n∑ j=1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥22 ≤ O ( nmω ) ‖u ( k ) − y‖22 . ( 28 ) Also , observe that | ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) jj | ≤ 1 for all j ∈ [ m ] , so by Lemma A.3 of Zou et al . [ 2020 ] , we know ‖D̃ ( k ) ` , i ‖2 ≤ O ( √ L ) . Combining all the facts , we can conclude : ‖u ( k + 1 ) − u ( k ) ‖22 = m · n∑ i=1 ( vTx ( k+1 ) L , i − v Tx ( k ) L , i ) 2 ≤ m · ‖v‖22 n∑ i=1 ∥∥∥x ( k+1 ) L , i − x ( k ) L , i∥∥∥2 2 ≤ η2m · ‖v‖22 n∑ i=1 [ L∑ ` =1 ∥∥∥D̃ ( k ) ` , i ∥∥∥2 2 · ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥2 2 · ∥∥∥x ( k ) ` −1 , i∥∥∥2 2 ] ≤ O ( η2n2L2m2ω2 ) ‖u ( k ) − y‖22 ≤ O ( η2n2L2m2 ) ‖u ( k ) − y‖22 , ( 29 ) where in the third inequality , we additionally used the fact ‖v‖22 = O ( ω ) with probability at least 1− exp ( −Ω ( m ) ) , and the inequality ( 28 ) . In the last inequality , we used the assumption ω ≤ 1 . Step 3. λmin ( H ( k ) ) ≥ λ02 with sufficiently large m. Denote ρ ( A ) as a sprectral radius of a matrix A . Then , we have ‖H ( k ) −H∞L ‖2 ≥ ρ ( H ( k ) −H∞L ) ≥ −λmin ( H ( k ) −H∞L ) ≥ λmin ( H∞L ) − λmin ( H ( k ) ) ≥ λ0 − λmin ( H ( k ) ) , ( 30 ) where , in the second inequality , we used a triangle inequality , λmin ( H ( k ) −H∞L ) + λmin ( H∞L ) ≤ λmin ( H ( k ) ) . By Lemmas 4.10 and 4.11 , setting m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ ) and Õ ( λ 4/3 0 δ 1/3 n4/3L4 ) ≤ ω ≤ 1 , we have ‖H ( k ) −H∞L ‖2 ≤ ‖H ( k ) −H ( 0 ) ‖2 + ‖H ( 0 ) −H ∞ L ‖2 ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) +O ( ωn2L5/2 4 √ log ( nL/δ ) m ) ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) ≤ λ0 2 . ( 31 ) Thus , combining ( 30 ) and ( 31 ) yields that λmin ( H ( k ) ) ≥ λ02 . Step 4 . Concluding the proof . Recall that I ( k ) 1 = −mη · H ( k ) ( u ( k ) − y ) . Then observe that ( u ( k ) − y ) > I ( k ) 1 = −ηm · ( u ( k ) − y ) > H ( k ) ( u ( k ) − y ) ≤ −ηm · λmin ( H ( k ) ) ‖u ( k ) − y‖22 ≤ −ηm · λ0 2 ‖u ( k ) − y‖22 . ( 32 ) We set the step size η , radius of perturbation region τ and network width m as follows , η = Ω ( λ0 n2L2m ) , τ = O ( n √ ω√ mδλ0 ) , m ≥ Ω ( ω7n8L18 log3 ( m ) λ80δ ) . With the above settings , we can control the ‖u ( k + 1 ) − y‖22 by combining ( 27 ) , ( 29 ) and ( 32 ) as follows , ‖u ( k + 1 ) − y‖22 = ∥∥u ( k ) − y + ( u ( k + 1 ) − u ( k ) ) ∥∥2 2 = ‖u ( k ) − y‖22 − 2ηm · ( u ( k ) − y ) > H ( k ) ( u ( k ) − y ) − ( u ( k ) − y ) > I ( k ) 2 + ‖u ( k + 1 ) − u ( k ) ‖ 2 2 ≤ ( 1− ηmλ0 +O ( ηnL3τ1/3mω √ log ( m ) ) +O ( η2n2L2m2 ) ) ‖u ( k ) − y‖22 ≤ ( 1− ηmλ0 2 ) ‖u ( k ) − y‖22 . So far , we have shown from Step 1 to Step 4 that given the radius of perturbation region τ has the order O ( n√ω√ mδλ0 ) , then we can show the training error drops linearly to 0 with the discount factor ( 1 − ηmλ02 ) along with the proper choices of η and m. It remains us to prove the iterates W ( k ) ` for all ` ∈ [ L ] generated by GD algorithm indeed stay in the perturbation region B ( W ( 0 ) , τ ) over k ≥ 0 with τ = O ( n√ω√ mδλ0 ) . Step 5 . The order of the radius of perturbation region . We employ the induction process for the proof . The induction hypothesis is : ∀s ∈ [ k + 1 ] , ∥∥∥W ( s ) ` −W ( 0 ) ` ∥∥∥ 2 ≤ η · O ( √ nmω ) s−1∑ t=0 ( 1− ηmλ0 2 ) t 2 O ( √ n δ ) ≤ O ( n √ ω√ mδλ0 ) . ( 33 ) First , it is easy to see it holds for s = 0 . Now , suppose it holds for s = 0 , . . . , k , we consider s = k + 1.∥∥∥W ( k+1 ) ` −W ( k ) ` ∥∥∥ 2 = ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 = η · ∥∥∥∥∥∥ n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] ∥∥∥∥∥∥ 2 ≤ η · √√√√ n∑ j=1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥22 √√√√ n∑ j=1 ( uj ( k ) − yj ) 2 ≤ η · O ( √ nmω ) √ 2LS ( W ( k ) ) ≤ η · O ( √ nmω ) ( 1− ηmλ0 2 ) k 2 O ( √ n δ ) , ( 34 ) where in the second inequality , we used Lemmas 4.4 . Note that since it is assumed that W ( k ) ` ∈ B ( W ( 0 ) , τ ) , the Lemma is applicable with m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ ) . Simi- larly , since it is assumed that the induction hypothesis holds for s = 0 , . . . , k , we can see ‖u ( k ) − y‖22 ≤ ( 1 − ηmλ02 ) k‖u ( 0 ) − y‖22 . This inequality is plugged in the last inequality with Lemma 4.9 . By combining the inequalities ( 33 ) for s ∈ [ k ] and ( 34 ) , and triangle inequality , we conclude the proof : ∥∥∥W ( k+1 ) ` −W ( 0 ) ` ∥∥∥ 2 ≤ η · O ( √ nmω ) k∑ t=0 ( 1− ηmλ0 2 ) t 2 O ( √ n δ ) ≤ O ( n √ ω√ mδλ0 ) . Proposition 5.1 . For some δ ∈ [ 0 , 1 ] , if we set the width of the network as m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ 2 ) , and we set the step-size of gradient descent as η = O ( λ0 n2L2m ) . Then , with probability at least 1 − δ over the randomness of initialized parameters W ( 0 ) , we have for k = 0 , 1 , 2 , . . . , u ( k ) − y = ( I− ηmH∞L ) k ( u ( 0 ) − y ) + ξ ( k ) , where ‖ξ ( k ) ‖2 = k ( 1− ηmλ0 2 ) k−1 O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖y − u ( 0 ) ‖2 . Proof . Define ui ( k ) : = fW ( k ) ( xi ) , then we have u ( k + 1 ) − u ( k ) = −ηm ·H ( k ) ( u ( k ) − y ) + I ( k ) 1 = −ηm ·H∞L ( u ( k ) − y ) − ηm · ( H ( k ) −H∞L ) ( u ( k ) − y ) + I ( k ) 1 = −ηm ·H∞L ( u ( k ) − y ) + e ( k ) . By recursively applying the above equality , we can easily derive a following for any k ≥ 0 , u ( k ) − y = ( I− ηmH∞L ) k ( u ( 0 ) − y ) + k−1∑ t=0 ( I− ηmH∞L ) t e ( k − 1− t ) ︸ ︷︷ ︸ =ξ ( k ) . ( 35 ) Now , we want to show ξ ( k ) can be controlled in arbitrarily small number . First , e ( k ) needs to be bounded in an ` 2 norm : ‖e ( k ) ‖2 ≤ ηm ‖H ∞ L −H ( k ) ‖2 ‖u ( k ) − y‖2 + ∥∥∥I ( k ) 2 ∥∥∥ 2 ≤ ηm · O ( ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖u ( k ) − y‖2 , where , in the second inequality , τ = O ( n √ ω√ mδλ0 ) is plugged in ( 27 ) . Equipped with the bound on ‖e ( k ) ‖2 , we can easily bound the ‖ξ ( k ) ‖2 as follows : ∥∥∥∥∥ k−1∑ t=0 ( I− ηmH∞L ) t e ( k − 1− t ) ∥∥∥∥∥ 2 ≤ k−1∑ t=0 ‖I− ηmH∞L ‖ t 2 ‖e ( k − 1− t ) ‖2 ≤ k−1∑ t=0 ( 1− ηmλ0 ) t O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖u ( k − 1− t ) − y‖2 ≤ k−1∑ t=0 ( 1− ηmλ0 ) t O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ( 1− ηmλ0 2 ) k−1−t ‖u ( 0 ) − y‖2 = k ( 1− ηmλ0 2 ) k−1 O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ 2 ) ‖u ( 0 ) − y‖2 . ( 36 ) Note that in the third inequality , we used the result from Theorem 1 . F PROOF OF THEOREM 3.8 We begin the proof by decomposing the error f̂W ( k ) ( x ) − f∗ ( x ) for any fixed x ∈ Unif ( Sd−1 ) into two terms as follows : f̂W ( k ) ( x ) − f∗ ( x ) = ( f̂W ( k ) ( x ) − g∗ ( x ) ) ︸ ︷︷ ︸ ∆1 + ( g∗ ( x ) − f∗ ( x ) ) ︸ ︷︷ ︸ ∆2 . ( 37 ) Here , we denote the solution of kernel regression with kernel H∞L as g ∗ ( x ) , which is a minimum RKHS norm interpolant of the noise-free data set { xi , f ? ρ ( xi ) } ni=1 . To avoid the confusion of the notation , we write Ker ( x , X ) = ( H∞L ( x , x1 ) , . . . , H ∞ L ( x , xn ) ) n i=1 ∈ Rn and let y∗ = [ f ? ρ ( x1 ) , . . . , f ? ρ ( xn ) ] > . Then , we have a following closed form solution g∗ ( x ) as , g∗ ( x ) : = Ker ( x , X ) ( H∞L ) −1 y∗ . With the decomposition ( 37 ) , the proof sketch of Theorem 3.8 is as follows . 1 . Note that for any ` ∈ [ L ] , we have f̂W ( k ) ( x ) = 〈vec ( ∇W ` [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ) 〉 . We can write the term vec ( W ( k ) ` ) with respect to vec ( W ( 0 ) ` ) , H∞L and the residual term via recursive applications of GD update rule and the result from proposition 2.1 . Readers can refer ( 38 ) . Using the equality ( 38 ) , we can further decompose ∆1 into three terms . That is , ∆1 = ∆11 + ∆12 + ∆13 . Then , using the boundedness of ` 2-norm of network gradient and the fact that the size of ‖ξ ( k ) ‖2 can be controlled with wide enough network , we can control the size of ‖∆12‖2 and ‖∆13‖2 aribtarily small . 2 . In the term ∆2 , the g ? is an interpolant based on noiseless data . For large enough data points , g ? converges fastly to f ? at the rate OP ( 1√n ) . 3 . Lastly , the ∆11 is the only term that is involved with random error ε , and we show that ‖∆11‖2 is bounded away from 0 for small and large GD iteration index k. Step 1 . Control on ∆1 . For n data points ( x1 , . . . , xn ) and for the kth updated parameter W ( k ) , denote : ∇W ` [ fW ( k ) ( X ) ] = [ vec ( ∇W ` [ fW ( k ) ( x1 ) ] ) , · · · , vec ( ∇W ` [ fW ( k ) ( xn ) ] ) ] . Note that when ` = 1 , ∇W ` [ fW ( k ) ( X ) ] ∈ Rmd×n and when ` = 2 , . . . , L , ∇W ` [ fW ( k ) ( X ) ] ∈ Rm2×n . With this notation , we can rewrite the Gradient Descent update rule as vec ( W ( k+1 ) ` ) = vec ( W ( k ) ` ) − η∇W ` [ fW ( k ) ( X ) ] ( u ( k ) − y ) , k ≥ 0 . Applying Proposition 3.8 , we can get : vec ( W ( k ) ` ) − vec ( W ( 0 ) ` ) = k−1∑ j=0 ( vec ( W ( j+1 ) ` ) − vec ( W ( j ) ` ) ) = −η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ( u ( j ) − y ) = η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) − η · k−1∑ j=0 ∇W ` [ fW ( k ) ( X ) ] ξ ( j ) = η · k−1∑ j=0 ∇W ` [ fW ( 0 ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) − η · k−1∑ j=0 ∇W ` [ fW ( k ) ( X ) ] ξ ( j ) + η · k−1∑ j=0 ( [ ∇W ` [ fW ( j ) ( X ) ] −∇W ` [ fW ( 0 ) ( X ) ] ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) ) = η · k−1∑ j=0 ∇W ` [ fW ( 0 ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) + ξ ′ ( k ) . ( 38 ) First , we control ` 2-norm of the first term of ξ ′ ( k ) as follows : Note that ‖∇W ` [ fW ( j ) ( X ) ] ‖F ≤ O ( √ nmω ) by Lemma 4.4 for 0 ≤ j ≤ k − 1 . Then , we have∥∥∥∥∥∥η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ξ ( j ) ∥∥∥∥∥∥ 2 ≤ k−1∑ j=0 O ( η √ nmω ) O ( j ( 1− ηmλ0 2 ) j−1 ) O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖y − u ( 0 ) ‖2 ≤ O ( n11/6L3ω5/3 m2/3λ 7/3 0 δ 1/6 √ log ( m ) ) ‖y − u ( 0 ) ‖2 . ( 39 ) In the second inequality , ∑∞ j=1 j ( 1− ηmλ02 ) j = O ( 1 η2m2λ20 ) is used . Then , we control ` 2-norm of the second term of ξ ′ ( k ) as follows : ∥∥∥∥∥∥η · k−1∑ j=0 [ ∇W ` [ fW ( j ) ( X ) ] −∇W ` [ fW ( 0 ) ( X ) ] ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) ∥∥∥∥∥∥ 2 ≤ k−1∑ j=0 η ‖I− ηmH∞L ‖ j 2 ‖y − u ( 0 ) ‖2 √√√√ n∑ i=1 ∥∥∇W ` [ fW ( j ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥22 ≤ k−1∑ j=0 η ( 1− ηmλ0 ) jO ( n1/3m1/3L2ω2/3 λ 1/3 0 δ 1/6 √ log ( m ) ) O ( √ n ) ‖y − u ( 0 ) ‖2 ≤ O ( n5/6L2ω2/3 m2/3λ 4/3 0 δ 1/6 √ log ( m ) ) ‖y − u ( 0 ) ‖2 , ( 40 ) where in the second inequality , we used Lemmas 4.6 with τ = O ( n√ω√ mδλ0 ) . Now , we are ready to control ∆1 term . By using the equality ( 38 ) , we can decompose the term ∆1 as follows : Let us denote Gk = ∑k−1 j=0 ηm ( I − ηmH∞L ) j . Note that for any ` ∈ [ L ] , f̂W ( k ) ( x ) = 〈vec ( ∇W ` [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ) 〉 and recall that y = y∗ + ε . Then , for any fixed ` ′ ∈ [ L ] , we have : ∆1 = [ 〈 vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ′ ) 〉 −Ker ( x , X ) ( H∞L ) −1 y∗ ] + Ker ( x , X ) Gky −Ker ( x , X ) Gky = [ Ker ( x , X ) [ Gk − ( H∞L ) −1 ] y∗ + Ker ( x , X ) Gkε ] ︸ ︷︷ ︸ =∆11 + [ 1 m L∑ ` =1 vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] −Ker ( x , X ) ] Gky − 1 m ∑ ` : ` 6= ` ′ vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] Gky︸ ︷︷ ︸ =∆12 + [ 〈 vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( 0 ) ` ′ ) 〉 + vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ξ ′ ( k ) − 1 m vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ] .︸ ︷︷ ︸ =∆13 ( 41 ) Our goal in this step is to control ‖∆12‖2 and ‖∆13‖2 . Then , in the third step , we will show ‖∆11‖2 is the term , which governs the behavior of the prediction risk with respect to algorithm iteration k. First , we bound the ` 2 norm of the first term in ∆12 as : ∥∥∥∥∥ [ 1 m L∑ ` =1 vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] −Ker ( x , X ) ] Gky ∥∥∥∥∥ 2 ≤ 1 mL L∑ ` =1 ∥∥vec ( ∇W ` [ fW ( k ) ( x ) ] ) − vec ( ∇W ` [ fW ( 0 ) ( x ) ] ) ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) : Lemma 4.6 ∥∥∇W ` [ fW ( 0 ) ( X ) ] ∥∥F︸ ︷︷ ︸ ≤O ( √ ωnm ) : Lemma 4.4 ‖Gky‖2 + 1 L √√√√ n∑ i=1 ( 1 m L∑ ` =1 〈 ∇W ` [ fW ( 0 ) ( x ) ] , ∇W ` [ fW ( 0 ) ( xi ) ] 〉 Tr −Ker ( x , xi ) ) 2 ‖Gky‖2 ≤ { O ( n5/6L2ω7/6 m1/6δ1/6λ 1/3 0 √ log ( m ) ) +O ( ωn1/2L3/2 4 √ log ( nL/δ ) m ) } ‖Gk‖2 ‖y‖2 ≤ O ( n5/6L2ω7/6 m1/6δ1/6λ 4/3 0 √ log ( m ) · ‖y‖2 ) +O ( ωn1/2L3/2 λ0 4 √ log ( nL/δ ) m · ‖y‖2 ) , ( 42 ) where , in the second inequality , we plugged τ = O ( n√ω√ mδλ0 ) in the result of Lemma 4.6 and used Lemma 4.11 . In the last inequality , we used ‖Gk‖2 ≤ O ( 1 λ0 ) . Similarly , we can control the ` 2 norm of the second term in ∆12 as follows : ∥∥∥∥∥∥ 1m ∑ ` : ` 6= ` ′ vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] Gky ∥∥∥∥∥∥ 2 ≤ 1 m ∑ ` : ` 6= ` ′ ‖vec ( ∇W ` [ fW ( k ) ( x ) ] ) ‖2︸ ︷︷ ︸ ≤O ( √ ωm ) · ‖∇W ` [ fW ( 0 ) ( X ) ] ‖F︸ ︷︷ ︸ ≤O ( √ ωmn ) · ‖Gk‖2︸ ︷︷ ︸ ≤O ( 1 λ0 ) ‖y‖2 ≤ O ( ωL √ n λ0 ) · ‖y‖2 . ( 43 ) We turn our attention to controlling ‖∆13‖2 . The first term in ∆13 ; Recall that∥∥∥vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) ∥∥∥ 2 ≤ O ( √ mω ) by Lemma 4.4 . Then , the random variable vec ( ∇W ` [ fW ( k ) ( x ) ] ) > vec ( W ( 0 ) ` ) is simply a N ( 0 , O ( ω ) ) for 1 ≤ ` ≤ L. A straightforward application of Chernoff bound for normal random variable and taking union bound over the layer 1 ≤ ` ≤ L yield that : with probability at least 1− δ , ∣∣∣vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > vec ( W ( 0 ) ` ′ ) ∣∣∣ ≤ O ( √ω log ( L δ ) ) . ( 44 ) The ` 2 norm of the third term in ∆13 can be bounded as follows : ∥∥∥∥ 1mvec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ∥∥∥∥ 2 ≤ 1 m ∥∥∥vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) ∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ mω ) ∥∥∥∇W ` ′ [ fW ( 0 ) ( X ) ] ∥∥∥ F︸ ︷︷ ︸ ≤O ( √ ωmn ) ‖Gku ( 0 ) ‖2︸ ︷︷ ︸≤O ( √nωλ0δ ) ≤ O ( nω3/2 λ0δ ) . ( 45 ) In the last inequality , we used the Lemma 4.8 and ‖Gk‖2 ≤ O ( 1 λ0 ) . By combin- ing ( 39 ) , ( 40 ) , ( 44 ) , ( 45 ) with ∥∥∥∇W ` ′ [ fW ( 0 ) ( x ) ] ∥∥∥ F ≤ O ( √ mω ) , we have a following : ‖∆13‖2 ≤ ∥∥∥〈vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( 0 ) ` ′ ) 〉∥∥∥ 2 + ∥∥∥ ( vec ( ∇W ` ′ [ fW ( 0 ) ( x ) ] ) ) > ξ′ ( k ) ∥∥∥ 2 + ∥∥∥∥ 1m ( vec ( ∇W ` ′ [ fW ( 0 ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ∥∥∥∥ 2 ≤ O ( √ ω log ( L δ ) ) +O ( n11/6L3ω13/6 ‖y − u ( 0 ) ‖2 m1/6λ 4/3 0 δ 1/6 √ log ( m ) ) +O ( n5/6L2ω7/6 ‖y − u ( 0 ) ‖2 m1/6λ 7/3 0 δ 1/6 √ log ( m ) ) +O ( nω3/2 λ0δ ) = O ( n11/6L3ω13/6 ‖y − u ( 0 ) ‖2 m1/6λ 4/3 0 δ 1/6 √ log ( m ) ) +O ( n5/6L2ω7/6 ‖y − u ( 0 ) ‖2 m1/6λ 7/3 0 δ 1/6 √ log ( m ) ) +O ( nω3/2 λ0δ ) . ( 46 ) Step 2 . Control on ∆2 . First , note that there is a recent finding that the reproducing kernel Hilbert spaces of NTKs with any number of layers ( i.e. , L ≥ 1 ) have the same set of functions , if kernels are defined on Sd−1 . See Chen & Xu [ 2020 ] . Along with this result , we can apply the proof used in Lemma.D.2 . in Hu et al . [ 2021 ] for proving a following : ‖∆2‖2 = OP ( 1√ n ) . ( 47 ) Step 3 . The behavior of L2 risk is characterized by the term ∆11 . Recall the decompositions ( 37 ) and ( 41 ) , then we have : f̂W ( k ) ( x ) − f∗ ( x ) = ∆11 + ( ∆12 + ∆13 + ∆2 ) : = ∆11 + Θ . ( 48 ) Our goal in this step is mainly two-folded : ( i ) Control Eε ‖Θ‖22 arbitrarily small with proper choices of step-size of GD η and width of the network m. ( ii ) Show that how Eε ‖∆11‖22 affect the behavior of prediction risk over the GD iterations k. First , note that we have Eε ‖y‖22 = Eε ‖y ∗ + ε‖22 ≤ 2 ( y∗ ) > y∗ + 2Eε ‖ε‖22 = O ( n ) . ( 49 ) Second , recall Lemma 4.9 and note that over the random initialization , with probability at least 1−δ , Eε ‖y − u ( 0 ) ‖22 ≤ O ( n δ ) . ( 50 ) Now , by combining the bounds ( 42 ) , ( 46 ) and ( 47 ) , we have Eε ‖Θ‖22 ≤ 3Eε ( ‖∆12‖22 + ‖∆13‖ 2 2 + ‖∆2‖ 2 2 ) ≤ Eε [ O ( n5/3L4ω7/3 m1/3λ 8/3 0 δ 1/3 log ( m ) · ‖y‖22 ) +O ( ω2nL3 λ20 √ log ( nL/δ ) m · ‖y‖22 ) +O ( n11/3L6ω13/3 ‖y − u ( 0 ) ‖22 m1/3λ 8/3 0 δ 1/3 log ( m ) ) +O ( n5/3L4ω7/3 ‖y − u ( 0 ) ‖22 m1/3λ 14/3 0 δ 1/3 log ( m ) ) ] +O ( n2ω3 λ20δ 2 ) +O ( 1 n ) ≤ O ( ω2n2L3 λ20 √ log ( nL/δ ) m ) +O ( n14/3L6ω13/3 m1/3λ 8/3 0 δ 4/3 log ( m ) ) +O ( n8/3L4ω7/3 m1/3λ 14/3 0 δ 4/3 log ( m ) ) +O ( n2ω3 λ20δ 2 ) +O ( 1 n ) , ( 51 ) where in the third inequality , we used ( 49 ) and ( 50 ) . Case 1 . When k is large , the L2 risk is bounded away from zero by some constant . Now we control Eε ‖∆11‖22 . Recall the definitions ‖f‖ 2 2 : = ∫ x∈Sd−1 |f ( x ) | 2dx and Gk = ∑k−1 j=0 ηm ( I− ηmH∞L ) j . Let us denote S = y∗y∗ > . Then , we have Eε ‖∆11‖22 = ∫ x∈Sd−1 Ker ( x , X ) [ ( Gk − ( H∞L ) −1 ) y∗y∗ > ( Gk − ( H∞L ) −1 ) +G2k ] Ker ( X , x ) dx = ∫ x∈Sd−1 Ker ( x , X ) ( H∞L ) −1 Mk ( H∞L ) −1Ker ( X , x ) dx where Mk = ( I− ηmH∞L ) k S ( I− ηmH∞L ) k + ( I− ( I− ηmH∞L ) k ) 2 = [ ( I− ηmH∞L ) k − ( S + I ) −1 ] ( S + I ) [ ( I− ηmH∞L ) k − ( S + I ) −1 ] + I − ( S + I ) −1 . For the algorithm iterations k ≥ ( log ( n ) ηmλ0 ) C0 with some constant C0 > 1 , we have ( I − ηmH∞L ) k ( 1− ηmλ0 ) k · I exp ( −ηmλ0k ) · I exp ( −C0 log ( n ) ) = 1 nC0 · I . Since 1 + ‖y‖22 ≤ C1n for some constant C1 , we have λmax ( 1 nC0 · ( S + I ) ) = 1 + ‖y‖22 nC0 ≤ C1 nC0−1 < 1 . ( 52 ) Using the first item of Lemma ( 4.12 ) with the inequality ( 52 ) , we have ( I − ηmH∞L ) k 1 nC0 · I ≺ ( S + I ) −1 . ( 53 ) The above inequality ( 53 ) lead to a following result : ( S + I ) −1 − ( I − ηmH∞L ) k ( S + I ) −1 − 1nC0 · I . ( 54 ) It is obvious that both ( S + I ) −1 − ( I − ηmH∞L ) k and ( S + I ) −1 − 1nC0 · I are positive definite matrices due to ( 54 ) , and it is also easy to see that they are exchangeable . By using the second item of Lemma ( 4.12 ) , we have Mk = [ ( I− ηmH∞L ) k − ( S + I ) −1 ] ( S + I ) [ ( I− ηmH∞L ) k − ( S + I ) −1 ] + I − ( S + I ) −1 [ ( S + I ) −1 − 1 nC0 · I ] ( S + I ) [ ( S + I ) −1 − 1 nC0 · I ] + I − ( S + I ) −1 = 1 n2C0 S + ( 1− 1 nC0 ) 2 · I . Then , we have Eε ‖∆11‖22 1 n2C0 A+ ( 1− 1 nC0 ) 2 B c0B , where c0 ∈ ( 0 , 1 ) is a constant and A = ∫ x∈Sd−1 [ Ker ( x , X ) ( H∞L ) −1 y∗ ] 2 dx , and B = ∫ x∈Sd−1 [ Ker ( x , X ) ( H∞L ) −1 ] 2 dx . ( 55 ) By triangle inequality with the decomposition ( 48 ) and the bound on Eε ‖Θ‖22 in ( 51 ) , we have : Eε ∥∥∥f̂W ( k ) − f∗∥∥∥2 2 = Eε ‖∆11 + Θ‖22 ≥ 1 2 Eε ‖∆11‖2 − Eε ‖Θ‖ 2 2 ≥ c0 2 B −O ( 1 n ) −O ( n2ω3 λ20δ 2 ) − Õ ( 1 m1/3 poly ( ω , n , L , 1 λ0 , 1 δ ) ) . ( 56 ) For the third term in ( 56 ) , we can choose ω ≤ C2 ( λ0δ n ) 2/3 for some constant C2 > 0 such that the term can be bounded by c08 ∥∥∥Ker ( · , X ) ( H∞L ) −1∥∥∥2 2 . Similarly , the width m can be chosen large enough such that the fourth term in ( 56 ) is upper-bounded by c08 ‖Ker ( · , X ) ( H∞L ) −1‖22 . Using the above choices of k , ω , and m , we can further bound ( 56 ) : Eε ∥∥fW ( k ) − f∗∥∥22 ≥ c04 ∥∥∥Ker ( · , X ) ( H∞L ) −1∥∥∥22 −O ( 1 n ) . ( 57 ) Note that Eε‖f̂∞ − g∗‖22 = ‖Ker ( · , X ) ( H∞L ) −1‖22 where g∗ : = 0 and f̂∞ denotes the noise interpolator . Then , by Theorem 4.2. of Hu et al . [ 2021 ] , we know that Eε‖f̂∞−g∗‖22 ≥ c1 for some constant c1 > 0 . Then , we can take n large enough such that the term O ( 1 n ) is upper-bounded by c0c1 8 , and finish the proof . Case 2 . When k is small , the L2 risk is bounded away from zero by some constant . Recall the definition of ∆11 in the decomposition ( 41 ) , ∆11 : = Ker ( x , X ) Gk [ y∗ + ε ] −Ker ( x , X ) H∞L y∗ : = ∆∗11 −Ker ( x , X ) H∞L y∗ . ( 58 ) We denote the eigen-decomposition of the matrix H∞L : = ∑n i=1 λivivi > , then we can easily see a following : Gk = ηm k−1∑ j=0 ( n∑ i=1 ( 1− ηmλi ) jvivi > ) ηm k−1∑ j=0 n∑ i=1 vivi > ηmk · I . By using the above inequality , we have Eε ‖∆∗11‖ 2 2 = ∫ x∈Sd−1 Ker ( x , X ) Gk ( S + I ) GkKer ( X , x ) dx ≤ η2m2k2 ( ∫ x∈Sd−1 [ Ker ( x , X ) y∗ ] 2 dx+ ‖Ker ( · , X ) ‖22 ) = O ( η2m2k2ω2n2L2 ) . Recall the decompositions ( 37 ) and ( 41 ) , then we have : Eε ∥∥∥f̂W ( k ) − f∗∥∥∥2 2 = Eε ‖∆∗11 + Θ−Ker ( · , X ) H∞L y∗‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 − Eε ‖∆ ∗ 11 + Θ‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 − 2Eε ‖∆ ∗ 11‖ 2 2 − 2Eε ‖Θ‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 −O ( η2m2k2ω2n2L2 ) −O ( 1 n ) −O ( n2ω3 λ20δ 2 ) − Õ ( 1 m1/3 poly ( ω , n , L , 1 λ0 , 1 δ ) ) . ( 59 ) For some constant C ′ 1 > 0 , let k ≤ C ′ 1 ( 1 ηmnωL ) such that the second term in the bound ( 59 ) can be bounded by 18‖Ker ( · , X ) ( H∞L ) −1 y∗‖22 . For the fourth term in ( 59 ) , we can choose ω ≤ C ′ 2 ( λ0δ n ) 2/3 for some constant C ′ 2 > 0 such that the term can be bounded by 1 8 ∥∥∥Ker ( · , X ) ( H∞L ) −1y∗∥∥∥2 2 . Sim- ilarly , the width m can be chosen large enough such that the fifth term in ( 59 ) is upper-bounded by 1 8‖Ker ( · , X ) ( H∞L ) −1 y∗‖22 . Using the above choices of k , ω , and m , we can further bound ( 59 ) : Eε ∥∥fW ( k ) − f∗∥∥22 ≥ 14 ∥∥∥Ker ( · , X ) ( H∞L ) −1y∗∥∥∥22 −O ( 1 n ) ≥ C ′ 3 ∥∥f ? ρ∥∥22 −O ( 1n ) . ( 60 ) In the second inequality , we used ( 47 ) with triangle inequality . In the third inequality , we can take n large enough such that the term O ( 1 n ) is upper-bounded by C ′ 3 2 ∥∥f ? ρ∥∥22 . Lastly , by using the assumption that f ? ρ is a square-integrable function , we finish the proof . G PROOF OF THEOREM 3.11-TRAINING ERROR For the convenience of notation , we denote uD , i ( k ) = fW ( k ) D ( xi ) and let uD ( k ) = [ u1 , D ( k ) , . . . , un , D ( k ) ] > . In order to analyze the training error of ` 2-regularized estimator , ‖uD ( k ) − y‖22 , we decompose the term as follows : ‖uD ( k + 1 ) − y‖22 = ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖ 2 2 + ‖ ( 1− η2µL ) uD ( k ) − y‖ 2 2 − 2 ( y − ( 1− η2µL ) uD ( k ) ) > ( uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ) ( 61 ) Equipped with this decomposition , the proof consists of the following steps : 1 . We decompose the decayed prediction difference uD ( k+ 1 ) − ( 1− η2µL ) uD ( k ) into two terms . We note that the first term is related with a gram matrix HD ( k ) and denote a second term as I ( k ) D . 2 . The term I ( k ) D can be further decomposed into three terms , where we denote them as I ( k ) 2 , D , I ( k ) 3 , D and I ( k ) 5 , D . The crux for controlling the ` 2-norm of the above three terms is to utilize the results from the Appendix A.4 . The applications of Lemmas in the Appendix A.4 is possible , since we can inductively guarantee that ‖W ( k ) D , ` − W ( 0 ) D , ` ‖2 is sufficiently small enough for large enough m. 3 . Given the decomposition ( 61 ) , we further decompose it into four terms as follows : ( 61 ) = ‖ ( 1− η2µL ) uD ( k ) − y‖22︸ ︷︷ ︸ : =T1 + ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖22︸ ︷︷ ︸ : =T2 + 2mη1 ( y − ( 1− η2µL ) uD ( k ) ) > HD ( k ) ( uD ( k ) − y ) ︸ ︷︷ ︸ : =T3 −2 ( y − ( 1− η2µL ) uD ( k ) ) > I ( k ) D︸ ︷︷ ︸ : =T4 . ( 62 ) In this step , we obtain the upper-bound of ‖Ti‖2 for i = 1 , 2 , 3 , 4 obtained in Step 4 . 4 . We combine the upper-bounds of ‖Ti‖2 for i = 1 , 2 , 3 , 4 in step 3 and obtain the bound on ‖uD ( k + 1 ) − y‖22 with respect to ‖uD ( k ) − y‖22 and ‖y‖2 . 5 . Lastly , we inductively show that the weights generated from regularized gradient descent stay within a perturbation region B ( W ( 0 ) , τ ) , irrespective with the number of iterations of algorithm . We start the proof by analyzing the term u ( k + 1 ) − ( 1− η2µL ) u ( k ) . Step 1 . Dynamics of uD ( k + 1 ) − ( 1−η2µL ) ·uD ( k ) . Recall ( Σ ( k ) D , ` , i ) jj = 1 ( 〈w ( k ) D , ` , j , x ( k ) D , ` −1 , i〉 ≥ 0 ) and we introduce a diagonal matrix Σ̃ ( k ) D , ` , i , whose jth entry is defined as follows : ( Σ̃ ( k ) D , ` , i ) jj = ( Σ ( k+1 ) D , ` , i −Σ ( k ) D , ` , i ) jj · 〈w ( k+1 ) D , ` , j , x ( k+1 ) D , ` −1 , i〉 〈w ( k+1 ) D , ` , j , x ( k+1 ) D , ` −1 , i〉 − 〈w ( k ) D , ` , j , x ( k ) D , ` −1 , i〉 . With this notation , the difference x ( k+1 ) D , L , i − x ( k ) D , L , i can be rewritten via the recursive applications of Σ̃ ( k ) D , ` , i : Then , we introduce following notations : D ( k ) D , ` , i = ( L∏ r= ` +1 Σ ( k ) D , r , iW ( k ) D , r ) Σ ( k ) D , ` , i , D̃ ( k ) D , ` , i = ( L∏ r= ` +1 ( Σ ( k ) D , r , i + Σ̃ ( k ) D , r , i ) W ( k+1 ) D , r ) ( Σ ( k ) D , ` , i + Σ̃ ( k ) D , ` , i ) . Now , we can write uD , i ( k + 1 ) − uD , i ( k ) by noting that uD , i ( k ) = √ m · vTx ( k ) D , L , i : uD , i ( k + 1 ) − uD , i ( k ) = √ m · vT ( x ( k+1 ) D , L , i − x ( k ) D , L , i ) = √ m · vT L∑ ` =1 D̃ ( k ) D , ` , i ( W ( k+1 ) D , ` −W ( k ) D , ` ) x ( k ) D , ` −1 , i = √ m · vT L∑ ` =1 D̃ ( k ) D , ` , i ( − η1∇W ` [ LS ( W ( k ) D ) ] − η2µW ( k ) D , ` + η2µW ( 0 ) D , ` ) x ( k ) D , ` −1 , i = −η1 √ m · vT L∑ ` =1 D ( k ) D , ` , i∇W ` [ LS ( W ( k ) D ) ] x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 1 , D , i −η1 √ m · vT L∑ ` =1 ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ∇W ` [ LS ( W ( k ) D ) ] x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 2 , D , i −η2µ √ m · vT L∑ ` =1 ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ( W ( k ) D , ` −W ( 0 ) D , ` ) x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 3 , D , i −η2µ √ m · vT L∑ ` =1 D ( k ) D , ` , iW ( k ) D , ` x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 4 , D , i +η2µ √ m · vT L∑ ` =1 D ( k ) D , ` , iW ( 0 ) D , ` x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 5 , D , i ( 63 ) where in the second equality , we used the recursive relation ( 24 ) , and in the third equality , modified GD update rule ( 6 ) is applied . Furthermore , I ( k ) 1 , D , i can be rewritten as follows : I ( k ) 1 , D , i = −η1 √ m · vT L∑ ` =1 D ( k ) D , ` , i n∑ j=1 ( uD , j ( k ) − yj ) ∇W ` [ f W ( k ) D ( xj ) ] x ( k ) D , ` −1 , i = −η1 · n∑ j=1 ( uD , j ( k ) − yj ) · ( √ m L∑ ` =1 vTD ( k ) D , ` , i∇W ` [ f W ( k ) D ( xj ) ] x ( k ) D , ` −1 , i ) = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) · 1 m L∑ ` =1 〈 ∇W ` [ f W ( k ) D ( xi ) ] , ∇W ` [ f W ( k ) D ( xj ) ] 〉 Tr = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) ·HD , i , j ( k ) . ( 64 ) With I ( k ) 4 , i = ( −η2µL ) · uD , i ( k ) and ( 64 ) , we can rewrite ( 63 ) as follows : uD , i ( k + 1 ) − ( 1− η2µL ) uD , i ( k ) = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) ·HD , i , j ( k ) + I ( k ) 2 , D , i + I ( k ) 3 , D , i + I ( k ) 5 , D , i . ( 65 ) Step 2 . Control of the size ∥∥∥I ( k ) D ∥∥∥ 2 . Let I ( k ) D = [ I ( k ) 2 , D,1 + I ( k ) 3 , D,1 + I ( k ) 5 , D,1 , . . . , I ( k ) 2 , D , n + I ( k ) 3 , D , n + I ( k ) 5 , D,1 ] > . Now , we control the bound on the ∥∥∥I ( k ) D ∥∥∥2 2 . Recall that in Eq . ( 27 ) , we have ∥∥∥I ( k ) 2 , D∥∥∥ 2 ≤ O ( η1nL 3τ1/3ωm √ log ( m ) ) ‖uD ( k ) − y‖2 . ( 66 ) Similarly , ∥∥∥I ( k ) 3 , D∥∥∥ 2 can be bounded : ∥∥∥I ( k ) 3 , D∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 3 , D , i∣∣∣ ≤ η2µ√m · n∑ i=1 [ L∑ ` =1 ∥∥∥∥v > ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( L2τ1/3 √ ω log ( m ) ) · ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2︸ ︷︷ ︸ ≤τ · ∥∥∥x ( k ) D , ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) ] ≤ O ( η2µnL 3τ4/3 √ ωm log ( m ) ) . ( 67 ) Lastly ∥∥∥I ( k ) 5 , D∥∥∥ 2 can be bounded : ∥∥∥I ( k ) 5 , D∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 5 , D , i∣∣∣ ≤ n∑ i=1 ∣∣∣∣∣η2µ√m · vT L∑ ` =1 D ( k ) D , ` , iW ( k ) D , ` x ( k ) D , ` −1 , i ∣∣∣∣∣+ n∑ i=1 ∣∣∣∣∣η2µ√m · vT L∑ ` =1 D ( k ) D , ` , i ( W ( k ) D , ` −W ( 0 ) D , ` ) x ( k ) D , ` −1 , i ∣∣∣∣∣ ≤ η2µL · n∑ i=1 |ui , D ( k ) |+ η2µ √ m · n∑ i=1 [ L∑ ` =1 ‖v‖2︸ ︷︷ ︸ ≤O ( √ ω ) · ∥∥∥D ( k ) D , ` , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ L ) · ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2︸ ︷︷ ︸ ≤τ · ∥∥∥x ( k ) D , ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) ] ≤ O ( η2µnL √ ω log ( L/δ ) ) +O ( η2µnL 3/2τ √ mω ) , ( 68 ) where in the last inequality , we employed the same logic used in ( 44 ) with the Lemma 4.2 to obtain the upper-bound on the |ui , D ( k ) | . We set the orders of the parameters µ , η1 , η2 , τ , and ω as follows : µ = Θ ( n d−1 2d−1 ) , η1 = Θ ( 1 m n− 3d−2 2d−1 ) , η2 = Θ ( 1 L n− 3d−2 2d−1 ) , τ = O ( L √ ω√ mδ n d 2d−1 ) , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) . ( 69 ) Plugging the choices of parameters ( 69 ) with sufficiently large m in ( 66 ) , ( 67 ) and ( 68 ) yields ∥∥∥I ( k ) D ∥∥∥ 2 ≤ O ( L37/12n− 9d−8 12d−6 √ log ( m ) m1/6δ1/3 ) · ‖uD ( k ) − y‖2 +OP ( 1 n2 ) . ( 70 ) Step 3 . Upper-bound of ‖Ti‖2 on i = 1 , 2 , 3 , 4 . First , we work on getting the upper-bound on λmax ( HD ( k ) ) . By the Gershgorin ’ s circle theorem [ Varga , 2004 ] , we know the maximum eigenvalue of symmetric positive semi-definite matrix is upper-bounded by the maximum absolute column sum of the matrix . Using this fact , we can bound the λmax ( HD ( k ) ) as : λmax ( HD ( k ) ) ≤ max i=1 , ... , n n∑ j=1 |HD , i , j ( k ) | ≤ max i=1 , ... , n n∑ j=1 ∣∣∣∣∣ 1m L∑ ` =1 〈 ∇W ` [ f W ( k ) D ( xi ) ] , ∇W ` [ f W ( k ) D ( xj ) ] 〉 Tr ∣∣∣∣∣ ≤ max i=1 , ... , n n∑ j=1 1 m L∑ ` =1 ∥∥∥∇W ` [ fW ( k ) D ( xi ) ] ∥∥∥F︸ ︷︷ ︸ ≤O ( √ mω ) ∥∥∥∇W ` [ fW ( k ) D ( xj ) ] ∥∥∥F︸ ︷︷ ︸ ≤O ( √ mω ) ≤ O ( nLω ) . ( 71 ) Recall the decomposition ( 62 ) . Our goal is to obtain the upper-bound on Ti for i = 1 , 2 , 3 , 4 . Control on T1 . By using the inequality 2η2µL ( 1−η2µL ) y > ( y−uD ( k ) ) ≤ η2µL ‖y‖22+η2µL ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 , we have ‖y − ( 1− η2µL ) uD ( k ) ‖22 = ∥∥ ( 1− η2µL ) ( y − uD ( k ) ) + η2µLy∥∥22 = ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 + η 2 2µ 2L2 ‖y‖22 + 2η2µL ( 1− η2µL ) y > ( y − uD ( k ) ) ≤ ( η2µL+ η22µ2L2 ) ‖y‖ 2 2 + ( 1 + η2µL ) ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 . ( 72 ) Control on T2 . Recall the equality ( 65 ) . Then , through applications of the Young ’ s inequality ‖a+ b‖22 ≤ 2 ‖a‖ 2 2 + 2 ‖b‖ 2 2 for a , b ∈ Rn , we have ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖22 = ∥∥∥−mη1 ·HD ( k ) ( uD ( k ) − y ) + I ( k ) D ∥∥∥2 2 ≤ 2m2η21λmax ( HD ( k ) ) 2 ‖y − uD ( k ) ‖22 + 2 ∥∥∥I ( k ) D ∥∥∥2 2 . ( 73 ) Similarly with T1 and T2 , we can control T3 and T4 as follows : Control on T3 . Recall HD ( k ) is a Gram matrix by definition . Then , by using the fact λmin ( HD ( k ) ) ≥ 0 and Cauchy-Schwarz inequality , we have 2mη1 ( y − ( 1− η2µL ) uD ( k ) ) > HD ( k ) ( uD ( k ) − y ) = −2mη1 ( 1− η2µL ) ( y − uD ( k ) ) > HD ( k ) ( y − uD ( k ) ) + ( 2mη1η2µL ) · y > HD ( k ) ( uD ( k ) − y ) ≤ ( 2mη1η2µL ) · λmax ( HD ( k ) ) ‖y − uD ( k ) ‖22 + ( 2mη1η2µL ) · ( λmax ( HD ( k ) ) ‖y‖2 ‖y − uD ( k ) ‖2 ) − 2mη1λmin ( HD ( k ) ) ‖y − uD ( k ) ‖22 = ( 4mη1η2µL ) · λmax ( HD ( k ) ) ‖y − uD ( k ) ‖22 + ( 4mη1η2µL ) · λmax ( HD ( k ) ) ‖y‖22 . ( 74 ) Control on T4 . By a simple Cauchy-Schwarz and Young ’ s inequality , we have − 2 ( y − ( 1− η2µL ) uD ( k ) ) > I ( k ) D = −2 ( 1− η2µL ) ( y − uD ( k ) ) > I ( k ) D + 2η2µL · y > ID ( k ) ≤ 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ‖y‖22 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ( 75 ) Step 4 . Upper-bound of the decomposition on training error ( 62 ) . Before getting the upper bound of the decomposition ( 62 ) , we first work on obtaining the bound of ( 76 ) . Set κ = O ( 1 n2 ) and notice η2µL = O ( 1 n ) by ( 69 ) , then we have 2 ∥∥∥I ( k ) D ∥∥∥2 2 + 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ( 76 ) = ( 2 + η2µL ) ∥∥∥I ( k ) D ∥∥∥2 2 + 2κ ( 1− η2µL ) ‖y − uD ( k ) ‖2 · 1 κ ∥∥∥I ( k ) D ∥∥∥ 2 ≤ ( 2 + η2µL+ 1 κ2 ) ∥∥∥I ( k ) D ∥∥∥2 2 + κ2 ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 = 1 κ2 · ∥∥∥I ( k ) D ∥∥∥2 2 + κ2 ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 ≤ { 1 κ2 · O ( L37/6n− 9d−8 6d−3 log ( m ) m1/3δ2/3 ) + κ2 ( 1− η2µL ) 2 } · ‖y − uD ( k ) ‖22 + 1κ2 · OP ( 1 n4 ) ≤ ( η2µL ) 4 ( 1− η2µL ) 2 · ‖y − uD ( k ) ‖22 + η2µL · ‖y‖22 , ( 77 ) where in the second inequality , the Eq . ( 70 ) is used with ( a + b ) 2 ≤ 2a2 + 2b2 for a , b ∈ R , and in the last inequality , we used ‖y‖22 = O ( n ) and the sufficiently large m to control the order of the coefficient terms of ‖y − uD ( k ) ‖22 . Specifically , we choose m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Now , by combining the inequalities ( 72 ) , ( 73 ) , ( 74 ) , ( 75 ) , ( 71 ) and ( 77 ) , we obtain the upper-bound on the decomposition ( 62 ) ; ‖uD ( k + 1 ) − y‖22 ≤ ( 2η2µL+ η 2 2µ 2L2 + 4mη1η2µL · λmax ( HD ( k ) ) ) · ‖y‖22 + ( ( 1 + η2µL ) ( 1− η2µL ) 2 + 2m2η21λmax ( HD ( k ) ) 2 + 4mη1η2µL · λmax ( HD ( k ) ) ) · ‖y − uD ( k ) ‖22 + ( 2 ∥∥∥I ( k ) D ∥∥∥2 2 + 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ) ≤ { 3η2µL+ η 2 2µ 2L2 +O ( ωmnη1η2µL 2 ) } · ‖y‖22 + { ( 1 + η2µL+ η 4 2µ 4L4 ) ( 1− η2µL ) 2 +O ( ω2m2n2η21L 2 ) +O ( ωmnη1η2µL 2 ) } · ‖y − uD ( k ) ‖22 : = A · ‖y‖22 + ( 1− B ) · ‖y − uD ( k ) ‖ 2 2 . ( 78 ) With the order choices of µ , η1 and η2 as in ( 69 ) , it is easy to see the leading terms of both A and B are same as η2µL = o ( 1n ) . Then , by recursively applying the inequality ( 78 ) , we can get the upper-bound on the training error . ‖y − uD ( k + 1 ) ‖22 ≤ A · ‖y‖ 2 2 + ( 1− B ) · ‖y − uD ( k ) ‖ 2 2 ≤ A‖y‖22 · ( k∑ j=0 ( 1− B ) j ) + ( 1− B ) k+1 · ‖y − uD ( 0 ) ‖22 ≤ A B · ‖y‖22 + ( 1− B ) k+1 · ‖y − uD ( 0 ) ‖22 ≤ O ( n ) + ( 1− η2µL ) k+1 · ‖y − uD ( 0 ) ‖22 . ( 79 ) In the last inequality , we used AB = o ( 1 ) , B ≥ η2µL and ‖y‖ 2 2 = O ( n ) . Step 5 . The order of the radius of perturbation region . It remains us to prove the radius of perturbation region τ has the order OP ( L √ ω√ m n d 2d−1 ) . First , recall that the ` 2-regularized GD update rule is as : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] + η2µW ( 0 ) D , ` , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . ( 80 ) Similarly with the proof in the Theorem 3.5 , we employ the induction process for the proof . The induction hypothesis is∥∥∥W ( s ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 ≤ O ( η1n √ mω√ δη2µ ) , ∀s ∈ [ k + 1 ] . ( 81 ) It is easy to see it holds for s = 0 , and suppose it holds for s = 0 , 1 , . . . , k , we consider k+1 . Using the update rule ( 80 ) , we have∥∥∥W ( k+1 ) D , ` −W ( k ) D , ` ∥∥∥ 2 ≤ η2µ ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 + η1 ∥∥∥∇W ` [ LS ( W ( k ) D ) ] ∥∥∥ 2 = η2µ ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 + η1 ∥∥∥∥∥ n∑ i=1 ( yi − uD , i ( k ) ) ∇W ` [ fWD ( k ) ( xi ) ] ∥∥∥∥∥ 2 ≤ O ( η1n √ mω√ δ ) +O ( η1 √ nmω ) · ‖y − uD ( k ) ‖2 ≤ O ( η1n √ mω√ δ ) +O ( η1 √ nmω ) · { O ( √ n ) + ( 1− η2µL ) k 2O ( √ n δ ) } ≤ O ( η1n √ mω√ δη2µ ) . In the first inequality , we use the induction hypothesis for s = k , and Lemma 4.4 . In the second inequality , since the induction hypothesis holds for s = 0 , 1 , . . . , k , we employ ‖y − uD ( k ) ‖2 ≤ O ( √ n ) + ( 1− η2µL ) k 2 ‖y − uD ( 0 ) ‖2 with the Lemma 4.9 . In the last inequality , we use η2µ < 1 . By triangle inequality , the induction holds for s = k + 1 . Plugging the proper choices of η1 , η2 and µ as suggested in ( 69 ) to O ( η1n √ mω√ δη2µ ) yields ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ OP ( L √ ω√ m n d 2d−1 ) . H PROOF OF THEOREM 3.11-KERNEL RIDGE REGRESSOR APPROXIMATION We present a following proof sketch on the approximation of regularized DNN estimator to kernel ridge regressor . 1 . The key idea for proving the second result in Theorem 3.8 is to write the distance between ui , D ( k ) ( where D is to denote the prediction is obtained from regularized GD rule ) and kernel regressor B : = H∞L ( Cµ · I + H∞L ) −1 y in terms of NTK matrix H∞L , which is as follows : uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) . Above equality describes how the regularized estimator evolves to fit the kernel regressor as iteration of algorithm goes by . 2 . We can bound the ` 2-norm of residual term eD ( k ) as O ( 1/n ) , and show that the ` 2 norm of the first term on the RHS of equation ( 4.3 ) decays at the rateO ( √ n ( 1− η2µL ) k ) . Here the √ n comes from the bound ‖B‖2 ≤ O ( √ n ) , since we know ‖u ( 0 ) ‖2 hasO ( √ nω ) with small ω ≤ 1 . This yields the claim . Recall the equality ( 65 ) . Then , we have uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) = −mη1 ·HD ( k ) ( uD ( k ) − y ) + I ( k ) 2 , D + I ( k ) 3 , D + I ( k ) 5 , D = −mη1 ·H∞L ( uD ( k ) − y ) −mη1 · ( HD ( k ) −H∞L ) ( uD ( k ) − y ) + I ( k ) 2 , D + I ( k ) 3 , D + I ( k ) 5 , D = −mη1 ·H∞L ( uD ( k ) − y ) + ξD ( k ) . ( 82 ) With τ = O ( L √ ω√ mδ n d 2d−1 ) , similarly with Lemma 4.10 and a direct employment of the result from Lemma 4.11 , we can control the distance from HD ( k ) to H∞L under operator norm as follows : ‖HD ( k ) −H∞L ‖2 ≤ ‖HD ( k ) −H ( 0 ) ‖2 + ‖H ( 0 ) −H ∞ L ‖2 ≤ O ( ω7/6L10/3n 7d−3 6d−3 6 √ log3 ( m ) mδ2 ) +O ( ωL5/2n 4 √ log ( nL/δ ) m ) ≤ O ( L19/12n− 21d−8 12d−6 6 √ log3 ( m ) mδ2 ) +O ( Ln− 18d−6 12d−6 4 √ log ( nL/δ ) m ) ≤ O ( L19/12n− 21d−8 12d−6 6 √ log3 ( m ) mδ2 ) , ( 83 ) where in the third inequality , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) is plugged-in . The last inequality holds with d ≥ 2 with large enough n and the condition on width m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Then , the ` 2 norm of ξD ( k ) can be bounded as : ‖ξD ( k ) ‖2 ≤ mη1 · ‖H ∞ L −HD ( k ) ‖2 ‖uD ( k ) − y‖2 + ∥∥∥I ( k ) D ∥∥∥ 2 ≤ O ( L19/12n− 12d−5 6d−3 √ log ( m ) m1/6δ1/3 ) · ‖uD ( k ) − y‖2︸ ︷︷ ︸ ≤O ( √ n/δ ) +OP ( 1 n2 ) ≤ O ( L19/12n− 18d−7 12d−6 √ log ( m ) m1/6δ5/6 ) +OP ( 1 n2 ) = OP ( 1 n2 ) , ( 84 ) where in the second inequality , we used ( 83 ) with η1 = O ( 1 mn − 3d−22d−1 ) to control the first term and employed Eq . ( 70 ) to control the second term . In the last equality , we usedm ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Now , by setting B : = ( η2µL η1m I + H∞L ) −1 H∞L y , we can easily convert the equality ( 82 ) as follows : for k ≥ 1 , uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) ( uD ( k − 1 ) −B ) + ξD ( k − 1 ) . ( 85 ) The recursive applications of the equality ( 85 ) yields uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + k∑ j=0 ( ( 1− η2µL ) · I −mη1H∞L ) j ξD ( k − j − 1 ) = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) . ( 86 ) Now , we bound the ` 2 norm of eD ( k ) in ( 86 ) : ‖eD ( k ) ‖2 = ∥∥∥∥∥∥ k∑ j=0 ( ( 1− η2µL ) · I −mη1H∞L ) j ξD ( k − j − 1 ) ∥∥∥∥∥∥ 2 ≤ k∑ j=0 ∥∥ ( 1− η2µL ) · I −mη1H∞L ∥∥j2 ‖ξD ( k − j − 1 ) ‖2 ≤ k∑ j=0 ( 1− η2µL ) j ‖ξD ( k − j − 1 ) ‖2 = O ( 1n ) , ( 87 ) in the last inequality , we used η2µL = O ( 1 n ) and Eq . ( 84 ) . Now , we control the ` 2-norm of the first term in ( 86 ) as : ∥∥∥∥∥ ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) ∥∥∥∥∥ 2 ≤ ( 1− η2µL ) k ‖uD ( 0 ) −B‖2 ≤ O ( √ n ( 1− η2µL ) k ) , ( 88 ) where in the second inequality , we used ‖uD ( 0 ) ‖2 ≤ O ( √ nω/δ ) and the fact that ‖B‖2 ≤ ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 H∞L ∥∥∥∥∥ 2 · ‖y‖2 ≤ O ( √ n ) . By combining ( 87 ) and ( 88 ) and using a fact ( 1− η2µL ) k ≤ exp ( −η2µLk ) , we conclude that after k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) , the error ‖uD ( k ) −B‖2 decays at the rate O ( 1 n ) . I PROOF OF THEOREM 3.12 We begin the proof by decomposing the error f̂ W ( k ) D ( x ) − f∗ ( x ) for any fixed x ∈ Unif ( Sd−1 ) into two terms as follows : f̂ W ( k ) D ( x ) − f∗ ( x ) = ( f̂ W ( k ) D ( x ) − g∗µ ( x ) ) ︸ ︷︷ ︸ ∆D,1 + ( g∗µ ( x ) − f∗ ( x ) ) ︸ ︷︷ ︸ ∆D,2 . ( 89 ) Here , we devise a solution of kernel ridge regression g∗µ ( x ) in the decomposition ( 89 ) : g∗µ ( x ) : = Ker ( x , X ) ( Cµ · I + H∞L ) −1 y , for some constant C > 0 . Specifically , in the proof to follow , we choose η1 and η2 such that C = η2Lη1m for the theoretical convenience . Our goal is to show that all the terms ‖∆D,1‖ 2 2 , and ‖∆D,2‖22 have the order either equal to or smaller than O ( n− d 2d−1 ) with the proper choices on m , µ , η1 and η2 . Since the high-level proof idea is similar with that of Theorem 3.8 , we omit the step-by-step proof sketch of Theorem 3.12 . The most notable difference between the proof strategies of the two theorems is that the regularized DNN approximate the kernel ridge regressor of noisy data , whereas in Theorem 3.8 , unregularized DNN approximate the interpolant based on noiseless data . Step 1 . Control on ∆D,2 . First , note that there is a recent finding that the reproducing kernel Hilbert spaces induced from NTKs with any number of layers ( i.e. , L ≥ 1 ) have the same set of functions , if kernels are defined on Sd−1 . See Chen & Xu [ 2020 ] . Along with this result , under the choice of model parameters as suggested in ( 69 ) , we can apply exactly the same proof used in Theorem.3.2 in Hu et al . [ 2021 ] for proving a following : ‖∆D,2‖22 : = ∥∥g∗µ − f∗∥∥22 = OP ( n− d2d−1 ) , ∥∥g∗µ∥∥2H = OP ( 1 ) . ( 90 ) Step 2 . Control on ∆D,1 . For n data points ( x1 , . . . , xn ) and for the kth updated parameter W ( k ) D , denote : ∇W ` [ f W ( k ) D ( X ) ] = [ vec ( ∇W ` [ f W ( k ) D ( x1 ) ] ) , · · · , vec ( ∇W ` [ f W ( k ) D ( xn ) ] ) ] . Note that when ` = 1 , ∇W ` [ f W ( k ) D ( X ) ] ∈ Rmd×n and when ` = 2 , . . . , L , ∇W ` [ f W ( k ) D ( X ) ] ∈ Rm2×n . With this notation , we can write the vectorized version of the update rule ( 80 ) as : vec ( W ( k ) D , ` ) = vec ( W ( 0 ) D , ` ) − η1 k−1∑ j=0 ( 1− η2µ ) j∇W ` [ fWD ( k−j−1 ) ( X ) ] ( uD ( k − j − 1 ) − y ) , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . Using the equality , we can get the decomposition : vec ( W ( k ) D , ` ) = vec ( W ( 0 ) D , ` ) ︸ ︷︷ ︸ : =E1 −η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( uD ( k − j − 1 ) − y ) ︸ ︷︷ ︸ : =E2 −η1 k−1∑ j=0 ( 1− η2µ ) j [ ∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ] ( uD ( k − j − 1 ) − y ) .︸ ︷︷ ︸ : =E3 ( 91 ) Let zD , k ( x ) : = vec ( ∇W ` [ f W ( k ) D ( x ) ] ) , and note that f W ( k ) D ( x ) = 〈zD , k ( x ) , vec ( W ( k ) D , ` ) 〉 . Then , by the definition of ∆D,1 and the decomposition ( 91 ) , we have ∆D,1 = 1 L L∑ ` =1 〈zD , k ( x ) , E1 + E2 + E3〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y = 1 L L∑ ` =1 〈zD , k ( x ) , E1〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E3〉 + 1 L L∑ ` =1 〈zD , k ( x ) , E2〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y︸ ︷︷ ︸ : =C ( 92 ) First , we focus on controlling the ` 2 bound on the first two terms in ( 92 ) . Observe that the first term can be bounded as : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E1〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E1〉|2 . ( 93 ) Recall that ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) by Lemma 4.4 . Then , the random variable zD , k ( x ) > vec ( W ( 0 ) D , ` ) | zD , k ( x ) is simply a N ( 0 , O ( ω ) ) for 1 ≤ ` ≤ L. A straightforward application of Chernoff bound for normal random variable and taking union bound over the layer 1 ≤ ` ≤ L yield that : with probability at least 1− δ , 1 L L∑ ` =1 ∣∣∣zD , k ( x ) > vec ( W ( 0 ) D , ` ) ∣∣∣2 ≤ O ( ω log ( L δ ) ) . ( 94 ) The ` 2 norm of the second term in ( 92 ) can be similarly bounded as ( 93 ) in addition with the CauchySchwarz inequality : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E3〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E3〉|2 ≤ 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 . ( 95 ) The ‖E3‖2 is bounded as : ‖E3‖2 = ∥∥∥∥∥∥η1 k−1∑ j=0 ( 1− η2µ ) j [ ∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ] ( uD ( k − j − 1 ) − y ) ∥∥∥∥∥∥ 2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · ∥∥∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ∥∥2 ‖uD ( k − j − 1 ) − y‖2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · ∥∥∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ∥∥F ‖uD ( k − j − 1 ) − y‖2 = η1 k−1∑ j=0 ( 1− η2µ ) j · √√√√ n∑ i=1 ∥∥∇W ` [ fWD ( k−j−1 ) ( xi ) ] −∇W ` [ fWD ( 0 ) ( xi ) ] ∥∥2F ‖uD ( k − j − 1 ) − y‖2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · √√√√2 n∑ i=1 ∥∥∇W ` [ fWD ( k−j−1 ) ( xi ) ] −∇W ` [ fWD ( 0 ) ( xi ) ] ∥∥22 ‖uD ( k − j − 1 ) − y‖2 ≤ η1 η2µ · O ( τ1/3L2 √ ωmn log ( m ) ) · O ( √ n ) ≤ O ( L10/3ω1/6 m2/3δ1/3 n 4d 6d−3 √ log ( m ) ) . ( 96 ) In the first , second and third inequalities , we used a simple fact that for the matrix A ∈ Rd1×d2 with rank r , then ‖A‖2 ≤ ‖A‖F ≤ √ r‖A‖2 . Recall that the rank of the matrix∇W ` [ fWD ( k−j−1 ) ( x ) ] − ∇W ` [ fWD ( 0 ) ( x ) ] is at most 2 . In the second to the last inequality , we use the result of Lemma 4.6 and the ‖uD ( i ) − y‖2 ≤ O ( √ n ) for any i ≥ 1 . In the last inequality , we plug the correct orders as set in ( 69 ) to τ , η1 , η2 and µ . Back to the inequality ( 95 ) , using the ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) and ( 96 ) , we can get 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 ≤ OP ( L20/3ω4/3 m1/3 n 8d 6d−3 log ( m ) ) . ( 97 ) Before controlling the ` 2 norm of C in ( 92 ) , recall that we set B : = ( η2µL η1m I + H∞L ) −1 H∞L y and the dynamics of uD ( k ) −B can be expressed in terms of H∞L as follows : For any k ≥ 1 , uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) , ( 98 ) with ‖eD ( k ) ‖2 ≤ O ( 1 n ) . Using ( 98 ) , we can further decompose the term E2 in ( 91 ) as : E2 : = −η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( uD ( k − j − 1 ) − y ) = η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 B − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 uD ( 0 ) − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) jeD ( k − j − 1 ) − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( B− y ) = E2,1 + E2,2 + E2,3 + E2,4 . ( 99 ) Then , we can re-write the error term C in ( 92 ) as : C = 1 L L∑ ` =1 〈zD , k ( x ) , E2,1〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E2,2〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E2,3〉 + { 1 L L∑ ` =1 〈zD , k ( x ) , E2,4〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y } ︸ ︷︷ ︸ : =D . ( 100 ) Our goal is to control the ` 2 norm of each summand in the equality ( 100 ) . For the first three terms in ( 100 ) , a simple Cauchy-Schwarz inequality can be applied : for i = 1 , 2 , 3 : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E2 , i〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E2 , i〉|2 ≤ 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 · ‖E2 , i‖ 2 2 . We work on obtaining the bound of ∑L ` =1 ‖E2,1‖ 2 2 . Let Tk be defined as Tk : = k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 . Then , we have L∑ ` =1 ‖E2,1‖22 = η 2 1 L∑ ` =1 ( B > T > k ∇W ` [ fWD ( 0 ) ( X ) ] > ∇W ` [ fWD ( 0 ) ( X ) ] TkB ) = mη21B > T > k H ( 0 ) TkB = mη21B > T > k ( H ( 0 ) −H∞L ) TkB +mη21B > T > k H∞L TkB ≤ mη21 ‖H ( 0 ) −H∞L ‖2 ·B > T 2k B +mη21B > T > k H∞L TkB . ( 101 ) To obtain the upper-bound on ( 101 ) , we need to control the terms T > k H∞L Tk and B > T 2k B . Let us denote H∞L = ∑n i=1 λiviv > i be the eigen-decomposition of H ∞ L . Using 1− η2µL ≤ 1− η2µ , note that Tk = k−1∑ j=0 ( 1− η2µ ) j ( 1− η2µL ) k−j−1 ( I − mη1 1− η2µL H∞L ) k−j−1 ( 1− η2µ ) k−1 k−1∑ i=0 ( I − mη1 1− η2µ H∞L ) i = ( 1− η2µ ) k−1 n∑ j=0 ( 1− ( 1− mη11−η2µλj ) k mη1 1−η2µλj ) vjv > j ( 1− η2µ ) k mη1λ0 · I . ( 102 ) A similar logic can be applied to bound T > k H∞L Tk : T > k H∞L Tk ( 1− η2µ ) k−1 n∑ j=0 ( 1− ( 1− mη11−η2µλj ) k mη1 1−η2µλj ) 2 λjvjv > j ( 1− η2µ ) 2k m2η21 · ( H∞L ) −1 . ( 103 ) Recall the definition of the notation B : = H∞L ( η2µL η1m I + H∞L ) −1 y . Then , we can bound the term B > T > k H∞L TkB : B > T > k H∞L TkB ≤ ( 1− η2µ ) 2k m2η21 ·B > ( H∞L ) −1 B = ( 1− η2µ ) 2k m2η21 · y > ( η2µL η1m I + H∞L ) −1 H∞L ( η2µL η1m I + H∞L ) −1 y = O ( ( 1− η2µ ) 2k m2η21 ) , ( 104 ) where in the last equality , we used ∥∥g∗µ∥∥2H = OP ( 1 ) in ( 90 ) . Now we turn our attention to bound the term B > T 2k B , B > T 2k B ≤ ( 1− η2µ ) 2k m2η21λ 2 0 y > ( η2µL η1m I + H∞L ) −1 ( H∞L ) 2 ( η2µL η1m I + H∞L ) −1 y = O ( ( 1− η2µ ) 2k n m2η21λ 2 0 ) , ( 105 ) where we used ‖y‖22 = O ( n ) in the last inequality . Combining the bounds ( 104 ) , ( 105 ) and the result from Lemma 4.11 , we can further bound ( 101 ) and have : L∑ ` =1 ‖E2,1‖22 ≤ O ( ω ( 1− η2µ ) 2k mλ20 n2L5/2 4 √ log ( nL/δ ) m + ( 1− η2µ ) 2k m ) ≤ O ( ( 1− η2µ ) 2k m ) , ( 106 ) where in the second inequality , we used m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Similarly , we can bound∑L ` =1 ‖E2,2‖ 2 2 : L∑ ` =1 ‖E2,2‖22 = η 2 1 L∑ ` =1 ( uD ( 0 ) > T > k ∇W ` [ fWD ( 0 ) ( X ) ] > ∇W ` [ fWD ( 0 ) ( X ) ] TkuD ( 0 ) ) = mη21uD ( 0 ) > T > k H ( 0 ) TkuD ( 0 ) = mη21uD ( 0 ) > T > k ( H ( 0 ) −H∞L ) TkuD ( 0 ) +mη21uD ( 0 ) > T > k H∞L TkuD ( 0 ) ≤ mη21 ‖H ( 0 ) −H∞L ‖2 · uD ( 0 ) > T 2k uD ( 0 ) +mη21uD ( 0 ) > T > k H∞L TkuD ( 0 ) ≤ mη21 ( 1− η2µ ) 2k m2η21λ 2 0 O ( ωnL5/2 4 √ log ( nL/δ ) m ) ‖uD ( 0 ) ‖22 +mη21 ( 1− η2µ ) 2k m2η21 uD ( 0 ) > ( H∞L ) −1uD ( 0 ) ≤ O ( ( 1− η2µ ) 2k n2ω2L5/2 mλ20δ 2 4 √ log ( nL/δ ) m + nω ( 1− η2µ ) 2k mλ0δ2 ) = OP ( nω ( 1− η2µ ) 2k mλ0 ) . ( 107 ) Here , in the second inequality , we used the inequalities ( 102 ) and ( 103 ) and Lemma 4.11 . In the third inequality , we used the Lemma 4.8 , ‖u ( 0 ) ‖2 = O ( √nω δ ) with probability at least 1− δ . In the last equality , we used m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Next , we bound ∑L ` =1 ‖E2,3‖ 2 2 as : L∑ ` =1 ‖E2,3‖22 = mη 2 1 · ( k−1∑ j=0 ( 1− η2µ ) j ( ek−j−1 ) ) > HD ( 0 ) ( k−1∑ j=0 ( 1− η2µ ) j ( ek−j−1 ) ) ≤ mη 2 1 η22µ 2 · λmax ( HD ( k ) ) · ‖ek−j−1‖22 ≤ mη21 η22µ 2 · O ( ωnL ) · O ( 1 n2 ) = O ( L3 m ω · n− 4d−3 2d−1 ) . ( 108 ) Now , we focus on obtaining the ` 2 norm bound on D in ( 100 ) . Recall the definition of the notation B : = H∞L ( η2µL η1m I + H∞L ) −1 y . A simple calculation yields that B− y = H∞L ( η2µL η1m I + H∞L ) −1 y − y = −η2µL mη1 ( η2µL η1m I + H∞L ) −1 y . Then , we can re-write the expression of the D as : D : = ( η2µL mη1 ) · η1 1 L L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 k−1∑ j=0 ( 1− η2µ ) j ( η2µL η1m I + H∞L ) −1 y −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y = ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y − ( 1− η2µ ) k 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y = ( 1 m L∑ ` =1 〈 z0 , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y − ( 1− η2µ ) k 1 m L∑ ` =1 〈 z0 , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y + ( 1− ( 1− η2µ ) k ) ( 1 m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ) ( η2µL η1m I + H∞L ) −1 y , ( 109 ) where in the second equality , ∑k−1 j=0 ( 1− η2µ ) j = 1− ( 1−η2µ ) k η2µ is used . The ` 2 norm of first term in the ( 109 ) can be bounded as : ∥∥∥∥∥ ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ∥∥∥∥∥ ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 = √√√√ n∑ i=1 ( 1 m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( xi ) ] 〉 −Ker ( x , xi ) ) 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ O ( ω √ nL5/2 4 √ log ( nL/δ ) m ) · O ( η1m η2µL √ n ) = O ( ωη1mn η2µL L5/2 4 √ log ( nL/δ ) m ) = O ( ωL5/2n d 2d−1 4 √ log ( nL/δ ) m ) , ( 110 ) where , in the second inequality , we used Lemma 4.11 , and also we used∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ √ y > ( η2µL η1m I + H∞L ) −2 y ≤ √ η21m 2 η22µ 2L2 · ‖y‖22 = O ( η1m η2µL √ n ) . ( 111 ) The ` 2 norm of the second term in ( 109 ) can be easily bounded as : ∥∥∥∥∥ ( 1− η2µ ) k 1m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ∥∥∥∥∥ ( 1− η2µ ) k ( 1 m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 + ∥∥∥∥∥ ( 1− η2µ ) kKer ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− η2µ ) k ∥∥∥∥∥ 1m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 + ( 1− η2µ ) k ∥∥∥∥∥Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− η2µ ) k · O ( ω√nL3/2 4√ log ( nL/δ ) m ) · O ( η1mη2µL√n ) +O ( ( 1− η2µ ) k ) ≤ ( 1− η2µ ) k · O ( ωL3/2n d2d−1 4√ log ( nL/δ ) m ) +O ( ( 1− η2µ ) k ) . ( 112 ) Lastly , the ` 2 norm of the third term in ( 109 ) is bounded as : ∥∥∥∥∥ ( 1− ( 1− η2µ ) k ) ( 1 m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ∥∥∥∥∥ 1m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ( 1 m L∑ ` =1 ‖zD , k ( x ) − zD,0 ( x ) ‖F ∥∥∇W ` [ fWD ( 0 ) ( X ) ] ∥∥F ) · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ( L m O ( τ1/3L2 √ ωm log ( m ) ) · O ( √ ωmn ) ) · O ( η1m η2µL √ n ) ≤ ( 1− ( 1− η2µ ) k ) · O ( ω7/6L10/3n 4d6d−3 √log ( m ) m1/6δ1/3 ) ≤ O ( ω7/6L10/3n 4d 6d−3 √ log ( m ) m1/6δ1/3 ) , ( 113 ) where in the fourth inequality , τ = OP ( L √ ω√ m n d 2d−1 ) is plugged in . Combining the inequali- ties ( 110 ) , ( 112 ) and ( 113 ) , we get the bound on ‖D‖2 in ( 109 ) : ‖D‖22 ≤ O ( ω2L5n 2d 2d−1 √ log ( nL/δ ) m ) + ( 1− η2µ ) 2kO ( ω2L3n 2d2d−1√ log ( nL/δ ) m ) +O ( ( 1− η2µ ) 2k ) +O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) ≤ O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) +O ( ( 1− η2µ ) 2k ) . ( 114 ) Step 3 . Combining all pieces . Recall ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) . With this fact , combining the bounds ( 94 ) , ( 97 ) , ( 106 ) , ( 107 ) , ( 108 ) and ( 114 ) , we can bound the ‖∆D,1‖22 via the decomposition ( 92 ) as follows : ‖∆D,1‖22 ≤ 1 L L∑ ` =1 ∣∣∣zD , k ( x ) > vec ( W ( 0 ) D , ` ) ∣∣∣2 + 1L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2,1‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2,2‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2.3‖ 2 2 + ∥∥∥∥∥ 1L L∑ ` =1 〈zD , k ( x ) , E2,4〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 2 ≤ O ( ω log ( L δ ) ) +OP ( L20/3ω4/3 m1/3 n 8d 6d−3 log ( m ) ) +OP ( ω ( 1− η2µ ) 2k L ) +OP ( nω2 ( 1− η2µ ) 2k Lλ0 ) +OP ( L2 m ω · n− 4d−3 2d−1 ) +O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) +OP ( ( 1− η2µ ) 2k ) ≤ OP ( n− d 2d−1 ) . | In this paper, the authors analyze the convergence rate of both the unregularized and the L2 regularized gradient descent for a regression problem. Under a positivity assumption of NTK, this paper shows that without early stopping, the vanilla GD may fail. This can be solved by using L2 regularization and could achieve a better convergence rate. | SP:232f05c3b3a9b5ca1cea27b2aea1293a7da190c8 |
Generalization of Overparametrized Deep Neural Network Under Noisy Observations | 1 INTRODUCTION Over the past few years , Neural Tangent Kernel ( NTK ) [ Arora et al. , 2019b ; Jacot et al. , 2018 ; Lee et al. , 2018 ; Chizat & Bach , 2018 ] has been one of the most seminal discoveries in the theory of neural network . The underpinning idea of the NTK-type theory comes from the observation that in a wide-enough neural net , model parameters updated by gradient descent ( GD ) stay close to their initializations during the training , so that the dynamics of the networks can be approximated by the first-order Taylor expansion with respect to its parameters at initialization . The linearization of learning dynamics on neural networks has been helpful in showing the linear convergence of the training error on both overparametrized shallow [ Li & Liang , 2018 ; Du et al. , 2018 ] and deep neural networks [ Allen-Zhu et al. , 2018 ; Zou et al. , 2018 ; 2020 ] , as well as the characterizations of generalization error on both models [ Arora et al. , 2019a ; Cao & Gu , 2019 ] . These findings clearly lead to the equivalence between learning dynamics of neural networks and the kernel methods in an reproducing kernel Hilbert spaces ( RKHS ) associated with NTK . 1 Specifically , Arora et al . [ 2019a ] provided the O ( n−1/2 ) generalization bound of shallow neural network , where n denotes the training sample size . Recently , in the context of nonparametric regression , two papers Nitanda & Suzuki [ 2020 ] and Hu et al . [ 2021 ] showed that neural network can obtain the convergence rate faster than O ( n−1/2 ) by specifying the complexities of target function and hypothesis space . Specifically , Nitanda & Suzuki [ 2020 ] showed that the shallow neural network with smoothly approximated ReLU ( swish , see Ramachandran et al . [ 2017 ] ) activation trained via ` 2-regularized averaged stochastic gradient descent ( SGD ) can recover the target function from RKHSs induced from NTK with swish activation . Similarly , Hu et al . [ 2021 ] showed that a shallow neural network with ReLU activation trained via ` 2-regularized GD can generalize well , when the target function ( i.e. , f ? ρ ) is fromHNTK1 . Notably , the rate that the papers Nitanda & Suzuki [ 2020 ] and Hu et al . [ 2021 ] obtained is minimax optimal , meaning that no estimators perform substantially better than the ` 2-regularized GD or averaged SGD algorithms for recovering functions from respective function spaces . Nevertheless , 1Henceforth , we denoteHNTK1 andHNTKL as RKHSs induced from NTK of shallow L = 1 and deep neural networks L ≥ 2 with ReLU activations , respecitvely . these results are restricted to shallow neural networks , and can not explain the generalization abilities of deep neural network ( DNN ) . Similarly with Arora et al . [ 2019a ] , Cao & Gu [ 2019 ] obtained the O ( n−1/2 ) generalization bound , showing that the SGD generalize well for f ? ρ ∈ HNTKL , when f ? ρ has a bounded RKHS norm . However , the rate they obtained is slower than the minimax rate we can actually achieve . Furthermore , their results become vacuous under the presence of additive noises on the data set . Motivated from these observations , the fundamental question in this study is as follows : When the noisy dataset is generated from a function fromHNTKL , does the overparametrized DNN obtained via ( ` 2-regularized ) GD provably generalize well the unseen data ? We consider a neural network that has L ≥ 2 hidden layers with width m n . ( i.e. , overparametrized deep neural network . ) We focus on the least-squares loss and assume that the activation function is ReLU . A positivity assumption of NTK from ReLU DNN is imposed , meaning that λ0 > 0 , where λ0 denotes the minimum eigenvalue of the NTK . We give a more formal mathematical definition of ReLU DNN in the following Subsection 2.2 . Under these settings , we provide an affirmative answer to the above question by investigating the behavior of L2-prediction error of the obtained neural network with respect to GD iterations . 1.1 CONTRIBUTIONS Our derivations of algorithm-dependent prediction risk bound require the analysis on training dynamics of the estimated neural network through ( regularized ) GD algorithm . We include these results as the contributions of our paper , which can be of independent interests as well . • In an unregulaized case , under the assumption λ0 > 0 , we show randomly initialized GD converges to 0 training loss at a linear rate . As will be detailed in subsection 3.3 , this is the different result from the seminal work of Allen-Zhu et al . [ 2018 ] , where they also prove a linear convergence of training loss of ReLU DNN , but under different data distribution assumption . • We show that the DNN updated via vanilla GD does not recover the ground truth function f ? ρ ∈ HNTKL under noisy observations , if the DNN is trained for either too short or too long : that is , the prediction error is bounded away from 0 by some constant as n goes to infinity . • In regularized case , we prove the mean-squared error ( MSE ) of DNN is upper bounded by some positive constant . Additionally , we proved the dynamics of the estimated neural network get close to the solution of kernel ridge regression associated with NTK from ReLU DNN . • We show that the ` 2-regularization can be helpful in achieving the minimax optimal rate of the prediction risk for recovering f ? ρ ∈ HNTKL under the noisy data . Specifically , it is shown that after some iterations of ` 2-regularized GD , the minimax optimal rate ( which is O ( n− d 2d−1 ) , where d is a feature dimension . ) can be achieved . Note that our paper is an extension of Hu et al . [ 2021 ] to DNN model , showing that the ` 2-regularized DNN can achieve a minimax optimal rate of prediction error for recovering f ? ρ ∈ HNTKL . However , we would like to emphasize that our work is not a trivial application of their work from at least two technical aspects . These aspects are more detailed in the following subsection . 1.2 TECHNICAL COMPARISONS WITH HU ET AL . [ 2021 ] Firstly , in the analysis of training loss of regularized shallow neural-net , Hu et al . [ 2021 ] begin the proof by decomposing the difference between two individual predictions into two terms : one that is related with the gram matrix evaluated at each iteration of the algorithm and the perturbation term . Henceforth , we name this decompostion as “ Gram+Pert ” decomposition . This decomposition can be checked with the equality ( E.2 ) in the supplementary PDF of Hu et al . [ 2021 ] . The key ingredients for the decomposition are ( i ) the simple gradient structure of the shallow neural net , and ( ii ) the partitioning of the nodes in the hidden-layer into two sets : a set of nodes whose activation patterns change from their initializations during training , and the complement of the set . This construction of the sets peels off the ReLU activation in the difference so that the GD algorithm can be involved in the analysis . However , because of the compositional structure of the network , the same nodes partitioning technique can not be applied for obtaining the decomposition in the DNN setting with ReLU activation . To avoid this difficulty , we employ a specially designed diagonal matrix Σ̃ and this matrix can peel off the ReLU function for each layer of the network . ( See the definition of Σ̃ in the proof of Theorem 3.5 in the Appendix . ) Recursive applications of this diagonal matrix across the entire hidden layers enable the Gram+Pert decomposition in our setting . It should be noted that the diagnoal matrix Σ̃ had been employed in Zou et al . [ 2020 ] , which analyzed the behavior of training loss of classification problem via ReLU DNN under logistic loss . However , since their result is dependent on different data distribution assumption under the different loss function from ours , they didn ’ t employ the Gram+Pert decomposition . Thus their technical approaches are different from ours . Secondly , Hu et al . [ 2021 ] directly penalized the weight parameter W by adding ‖W‖2F to the objective function . The ` 2-regularization solely on the W has an effect of pushing the weight towards the origin . This makes ‖W ( k ) −W ( 0 ) ‖2 ≤ O ( 1 ) 2 , allowing most activation patterns of the nodes in the hidden layer can change during the training , even in overparametrized setting . Here , W ( k ) denotes the updated weight parameter at kth itertaion of algorithm , and ‖ · ‖2 denotes the spectral norm of the matrix . Nonetheless , this doesn ’ t affect the analysis on obtaining the upper-bound of MSE in shallow neural net , since the network has only a single hidden layer . In contrast , in the DNN setting , we allow the non-convex interactions of parameters across the hidden layers . To the best of our knowledge , a technique for controlling the size of ` 2-norm of network gradient has not been developed under this setting , yet . We circumvent this difficulty by regularizing the distance between the updated and the initialized parameter , instead by directly regularizing the updated parameter . This ensures that the updated parameter by ` 2-regularized GD stays in a close neighborhood to its initialization , so that with heavy over-parametrization , the dynamics of network becomes linearized in parameter and we can ignore the non-convex interactions of parameters across the hidden layers . Specifically , under suitable model parameter setting , we prove that ‖W ( k ) ` −W ( 0 ) ` ‖2 ≤ ÕP ( 1√ m ) 3 over all ` ∈ { 1 , . . . , L } . Here , ÕP ( · ) hides the dependencies on the model parameters ; L , ω , and n. This result allows us to adopt the so-called “ Forward Stability ” argument developed by Allen-Zhu et al . [ 2018 ] , and eventually leads to the control of network gradient under ` 2 sense . 1.3 ADDITIONAL RELATED WORKS There has been another line of works trying to characterize the generalizabilities of DNN under noisy observation settings . Specifically , it has been shown that the neural network model can achieve minimax style optimal convergence rates of L2-prediction risk both in regression [ Bauer & Kohler , 2019 ; Liu et al. , 2019 ; Schmidt-Hieber , 2020 ] and classification [ Kim et al. , 2021 ] problems . Nonetheless , a limitation of the aforementioned papers is that they assume an adequate minimizer of the empirical risk can be obtained . In other words , the mathematical proofs of their theorems do not correspond to implementable algortihms . Recently , several papers , which study the generalization properties of neural network with algorithmic guarantees , appear online . Specifically , Kohler & Krzyzak [ 2019 ] showed that the data interpolants obtained through DNN by vanilla GD is inconsistent . This result is consistent with our result , but they consider the overparametrized DNN that is a linear combination of Ω ( n10d 2 ) smaller neural network , and the activation function they consider is sigmoid function , which is smooth and differentiable . Along this line of research , Kuzborskij & Szepesvári [ 2021 ] ( regression ) and Ji et al . [ 2021 ] ( classification ) showed that when training overparametrized shallow neural network , early stopping of vanilla GD enables us to obtain consistent estimators . Notation . We use the following notation for asymptotics : For sufficiently large n , we write f ( n ) = O ( g ( n ) ) , if there exists a constant K > 0 such that f ( n ) ≤ Kg ( n ) , and f ( n ) = Ω ( g ( n ) ) if f ( n ) ≥ K ′g ( n ) for some constant K ′ > 0 . The notation f ( n ) = Θ ( g ( n ) ) means that f ( n ) = O ( g ( n ) ) and 2This was empirically shown to be true in paper Wei et al . [ 2019 ] . See Figure 3 in their paper . We provide a brief mathematical explanation on why this result is hard to be shown in Appendix C. 3Readers can find the proof of this result in Appendix G. f ( n ) = Ω ( g ( n ) ) . Let 〈A , B〉Tr : = Tr ( A > B ) for the two matrices A , B ∈ Rd1×d2 . We adopt the shorthand notation denoting [ n ] : = { 1 , 2 , . . . , n } for n ∈ N. 2 PROBLEM FORMULATION 2.1 NON-PARAMETRIC REGRESSION Let X ⊂ Rd and Y ⊂ R be the measureable feature space and output space . We denote ρ as a joint probability measure on the product space X × Y , and let ρX be the marginal distribution of the feature space X . We assume that the noisy data-set D : = { ( xi , yi ) } ni=1 are generated from the non-parametric regression model yi = f ? ρ ( xi ) + εi , where εi i.i.d.∼ N ( 0 , 12 ) for i = 1 , . . . , n. Let f̂W ( k ) ( · ) be the value of neural network evaluated with the parameters W at the k-th iterations of GD update rule . At k = 0 , we randomly initialize the weight parameters in the model following He initialization [ He et al. , 2015 ] with a slight modification . Then , the L2 prediction risk is defined as the difference between two expected risks ( i.e. , excess risk ) R ( f̂W ( k ) ) : = Eρ∼ ( x , y ) [ ( y − f̂W ( k ) ( x ) ) 2 ] and R ( f ? ρ ) : = Eρ∼ ( x , y ) [ ( y − f ? ρ ( x ) ) 2 ] , where f ? ρ ( x ) : = E [ y|x ] . Then , we can easily show the prediction risk has a following form : R ( f̂k , f ? ρ ) : = R ( f̂W ( k ) ) −R ( f ? ) = Eρx , ε [ ( f̂W ( k ) ( x ) − f ? ρ ( x ) ) 2 ] . ( 1 ) Note that the expectation is taken over the marginal probability measure of feature space , ρx , and the noise of the data , ε . However , the ( 1 ) is still a random quantity due to the randomness of the initialized parameters ( W ( 0 ) ` ) ` =1 , ... , L . 2.2 DEEP NEURAL NETWORK WITH RELU ACTIVATION Following the setting introduced in Allen-Zhu et al . [ 2018 ] , we consider a fully-connected deep neural networks with L hidden layers and m network width . For L ≥ 2 , the output of the network fW ( · ) ∈ R with input data x ∈ X can be formally written as follows : fW ( x ) = √ m · vTσ ( WLσ ( WL−1 · · ·σ ( W1x ) · · · ) ) , ( 2 ) where Sd−1 is a unit sphere in d-dimensional euclidean space , σ ( · ) is an entry-wise activation function , W1 ∈ Rm×d , W2 , . . . , WL ∈ Rm×m denote the weight matrices for hidden layers and v ∈ Rm×1 denote the weight vector for the output layer . Following the existing literature , we will consider ReLU activation function σ ( x ) = max ( x , 0 ) , which is the most commonly used activation function by practitioners . Random Initialization . Each entries of weight matrices in hidden layers are assumed to be generated from ( Wi , j ) ` =1 , ... , L ∼ N ( 0 , 2m ) , and entries of the output layer are drawn from vj ∼ N ( 0 , ωm ) . This initialization scheme helps the forward propagation neither explode nor vanish at the initialization , seeing Allen-Zhu et al . [ 2018 ] ; Zou et al . [ 2018 ; 2020 ] . Note that we initialize the parameters in the last layer with variance ωm , where ω ≤ 1 is a model parameter to be chosen later for technical convenience . Unregularized GD update rule . We solve a following ` 2-loss function with the given dataset D : LS ( W ) = 1 2 n∑ i=1 ( yi − fW ( xi ) ) 2 . ( 3 ) Let W ( 0 ) 1 , . . . , W ( 0 ) L be the initialized weight matrices introduced above , and we consider a following gradient descent update rule : W ( k ) ` = W ( k−1 ) ` − η∇W ` ( LS ( W ( k−1 ) ` ) ) , ` ∈ [ L ] , k ≥ 1 , ( 4 ) where ∇W ` ( LS ( · ) ) is a partial gradient of the loss function LS ( · ) with respect to the ` -th layer parameters W ` , and η > 0 is the learning rate of the gradient descent . ` 2-regularized GD update rule . The estimator is obtained by minimizing a ` 2-regularized function ; ΦD ( W ) : = LS ( WD ) + µ 2 L∑ ` =1 ∥∥∥WD , ` −W ( 0 ) D , ` ∥∥∥2 F . ( 5 ) Naturally , we update the model parameters { WD , ` } ` =1 , ... , L via modified GD update rule : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] + η2µW ( 0 ) D , ` , ∀ ` ∈ [ L ] , ∀k ≥ 1 . ( 6 ) The notations η1 , η2 are step sizes , and µ > 0 is a tuning parameter on regularization . We adopt the different step sizes for the partial gradient and regularized term for the theoretical conveniences . Furthermore , we add the additional subscript D to the update rule ( 6 ) to denote the variables are under the regularized GD update rule . Recall that the W ( 0 ) D , ` are initialized parameters same with the unregularized case . For simplicity , we fix the output layer , and train L hidden layers for both unregularized and regularized cases . 3 MAIN THEORY First , we describe the neural tangent kernel ( NTK ) matrix of ( 2 ) , which is first proposed by Jacot et al . [ 2018 ] and further studied by Arora et al . [ 2019b ] ; Du et al . [ 2019 ] ; Lee et al . [ 2018 ] ; Yang [ 2019 ] . NTK matrix of DNN is a L-times recursively defined n × n kernel matrix , whose entries are the infinite-width limit of the gram matrix . Let ∇W ` [ fW ( 0 ) ( · ) ] be the gradient of the ReLU DNN ( 2 ) with respect to the weight matrix in ` th hidden layer at random initialization . Note that when ` = 1 , ∇W ` [ fW ( 0 ) ( · ) ] ∈ Rm×d and when ` ∈ { 2 , . . . , L } , ∇W ` [ fW ( 0 ) ( · ) ] ∈ Rm×m . Then , as m→∞ , H ( 0 ) : = ( 1 m L∑ ` =1 〈 ∇W ` [ fW ( 0 ) ( xi ) ] , ∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ) n×n → H∞L , ( 7 ) where H∞L : = { Ker ( xi , xj ) } n i , j=1 . Here , Ker ( · , · ) denotes a NTK function of ( 2 ) to be defined as follows : Definition 3.1 . ( NTK function of ( 2 ) ) . For any x , x′ ∈ X and ` ∈ [ L ] , define Φ ( 0 ) ( x , x′ ) = 〈x , x′〉 , Θ ( ` ) ( x , x′ ) = ( Φ ( ` −1 ) ( x , x ) Φ ( ` −1 ) ( x , x′ ) Φ ( ` −1 ) ( x′ , x ) Φ ( ` −1 ) ( x′ , x′ ) ) ∈ R2×2 , Φ ( ` ) ( x , x′ ) = 2 · E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ ( u ) · σ ( v ) ] , and Φ̇ ( ` ) ( x , x′ ) = 2 · E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ̇ ( u ) · σ̇ ( v ) ] , where σ̇ ( u ) = 1 ( u ≥ 0 ) . Then , we can derive the final expression of NTK function of ( 2 ) as follows : Ker ( x , x′ ) = ω 2 · L∑ ` =1 ( Φ ( ` −1 ) ( x , x′ ) · L∏ ` ′= ` Φ̇ ( ` ′ ) ( x , x′ ) ) . ( 8 ) The expression in ( 8 ) is adapted from Cao & Gu [ 2019 ] . As remarked in Cao & Gu [ 2019 ] , a coefficient 2 in Φ ( ` ) and Φ̇ ( ` ) remove the exponential dependence on the network depth L in the NTK function . However , when compared with the NTK formula in Cao & Gu [ 2019 ] , ( 8 ) is different from two aspects : ( i ) An additional factor ω in ( 8 ) ) comes from the difference in initialization settings of the output layer , in which Cao & Gu [ 2019 ] considers vj ∼ N ( 0 , 1m ) , whereas we consider vj ∼ N ( 0 , ωm ) . ( ii ) Φ ( L ) is not added in the final expression of ( 8 ) ) , whereas it is added in the definition provided in Cao & Gu [ 2019 ] . This is because we only train the L hidden layers but fix the output layer , while Cao & Gu [ 2019 ] train the entire layers of the network including the output layer . As already been pointed by several papers Cho & Saul [ 2009 ] ; Jacot et al . [ 2018 ] , it can be proved that the NTK function ( 8 ) is a positive semi-definite kernel function . Furthermore , Cho & Saul [ 2009 ] prove that the expectations in Φ and Φ̇ have closed form solutions , when the covariance matrices have the form ( 1 tt 1 ) with |t| ≤ 1 : E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ ( u ) · σ ( v ) ] = 1 2π ( t · ( π − arccos ( t ) ) + √ 1− t2 ) , E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ̇ ( u ) · σ̇ ( v ) ] = 1 2π ( π − arccos ( t ) ) . ( 9 ) Clearly , ( 8 ) is symmetric and continuous on the product space X ×X , from which it can be implied that Ker ( · , · ) is a Mercer kernel inducing an unique RKHS . Following Ghorbani et al . [ 2020 ] , we define the RKHS induced by ( 8 ) as : Definition 3.2 . ( NTK induced RKHS ) . For some integer p ∈ N , set of points { x̃j } pj=1 ⊂ X , and weight vector α : = { α1 , . . . , αp } ∈ Rp , define a complete vector space of functions , f : X → R , HNTKL : = cl ( { f ( · ) = p∑ j=1 αjKer ( · , x̃j ) } ) , ( 10 ) where cl ( · ) denotes closure . In the remaining of our work , we assume the regression function f ? ρ ( x ) : = E [ y|x ] belongs toHNTKL . 3.1 ASSUMPTIONS . In this subsection , we state the assumptions imposed on the data distribution with some remarks . ( A1 ) ρX is an uniform distribution on Sd−1 : = { x ∈ Rd | ‖x‖2 = 1 } , and noisy observations are assumed to be bounded . ( i.e. , ρx ∼ Unif ( Sd−1 ) , yi = O ( 1 ) , ∀i ∈ [ n ] . ) ( A2 ) Draw n i.i.d . samples { xi , f ? ρ ( xi ) } ni=1 from the joint measure ρ . Then , with probability at least 1− δ , we have λmin ( H∞L ) = λ0 > 0 . Remark 3.3 . • When the feature space is restricted on the unit sphere , the NTK function in ( 8 ) becomes rotationally invariant zonal kernel . This setting allows to adopt the results of spectral decay of ( 8 ) in the basis of spherical harmonic polynomials for measuring the complexity of hypothesis space , HNTKL . See the Appendix 3.2 and references therein . • Assumption ( A2 ) is commonly employed in NTK related literature for proving global convergence of training error and generalization error of both deep and shallow neural network , Du et al . [ 2018 ; 2019 ] ; Arora et al . [ 2019a ] . Note that the ( A2 ) holds as long as no two xi and xj are parallel to each other , which is true for most of the real-world distributions . See the proof of this claim in Du et al . [ 2019 ] . 3.2 MINIMAX RATE FOR RECOVERING f ? ρ ∈ HNTKL The obtainable minimax rate of L2-prediction error is directly related with the complexity of function space of interest . In our setting , the complexity of RKHS HNTKL can be characterized by the eigen-decay rate of the NTK function . Since Ker ( x , x′ ) is defined on the sphere , the decomposition can be given in the basis of spherical harmonics as follows : Ker ( x , x′ ) = ∞∑ k=0 µk N ( d , k ) ∑ j=1 Yk , j ( x ) Yk , j ( x ′ ) , where Yk , j , j = 1 , . . . , N ( d , k ) are spherical harmonic polynomials of degree k and { µk } ∞k=0 are non-negative eigenvalues . Recently , several researchers , both empirically [ Basri et al. , 2020 ] and theoretically [ Chen & Xu , 2020 ; Geifman et al. , 2020 ; Bietti & Bach , 2021 ] , showed that , for large enough harmonic function frequency k , the decay rate of the eigenvalues µk is in the order of Θ ( k−d ) 4 . Given this result and the fact N ( d , k ) = 2k+d−3k ( k+d−3 d−2 ) grows as kd−2 for large k , it can be easily shown λj = Θ ( j− d d−1 ) , when Ker ( x , x′ ) = ∑∞ j=1 λjφj ( x ) φj ( x ′ ) , for eigen-values λ1 ≥ λ2 ≥ · · · ≥ 0 and orthonormal basis { φj } ∞j=1 . Furthermore , it is a well known fact that if the eigenvalues decay at the rate λj = Θ ( j−2ν ) , then the corresponding minimax rate for estimating function in RKHS is O ( n− 2ν 2ν+1 ) , [ Raskutti et al. , 2014 ; Yuan & Zhou , 2016 ; Hu et al. , 2021 ] . By setting 2ν = dd−1 , we can see the minimax rate for recovering f ? ρ ∈ HNTKL isO ( n− d 2d−1 ) . Remark 3.4 . We defer all the technical proofs of the Theorems in subsections 3.3 and 3.4 in the Appendix for conciseness of the paper . At the beginning of each proof , a high-level proof idea is also provided . We also provide numerical experiments which can corroborate our theoretical findings in the Appendix A . 3.3 ANALYSIS OF UNREGULARIZED DNN In this subsection , we provide the results on the training loss of DNN estimator obtained via minimizing unregularized ` 2-loss ( 3 ) and on the corresponding estimator ’ s L2-prediction riskR ( f̂k , f ? ρ ) . Theorem 3.5 . ( Optimization ) For some δ ∈ [ 0 , 1 ] , if we set the width of the network as m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) , and we set the step-size of gradient descent as η = O ( λ0 n2L2m ) . Then , with probability at least 1−δ over the randomness of initialized parametersW ( 0 ) : = { W ( 0 ) ` } L+1 ` =1 with W ( 0 ) L+1 = v , we have for k = 0 , 1 , 2 , . . . , LS ( W ( k ) ) ≤ ( 1− ηmλ0 2 ) k LS ( W ( 0 ) ) . ( 11 ) In other words , the training loss drops to 0 at a linear rate . We acknowledge a series of past works Allen-Zhu et al . [ 2018 ] ; Du et al . [ 2019 ] have similar spirits with those in Theorem 3.5 . However , it is worth noting that their results are not applicable in our problem settings and data assumptions . Specifically , the result of Du et al . [ 2019 ] is based on the smooth and differentiable activation function , whereas the Theorem 3.5 is about the training error of ReLU activation function , which is not differentiable at 0 . Furthermore , the result of Allen-Zhu et al . [ 2018 ] relies on φ-separateness assumption stating that the every pair of feature vectors { xi , xj } n i 6=j is apart from each other by some constant φ > 0 in a Euclidean norm . In our work , the positivity assumption on the minimum eigenvalue of the NTK is imposed ( i.e. , λ0 > 0 ) . Remark 3.6 . Reducing the order of network width is definitely another line of interesting research direction . We are aware of some works in literature , but we chose not to adopt the techniques since this can make the analysis overly complicated . To the best of our knowledge , the paper that most neatly summarizes this line of literature is Zou & Gu [ 2019 ] . See the table in page 3 in their paper . The order of width they obtained is Ω ( n8L12 φ8 ) , where they impose φ-separateness assumption . Remark 3.7 . There has been an attempt to make a connection between the positivity and φseparateness assumptions . Recently , Zou & Gu [ 2019 ] proved the relation λ0 = Ω ( φn−2 ) 5 in a shallow-neural net setting . See Proposition 3.6. of their work . However , it is still an open question on whether this relation holds in DNN setting as well . The results in Theorem 3.5 suggest a positive conjecture on this question . Indeed , plugging the relation λ0 = Ω ( φn−2 ) in ( 11 ) and in the η = O ( λ0 n2L2m ) yield the discount factor ( 1− Ω ( ηmφ n2 ) ) k and step-size η = O ( φ n4L2m ) , which are exactly the same orders as presented in Allen-Zhu et al . [ 2018 ] . See Theorem 1 of their ArXiv version paper for the clear comparison . We leave the proof of this conjecture as a future work . 4In shallow neural network with ReLU activation without bias terms , it is shown that µk satisfy µ0 , µ1 > 0 , µk = 0 if k = 2j + 1 with j ≥ 1 , and otherwise µk = Θ ( k−d ) . See Bietti & Mairal [ 2019 ] . However , in ReLU DNN , it is shown that these parity constraints can be removed even without bias terms and µk achieves Θ ( k−d ) decay rate for large enough k. Readers can refer Bietti & Bach [ 2021 ] for this result . 5We conjecture that this is not the tightest lower bound on λ0 . Recently , Bartlett et al . [ 2021 ] proves that λ0 & d/n in shallow neural net setting . See Lemma 5.3 in their paper . Theorem 3.8 . ( Generalization ) Let f ? ρ ∈ HNTKL . Fix a failure probability δ ∈ [ 0 , 1 ] . Set the width of the network as m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) , the step-size of gradient descent as η = O ( λ0 n2L2m ) , and the variance parameter ω ≤ O ( ( λ0δ n ) 2/3 ) . Then , if the GD iteration k ≥ Ω ( log ( n ) ηmλ0 ) or k ≤ O ( 1 ηmωL ) , with probability at least 1− δ over the randomness of initialized parametersW ( 0 ) , we have R ( f̂k , f ? ρ ) = Ω ( 1 ) . This theorem states that if the network is trained for too long or too short , the L2-prediction error of f̂W ( k ) is bounded away from 0 by some constant factor . Specifically , the former scenario indicates that the overfitting can be harmful for recovering f ? ρ ∈ HNTKL given the noisy observations . Remark 3.9 . Readers should note that the Theorem 3.8 does not consider if the GD algorithm can achieve low prediction risk R ( f̂k , f ? ρ ) over the range of iterations ( ηmωL ) −1 . k . ( ηmλ0 ) −1 log ( n ) . In the numerical experiment to be followed in Section A , we observe that for some algorithm iterations k∗ , the risk indeed decreases to the same minimum as low as the ` 2- regularized algorithm can achieve , and increases again . This observation implies that the unregularized algorithm can achieve the minimax rate of prediction risk . However , analytically deriving a data-dependent stopping time k∗ in our scenario requires further studies , since we need a sharp characterization of eigen-distribution of NTK matrix of ReLU DNN , denoted as H∞L in this paper . Readers can refer the Theorem 4.2. of Hu et al . [ 2021 ] in shallow-neural network and equation ( 6 ) in Raskutti et al . [ 2014 ] in kernel regression context on how to compute k ? with the given eigenvalues of the associated kernel matrices . Remark 3.10 . From practitioner ’ s point of view , we can simply stop GD before it hits the nearlyzero empirical risk , but monitor the performance on a held-out validation sample , and stop training early when a minimum on the validation sample has been reached . 3.4 ANALYSIS OF ` 2-REGULARIZED DNN In this subsection , we study the training dynamics of ` 2-regularized DNN and the effects of the regularization for obtaining the minimax optimal convergence rate of L2-prediction risk . In the results to be followed , we set the orders of model parameters µ , η1 , η2 in ( 6 ) , and a variance parameter of output layer , ω as follows : µ = Θ ( n d−1 2d−1 ) , η1 = Θ ( 1 m n− 3d−2 2d−1 ) , η2 = Θ ( 1 L n− 3d−2 2d−1 ) , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) . ( 12 ) Theorem 3.11 . ( Optimization ) Suppose we minimize ` 2-regularized objective function ( 5 ) via modified GD ( 6 ) . Set the network width m log3 ( m ) ≥ Ω ( L20n24 δ2 ) and model parameters as in ( 12 ) . Then , with probability at least 1− δ , the mean-squared error follows LS ( W ( k ) D ) /n ≤ ( 1− η2µL ) k · LS ( W ( 0 ) D ) /n+OP ( 1 ) , ( 13 ) for k ≥ 0 . Additionally , after k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) iterations of ( 6 ) , for some constant C > 0 , we have ∥∥∥∥∥uD ( k ) −H∞L ( Cµ · I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ OP ( 1 n ) , ( 14 ) where we denote uD ( k ) : = [ f̂W ( k ) D ( x1 ) , . . . , f̂W ( k ) D ( xn ) ] > . Several comments are in sequel . Theorem 3.11 is , to our knowledge , the first result that rigorously shows the training dynamics of ` 2-regularized ReLU DNN in overparametrized setting . Observe that the first term on the right-hand side of the inequality ( 13 ) converges linearly to 0 , and the second term is some positive constant that is bounded away from 0 . This implies that the MSE of regularized DNN is upper-bounded by some positive constant . Note that we only provide the upper bound , but the results of our numerical experiments indicate that the MSE is lower-bounded by OP ( 1 ) as well . We leave the proof of this conjecture for the future work . The inequality ( 14 ) states that the trained dynamics of the regularized neural network can approximate the optimal solution ( denoted as g ? µ ) of the following kernel ridge regression problem : min f∈HNTK { 1 2 n∑ i=1 ( yi − f ( xi ) ) 2 + Cµ 2 ‖f‖2HNTKL } , ( 15 ) where ‖ · ‖HNTKL denotes a NTK-induced RKHS norm . ( Note that the optimization problem in ( 15 ) is not normalized by sample size n. ) The inequality ( 14 ) states that after approximately ( η2µL ) −1 iterations of ( 6 ) , the error rate becomes OP ( 1 n ) . The approximation error is computed at the training data points under ` 2 norm . This should be compared with the Theorem 5.1 of Hu et al . [ 2021 ] , where they showed that the similar approximation holds “ within ” a certain range of algorithm in shallow neural network setting . In contrast , we show that the approximation holds “ after ” k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) in deep neural network . It should be noted that the difference of results comes from the regularization scheme , where we penalize the ∑L ` =1 ‖W ` −W ( 0 ) ` ‖2F , whereas Hu et al . [ 2021 ] regularized the term ‖W1‖2F . As another important comparison , Hu et al . [ 2019 ] showed the equivalence of a solution of kernel ridge regression associated with NTK and first order Taylor expansion of the regularized neural network dynamics ; note , however , that the uD ( k ) in ( 14 ) is a full neural network dynamics . LetR ( f̂ W ( k ) D , f ? ρ ) be the L2-prediction risk of the regularized estimator f̂W ( k ) D via modified GD ( 6 ) . Next theorem states the result of generalization ability of f̂ W ( k ) D . Theorem 3.12 . ( Generalization ) Let f ? ρ ∈ HNTKL . Suppose the network width mlog3 ( m ) ≥ Ω ( L20n24 δ2 ) and model parameters are set as suggested in ( 12 ) . Then , with probability tending to 1 , we have R ( f̂ W ( k ) D , f ? ρ ) = OP ( n− d 2d−1 ) . The resulting convergence rate is O ( n− d 2d−1 ) with respect to the training sample size n. Note that the rate is always faster than O ( n−1/2 ) and turns out to be the minimax optimal [ Caponnetto & De Vito , 2007 ; Blanchard & Mücke , 2018 ] for recovering f ? ρ ∈ HNTKL in the following sense : lim r→0 lim inf n→∞ inf f̂ sup ρ P [ R ( f̂ , f ? ρ ) > rn− d 2d−1 ] = 1 , ( 16 ) where ρ is a data distribution class satisfying the Assumptions ( A1 ) , ( A2 ) and f ? ρ ∈ HNTKL , and infimum is taken over all estimators D → f̂ . It is worth noting that the minimax rate in ( 16 ) is same with the minimax rate for recovering f ? ρ ∈ HNTK1 . ( i.e. , Hu et al . [ 2021 ] ) This result can be derived from the recent discovery of the equivalence between two function spaces , HNTK1 = HNTKL . See Geifman et al . [ 2020 ] and Chen & Xu [ 2020 ] . Remark 3.13 . A particular choice of µ = Θ ( n d−1 2d−1 ) in ( 12 ) is for obtaining an optimal minimax rate for prediction error in Theorem 3.12 . Specifically , the order of µ determines the L2 distance between the f ? ρ and the kernel regressor g ? µ . That is , ‖f ? ρ − g ? µ‖22 = OP ( µ n ) . With the resultH NTK 1 = HNTKL , the same proof of Lemma D.2 . in Hu et al . [ 2021 ] can be applied for proving this result . 4 CONCLUSION We analyze the convergence rate of L2-prediction error of both the unregularized and the regularized gradient descent for overparameterized DNN with ReLU activation for a regression problem . Under a positivity assumption of NTK , we show that without the adoption of early stopping , the L2-prediction error of the estimated DNN via vanilla GD is bounded away from 0 ( Theorem 3.5 ) , whereas the prediction error of the DNN via ` 2-regularized GD achieves the optimal minimax rate ( Theorem 3.12 ) . The minimax rate O ( n− d 2d−1 ) is a faster than the O ( n−1/2 ) by specifying the complexities of target function and hypothesis space . 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Stochastic gradient descent optimizes over-parameterized deep ReLU networks . arxiv e-prints , art . arXiv preprint arXiv:1811.08888 , 2018 . Difan Zou , Yuan Cao , Dongruo Zhou , and Quanquan Gu . Gradient descent optimizes overparameterized deep ReLU networks . Machine Learning , 109 ( 3 ) :467–492 , 2020 . A NUMERICAL ILLUSTRATIONS In this section , we use synthetic data to corroborate our theoretical findings . We use the He initialization [ He et al. , 2015 ] and employ ( ` 2-regularized ) GD as introduced in subsection 2.2 . For the experiments , we run 1000 epochs of GD and use a fixed step size , setting η1 = η2 = 0.001 . We uniformly generate n feature data xitrain from Sd−1 with d = 2 and generate yi from f ? ρ ( xtraini ) with εi ∼ N ( 0 , 1 ) . To create a function f ? ρ ∈ HNTKL , we use the definition in ( 10 ) with α ∈ Unif ( Sp−1 ) and with p fixed points { x̃j } pj=1 ⊂ Unif ( Sd−1 ) , where p is simply set as 1 . Note that Ker ( · , · ) in ( 10 ) can be calculated via the formulas ( 8 ) and ( 9 ) with specified network depth L. We consider a scenario where we have a network with depth L = 8 and width m = 2000 . The variance parameter of the output layer ( ω ) is set as 1 for unregularized and 0.001 for regularized cases . In Fig 1 . ( a ) , we record the training errors of regularized networks over the GD epochs k ≤ 1000 , where we have n ∈ { 100 , 300 , 500 , 1000 , 5000 } training samples . This aims to verify the inequality ( 13 ) that the MSE of regularized network is bounded away from 0 by some constant . In Fig 1 . ( b ) , the prediction risks of both unregularized and regularized networks are displayed . We approximate the risk with 1500 ∑500 j=1 ( f̂k ( x test j ) − f ? ρ ( xtestj ) ) 2 with a new test data set { xtestj , f ? ρ ( xtestj ) } 500j=1 over k ≤ 1000 for both unregularized and regularized cases . In both cases , they reach the same minimal risks , but the risk of unregularized network increase after it hits the minimal point , whereas the risk of regularized network stays stable . Theorem 3.6 tells us that for the iteration less than the order O ( 1 ηmωL ) , the prediction error is bounded away from 0 . In the experiment for unregularized case , we set η = 0.01 , m = 2000 , L = 8 , and ω = 1 . Plugging in these parameters in the bound says that the minimum can be achieved within a very few iterations . Note that the optimal risk is non-zero as long as we have finite sample sizes n , but converges to 0 at the rate O ( n− d 2d−1 ) . In Fig 1 . ( c ) , we verify that the more training sample sizes we have , the closer the risks of the regularized networks get to 0 . The risk is evaluated at the sample sizes n = { 100 , 300 , 500 , 1000 , 5000 } . We have to acknowledge that there is a discrepancy between our experiment setting and theory . Specifically , due to the limited computing power , we could not run the experiment under the regime of width m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) . But the prediction risk behaves similarly as expected by our theorem , which can be a partial evidence that the statement in Theorem 3.6 still holds in the narrower width of the network . B PRELIMINARY NOTATIONS Before presenting the formal proofs of Lemmas and main results , we introduce several notations used frequently throughout the proofs . First , we denote x ` , i the output of the ` th hidden layer with the input data xi after applying entry-wise ReLU activation function . x ` , i = σ ( W ` σ ( W ` −1 · · ·σ ( W1xi ) · · · ) ) . Denote fW ( k ) ( x ) a value of neural network ( 2 ) evaluated at the collection of network parameters W ( k ) : = { W ( k ) ` } ` =1 , ... , L and W ( k ) ` denotes the ` th hidden layer parameter updated by kth GD iterations . Partial gradient of fW ( k ) ( x ) . We employ the following matrix product notation which was used in several other papers [ Zou et al. , 2018 ; Cao & Gu , 2019 ] : ` 2∏ r= ` 1 Ar : = { A ` 2A ` 2−1 · · ·A ` 1 if ` 1 ≤ ` 2 , I otherwise . ( 17 ) Then , the partial gradient of fW ( k ) ( x ) with respect to W ( k ) ` for 1 ≤ ` ≤ L has a following form : for i ∈ { 1 , . . . , n } , ∇W ` [ fW ( k ) ( xi ) ] = √ m · [ x ( k ) ` −1 , iv T ( L∏ r= ` +1 Σ ( k ) r , i W ( k ) r ) Σ ( k ) ` , i ] > , ` ∈ [ L ] , where Σ ( k ) ` , i : = Diag ( 1 ( 〈w ( k ) ` ,1 , x ( k ) ` −1 , i〉 ≥ 0 ) , . . . ,1 ( 〈w ( k ) ` , m , x ( k ) ` −1 , i〉 ≥ 0 ) ) ∈ Rm×m and w ( k ) ` , j denotes jth column of the matrix W ( k ) ` . Gram matrix H ( k ) . Each entries of empirical gram matrix evaluated at the kth GD update are defined as follows : Hi , j ( k ) = 1 m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr . Note that H ( 0 ) → H∞L as m → ∞ which is proved in Jacot et al . [ 2018 ] ; Yang [ 2019 ] ; Lee et al . [ 2018 ] ; Arora et al . [ 2019b ] . Perturbation region of weight matrices . Consider a collection of weight matrices W̃ = { W̃ ` } ` =1 , ... , L such that W̃ ∈ B ( W ( 0 ) , τ ) : = { W̃ ` : ‖W̃ ` −W ( 0 ) ` ‖2 ≤ τ , ∀ ` ∈ [ L ] } . ( 18 ) For all i ∈ { 1 , . . . , n } and ` = 1 , . . . , L , we denote x ` , i and x̃ ` , i as the outputs of the ` -th layer of the neural network with weight matrices W ( 0 ) and W̃ , and Σ ` , i and Σ̃ ` , i are diagonal matrices with ( Σ ` , i ) jj = 1 ( 〈w ( 0 ) ` , j , x ` −1 , i〉 ≥ 0 ) and ( Σ̃ ` , i ) jj = 1 ( 〈w̃ ` , j , x̃ ` −1 , i〉 ≥ 0 ) , respectively . C WHY IS IT HARD TO PROVE ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) ? In this subsection , we provide a heuristic argument on why it is hard to prove ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) , where W ( k ) D , ` is the model parameter of ` th layer in kth iteration of algorithm . Here , we regularize solely on the model parameter , instead on the relative to the initialization . In this case , we can write the update rule as follows : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . ( 19 ) By recursively applying above equation ( 4.3 ) , we can write W ( k ) D , ` with respect to W ( 0 ) D , ` as follows : W ( k ) D , ` = ( 1− η2µ ) kW ( 0 ) D , ` − η1 k−1∑ ` =0 ( 1− η2µ ) ` ∇W ` [ LS ( W ( k− ` −1 ) D ) ] . Then , we can control the bound as follows : ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ ( 1− ( 1− η2µ ) k ) ∥∥∥W ( 0 ) D , ` ∥∥∥ 2 + η1 η2µ max ` =0 , ... , k−1 ∥∥∥∇W ` [ LS ( W ( k− ` −1 ) D ) ] ∥∥∥ 2 . We know under the initialization setting in our paper , ‖W ( k ) D , ` ‖2 ≤ O ( 1 ) with high-probability ( see Vershynin [ 2018 ] ) , and as long as we can prove the ` 2-norm of gradient is bounded , then we can conclude ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) . However , we are not aware of works in which they control the size of ‖∇W ` [ LS ( W ( k− ` −1 ) D ) ] ‖2 where the non-convex interactions between model parameters across the hidden layers are allowed . To the best of our knowledge , we know the work Allen-Zhu et al . [ 2019 ] deals with the three layer case under this setting . But we need further investigations on whether the techniques employed in their paper can be generalized to arbitrary L-hidden layer setting . D USEFUL LEMMAS A simple fact . Suppose vj i.i.d∼ N ( 0 , ωm ) for j ∈ [ m ] . Then , with probability at least 1 − exp [ −Ω ( m ) ] , ‖v‖22 ≤ O ( ω ) . Proof . Since ∥∥v2j∥∥Ψ1 ≤ O ( ωm ) for j ∈ [ m ] , where ‖ · ‖Ψ1 denotes a sub-exponential norm , Bernstein ’ s inequality for i.i.d . centered sub-exponential random variables can be employed : For any t ≥ 0 , P ( ∣∣∣∣∣∣ m∑ j=1 ( v2j − ω m ) ∣∣∣∣∣∣ ≥ t ) ≤ 2 exp ( − cmin ( t2∑m j=1 ∥∥v2j∥∥2Ψ1 , t maxj ∥∥v2j∥∥Ψ1 ) ) , ( 20 ) where c > 0 is an absolute constant . Note that we used the fact centering does not hurt the subexponentiality of random variable . Choosing t = O ( ω ) concludes the proof . Lemma 4.1 ( Lemma 7.1 . Allen-Zhu et al . [ 2018 ] ) . With probability at least 1 − O ( nL ) · exp [ −Ω ( m/L ) ] , 3/4 ≤ ‖x ( 0 ) ` , i ‖2 ≤ 5/4 for all i ∈ { 1 , . . . , n } and ` ∈ { 1 , . . . , L } . Lemma 4.2 ( Lemma B.1 . Cao & Gu [ 2019 ] ) . If τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , then with probability at least 1 − O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , 1/2 ≤ ‖x̃ ` , i‖2 ≤ 3/2 for all W̃ ∈ B ( W ( 0 ) , τ ) , i ∈ { 1 , . . . , n } and ` ∈ { 1 , . . . , L } . Lemma 4.3 ( Allen-Zhu et al . [ 2018 ] ) . Uniformly over i ∈ { 1 , . . . , n } and 1 ≤ ` 1 ≤ ` 2 ≤ L , the following results hold : 1 . ( Lemma.7.3 , Allen-Zhu et al . [ 2018 ] ) Suppose m ≥ Ω ( nL log ( nL ) ) , then with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥∥∥ ` 2∏ r= ` 1 Σ ( 0 ) r , iW ( 0 ) r ∥∥∥∥∥ 2 ≤ O ( √ L ) . 2 . ( Lemma.7.4 , Allen-Zhu et al . [ 2018 ] ) Suppose m ≥ Ω ( nL log ( nL ) ) , then with probability at least , 1−O ( nL ) · exp [ −Ω ( m/L ) ] , ∥∥∥∥∥v > ( L∏ r= ` 1 Σ ( 0 ) r , iW ( 0 ) r ) ∥∥∥∥∥ 2 ≤ O ( √ w ) . 3 . ( Lemma.8.2 , Allen-Zhu et al . [ 2018 ] ) Suppose τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) . For all W̃ ∈ B ( W ( 0 ) , τ ) , with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥x̃ ` 1 , i − x ( 0 ) ` 1 , i∥∥∥2 ≤ O ( τL5/2√log ( m ) ) . 4 . ( Corollary.8.4 , Allen-Zhu et al . [ 2018 ] ) Suppose τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , then with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥Σ̃ ` 1 , i −Σ ( 0 ) ` 1 , i∥∥∥0 ≤ O ( mτ2/3L ) . 5 . ( Lemma.8.7 , Allen-Zhu et al . [ 2018 ] ) For all ` ∈ [ L ] , let Σ′′ ` , i ∈ [ −3 , 3 ] m×m be the diagonal matrices with at most s = O ( mτ2/3L ) non-zero entries . For all W̃ ∈ B ( W ( 0 ) , τ ) , where τ = O ( 1 L1.5 ) , with probability at least 1−O ( nL ) · exp [ −Ω ( s log ( m ) ) ] , ∥∥∥∥∥vT ( L∏ r= ` 1+1 ( Σ′′r , i + Σ ( 0 ) r , i ) W̃r , i ) ( Σ′′ ` 1 , i + Σ ( 0 ) r , i ) − vT ( L∏ r= ` 1+1 Σ ( 0 ) r , iW ( 0 ) r , i ) Σ ( 0 ) ` 1 , i ∥∥∥∥∥ 2 ≤ O ( τ1/3L2 √ ω log ( m ) ) . Lemma 4.4 ( Lemma B.3 . Cao & Gu [ 2019 ] ) . There exists an absolute constant κ such that , with probability at least 1 − O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , i ∈ 1 , . . . , n and ` ∈ 1 , . . . , L and for all W̃ ∈ B ( W ( 0 ) , τ ) , with τ ≤ κL−6 [ log ( m ) ] −3 , it holds uniformly that∥∥∇W ` [ fW̃ ( xi ) ] ∥∥2 ≤ O ( √ωm ) . Lemma 4.5 . Suppose W̃ ∈ B ( W ( 0 ) , τ ) and τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) . For all u ∈ Rm with a cardinality ‖u‖0 ≤ s , for any 1 ≤ ` ≤ L and i ∈ { 1 , . . . , n } , with probability at least 1−O ( nL ) · exp ( − Ω ( s log ( m ) ) ) −O ( nL ) · exp ( − Ω ( mτ2/3L ) ) , ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≤ √ ωs log ( m ) m · O ( ‖u‖2 ) . Proof . Recall Lemma 4.2 . For any fixed vector u ∈ Rm , with probability at least 1 − O ( nL ) · exp [ −Ω ( mτ2/3L ) ] for τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , for any 1 ≤ ` ≤ L and i ∈ { 1 , . . . , n } , we have the event T , ∥∥∥∥∥ ( L∏ r= ` Σ̃r , iW̃r , i ) u ∥∥∥∥∥ 2 ≤ 3 ‖u‖2 . ( 21 ) Conditioned on this event happens , it is easy to see the random variable v > ( ∏L r=a Σ̃r , iW̃r , i ) u ∼ SG ( 9ω m ‖u‖ 2 2 ) . Based on this observation , we have the probability , P ( ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≥ √ ωs log ( m ) m · O ( ‖u‖2 ) ) ≤ P ( ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≥ √ ωs log ( m ) m · O ( ‖u‖2 ) ∣∣∣ T ) + P ( T c ) ≤ O ( nL ) · exp ( − Ω ( s log ( m ) ) ) +O ( nL ) · exp ( − Ω ( mτ2/3L ) ) , where in the last inequality , union bounds over the indices ` and i , and over the vector u ∈ Rm with ‖u‖0 ≤ s are taken . Lemma 4.6 . Suppose τ ≤ 1 CL9/2 [ log ( m ) ] 3 for some constant C > 0 . Then , for all i ∈ [ n ] and ` ∈ [ L ] , with probability at least 1−O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , we have ∥∥∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥2 ≤ O ( τ1/3L2√ωm log ( m ) ) . Proof . By using the results from Lemma 4.3 , we can control the term : ‖∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ‖2 = √ m · ∥∥∥∥∥x ( k ) ` −1vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` − x ( 0 ) ` −1v T ( L∏ r= ` +1 Σ ( 0 ) r W ( 0 ) r ) Σ ( 0 ) ` ∥∥∥∥∥ 2 ≤ √ m · ‖x ( k ) ` −1 − x ( 0 ) ` −1 ‖2︸ ︷︷ ︸ ≤O ( τL5/2 √ log ( m ) ) · ∥∥∥∥∥vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` ∥∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ ω ) + √ m · ∥∥∥x ( 0 ) ` −1∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) · ∥∥∥∥∥vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` − v T ( L∏ r= ` +1 Σ ( 0 ) r W ( 0 ) r ) Σ ( 0 ) ` ∥∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ω log ( m ) ) ≤ O ( τ1/3L2 √ ωm log ( m ) ) , where , in the last inequality , we used the condition on τ ≤ 1 CL9/2 [ log ( m ) ] 3 < 1 . Remark 4.7 . Note that the results in Lemmas 6.3 ( second and fifth items ) , 6.4 , 6.5 , 6.6 are in the setting of vj ∼ N ( 0 , ωm ) for j ∈ [ m ] . For the notational convenience , in following Lemmas we denote fW ( k ) ( xi ) as ui ( k ) and let u ( k ) : = [ u1 ( k ) , . . . , un ( k ) ] > for k ≥ 0 . Lemma 4.8 . For some δ ∈ [ 0 , 1 ] , if m ≥ Ω ( L log ( nL/δ ) ) , then with probability at least 1 − δ , ‖u ( k ) ‖2 ≤ O ( √nω δ ) for any k ≥ 0 . Proof . Recall the Lemma 4.2 stating that ∥∥∥x ( k ) L , i∥∥∥ 2 = O ( 1 ) for any input data xi for i ∈ [ n ] . Also recall that vj ∼ N ( 0 , ωm ) for j ∈ [ m ] , xL , i ∈ R m and ui ( k ) = √ mv > xL , i ∼ N ( 0 , O ( ω ) ) . Then , we have a following via simple Markov inequality : for any t ≥ 0 , P ( ‖u ( k ) ‖2 ≥ t ) ≤ E [ ‖u ( k ) ‖2 ] t ≤ √ E [ ‖u ( k ) ‖22 ] t ≤ O ( √ nω ) t . Lemma 4.9 . For some δ ∈ [ 0 , 1 ] , if m ≥ Ω ( L log ( nL/δ ) ) , then with probability at least 1− δ , we have ‖u ( 0 ) − y‖2 ≤ O ( √ n δ ) . Proof . By Markov ’ s inequality , for any t ≥ 0 , P ( ‖u ( 0 ) − y‖2 ≥ t ) ≤ Eε , W ( 0 ) , v [ ‖u ( 0 ) − y‖22 ] t2 . ( 22 ) Note that the expectation in the nominator of ( 22 ) is taken over the random noise ε and initialized parameter W ( 0 ) , v. We can expand the nominator as follows : Eε , W ( 0 ) , v [ ‖u ( 0 ) − y‖22 ] = EW ( 0 ) , v‖u ( 0 ) ‖22 + Eε‖y‖22 − 2Eε , W ( 0 ) , v [ y > u ( 0 ) ] . ( 23 ) For the convenience of notation , let y∗ : = [ f ? ρ ( x1 ) , . . . , f ? ρ ( xn ) ] > and ε : = [ ε1 , . . . , εn ] > . Recall that we have y = y∗ + ε , and ‖y∗‖22 = O ( n ) . Also note that by Lemma 4.1 , with probability at least 1−O ( nL ) · exp [ −Ω ( m/L ) ] , for any i = 1 , . . . , n , ‖x ( 0 ) L , i‖22 = O ( 1 ) . Then , we have a random variable ui ( 0 ) = √ mv > xL , i ∼ N ( 0 , O ( ω ) ) . Now , we are ready to derive the orders of three terms on the RHS of ( 23 ) . EW ( 0 ) , v‖u ( 0 ) ‖22 = O ( n ) , Eε‖y‖22 = Eε [ ‖y∗‖22 + ‖ε‖22 − 2y > ε ] = O ( n ) , Eε , W ( 0 ) , v [ y > u ( 0 ) ] = Eε , W ( 0 ) , v [ ( y∗ + ε ) > u ( 0 ) ] = 0 . Combining the above three equalities , we conclude the proof . Lemma 4.10 . Suppose τ = O ( n√ω√ mδλ0 ) . For some δ ∈ [ 0 , 1 ] such that δ ≥ O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , then with probability at least 1− δ , we have ‖H ( k ) −H ( 0 ) ‖2 ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) . Proof . By the definition of gram matrix Hi , j ( k ) for any k ≥ 0 , we have |Hi , j ( k ) −Hi , j ( 0 ) | = ∣∣∣∣∣ 1m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr − 〈 ∇W ` [ fW ( 0 ) ( xi ) ] , ∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ∣∣∣∣∣ ≤ 1 m L∑ ` =1 { ∣∣∣∣〈∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] −∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ∣∣∣∣ + ∣∣∣∣〈∇W ` [ fW ( 0 ) ( xj ) ] , ∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] 〉 Tr ∣∣∣∣ } ≤ 1 m L∑ ` =1 { ∥∥∇W ` [ fW ( k ) ( xi ) ] ∥∥2︸ ︷︷ ︸ ≤O ( √ ωm ) · ∥∥∇W ` [ fW ( k ) ( xj ) ] −∇W ` [ fW ( 0 ) ( xj ) ] ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) + ∥∥∇W ` [ fW ( 0 ) ( xj ) ] ∥∥2︸ ︷︷ ︸ ≤O ( √ ωm ) · ∥∥∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) } ≤ O ( ω7/6n1/3L3 6 √ log3 ( m ) mδλ20 ) . In the second inequality , Lemmas 4.4 and 4.6 are used , and in the last inequality , τ = O ( n√ω√ mδλ0 ) is plugged in . With this , using the fact that Frobenius norm of a matrix is bigger than the operator norm , we bound the term ‖H ( k ) −H ( 0 ) ‖2 as follows : ‖H ( k ) −H ( 0 ) ‖2 ≤ ‖H ( k ) −H ( 0 ) ‖F ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) . Lemma 4.11 . For some δ ∈ [ 0 , 1 ] , with probability at least 1− δ , ‖H∞L −H ( 0 ) ‖2 ≤ O ( ωnL5/2 4 √ log ( nL/δ ) m ) Proof . For some δ ′ ∈ [ 0 , 1 ] , set ε = L3/2 4 √ log ( L/δ ′ ) m from Theorem 3.1. of Arora et al . [ 2019b ] . For any fixed i , j ∈ [ n ] , we have P [ ∣∣H∞i , j −Hi , j ( 0 ) ∣∣ ≤ O ( ωL5/2 4√ log ( L/δ′ ) m ) ] ≥ 1− δ′ . After applying the union bound over all i , j ∈ [ n ] , setting δ = δ ′ n2 , and using the fact that Frobenius norm of a matrix is bigger than the operator norm , we conclude the proof . For two positive semi-definite matrices A and B , if we write A B , then it means A − B is positive semi-definite matrix . Similarly , if we write A B , then it means A−B is positive definite matrix . With these notations , we introduce a following Lemma . Lemma 4.12 ( Lemma D.6 . Hu et al . [ 2021 ] ) . For two positive semi-definite matrices A and B , 1 . Suppose A is non-singular , then A B ⇐⇒ λmax ( BA−1 ) ≤ 1 and A B ⇐⇒ λmax ( BA −1 ) < 1 , where λmax ( · ) denotes the maximum eigenvalue of the input matrix . 2 . Suppose A , B and Q are positive definite matrices , A and B are exchangeable , then A B =⇒ AQA BQB . E PROOF OF THEOREM 3.5 For the convenience of notation , denote ui ( k ) = fW ( k ) ( xi ) and let u ( k ) = [ u1 ( k ) , u2 ( k ) , . . . , un ( k ) ] > . In order to achieve linear convergence rate of the training error , ‖u ( k ) − y‖22 , we decompose the term as follows : ‖u ( k + 1 ) − y‖22 = ∥∥u ( k ) − y + ( u ( k + 1 ) − u ( k ) ) ∥∥2 2 = ‖u ( k ) − y‖22 − 2 ( u ( k ) − y ) > ( u ( k + 1 ) − u ( k ) ) + ‖u ( k + 1 ) − u ( k ) ‖22 . Equipped with this decomposition , the proof consists of the following steps : 1 . Similarly with Du et al . [ 2019 ] , a term ( u ( k + 1 ) − u ( k ) ) is decomposed into two terms , where we denote them as I ( k ) 1 and I ( k ) 2 , respectively . We note that the first term I ( k ) 1 is related with a gram matrix H ( k ) and a second term I ( k ) 2 can be controlled small enough in ` 2 sense with proper choices of the step size and the width of network . 2 . A term ‖u ( k + 1 ) − u ( k ) ‖22 needs to be controlled small enough to ensure 2 ( u ( k ) − y ) > ( u ( k + 1 ) − u ( k ) ) > ‖u ( k + 1 ) − u ( k ) ‖22 so that the loss decreases . 3 . It is shown that the distance between the gram matrix H ( k ) and the NTK matrix H∞L is close enough in terms of operator norm . 4 . Lastly , we inductively show that the weights generated from gradient descent stay within a perturbation region B ( W ( 0 ) , τ ) , irrespective with the number of iterations of algorithm , We start the proof by analyzing the term u ( k + 1 ) − u ( k ) . Step 1 . Control on u ( k + 1 ) − u ( k ) . Recall ( Σ ( k ) ` , i ) jj = 1 ( 〈w ( k ) ` , j , x ( k ) ` −1 , i〉 ≥ 0 ) and we introduce a diagonal matrix Σ̃ ( k ) ` , i , whose jth entry is defined as follows : ( Σ̃ ( k ) ` , i ) jj = ( Σ ( k+1 ) ` , i −Σ ( k ) ` , i ) jj · 〈w ( k+1 ) ` , j , x ( k+1 ) ` −1 , i 〉 〈w ( k+1 ) ` , j , x ( k+1 ) ` −1 , i 〉 − 〈w ( k ) ` , j , x ( k ) ` −1 , i〉 . With this notation , the difference x ( k+1 ) L , i − x ( k ) L , i can be rewritten via the recursive applications of Σ̃ ( k ) ` , i : x ( k+1 ) L , i − x ( k ) L , i = ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) ( W ( k+1 ) L x ( k+1 ) L−1 , i −W ( k ) L x ( k ) L−1 , i ) = ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) W ( k+1 ) L ( x ( k+1 ) L−1 , i − x ( k ) L−1 , i ) + ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) ( W ( k+1 ) L −W ( k ) L ) x ( k ) L−1 , i = L∑ ` =1 ( L∏ r= ` +1 ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) W ( k+1 ) r ) ( Σ ( k ) ` , i + Σ̃ ( k ) ` , i ) ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i ( 24 ) Then , we introduce following notations : D ( k ) ` , i = ( L∏ r= ` +1 Σ ( k ) r , i W ( k ) r ) Σ ( k ) ` , i , D̃ ( k ) ` , i = ( L∏ r= ` +1 ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) W ( k+1 ) r ) ( Σ ( k ) ` , i + Σ̃ ( k ) ` , i ) . Now , we can write ui ( k + 1 ) − ui ( k ) by noting that ui ( k ) = √ m · vTx ( k ) L , i : ui ( k + 1 ) − ui ( k ) = √ m · vT ( x ( k+1 ) L , i − x ( k ) L , i ) = √ m · vT L∑ ` =1 D̃ ( k ) ` , i ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i ( 25 ) = −η √ m · vT L∑ ` =1 D ( k ) ` , i ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i︸ ︷︷ ︸ I ( k ) 1 , i −η √ m · vT L∑ ` =1 ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i︸ ︷︷ ︸ I ( k ) 2 , i Here , I ( k ) 1 , i can be rewritten as follows : I ( k ) 1 , i = −η √ m · vT L∑ ` =1 D ( k ) ` , i n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] x ( k ) ` −1 , i = −η · n∑ j=1 ( uj ( k ) − yj ) · ( √ m L∑ ` =1 vTD ( k ) ` , i ∇W ` [ fW ( k ) ( xj ) ] x ( k ) ` −1 , i ) = −mη · n∑ j=1 ( uj ( k ) − yj ) · 1 m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr = −mη · n∑ j=1 ( uj ( k ) − yj ) ·Hi , j ( k ) . For I ( k ) 2 , i , we need a more careful control . First , we pay our attention on bounding the term ‖v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ‖2 as follows : By triangle inequality , we have∥∥∥∥v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∥∥∥∥ 2 ≤ ∥∥∥∥v > ( D ( k ) ` , i −D ( 0 ) ` , i ) ∥∥∥∥ 2 + ∥∥∥∥v > ( D̃ ( k ) ` , i −D ( 0 ) ` , i ) ∥∥∥∥ 2 . ( 26 ) We control the first term of the right-hand side ( R.H.S ) in ( 26 ) . By the fourth item of the Lemma 4.3 , we know ‖Σ ( k ) r , i −Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) and | ( Σ ( k ) r , i −Σ ( 0 ) r , i ) j , j | ≤ 1 for j ∈ [ m ] . Then , we can plug Σ′′r , i = Σ ( k ) r , i −Σ ( 0 ) r , i in the inequality of the fifth item of Lemma 4.3 . So , the first term of the R.H.S in ( 26 ) can be bounded by O ( τ1/3L2 √ ω log ( m ) ) . The second term of the R.H.S in ( 26 ) can be similarly controlled as the first term . Observe that | ( Σ ( k ) r , i +Σ̃ ( k ) r , i ) jj | ≤ 1 , then we have | ( Σ ( k ) r , i +Σ̃ ( k ) r , i −Σ ( 0 ) r , i ) j , j | ≤ 2 for all j ∈ [ m ] . Note that by the definition of Σ̃ ( k ) r , i , we have ‖Σ̃ ( k ) r , i ‖0 = ‖Σ ( k+1 ) r , i −Σ ( k ) r , i ‖0 ≤ ‖Σ ( k+1 ) r , i −Σ ( 0 ) r , i ‖0+‖Σ ( k ) r , i −Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) . Thus , by the triangle inequality , we have ‖Σ ( k ) r , i + Σ̃ ( k ) r , i − Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) . These observations enable us to plug Σ′′r , i = Σ ( k ) r , i + Σ̃ ( k ) r , i −Σ ( 0 ) r , i in the inequality of the fifth item of Lemma 4.3 , and give the bound on the second term as O ( τ1/3L2 √ ω log ( m ) ) . We have ‖v > ( D̃ ( k ) ` , i − D ( k ) ` , i ) ‖2 ≤ O ( τ1/3L2 √ ω log ( m ) ) . Now , we control the ` 2-norm of the I ( k ) 2 as follows : ∥∥∥I ( k ) 2 ∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 2 , i ∣∣∣ ≤ η √ m · n∑ i=1 [ L∑ ` =1 ∥∥∥∥v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( L2τ1/3 √ ω log ( m ) ) · ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 · ∥∥∥x ( k ) ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) : Lemma 4.2 ] ≤ O ( ηnL2τ1/3 √ ωm log ( m ) ) L∑ ` =1 ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 ≤ O ( ηnL5/2τ1/3 √ ωm log ( m ) ) √√√√ L∑ ` =1 ∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥2F ≤ O ( ηnL5/2τ1/3 √ ωm log ( m ) ) √√√√ n∑ j=1 ( uj ( k ) − yj ) 2 L∑ ` =1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥2F ≤ O ( ηnL3τ1/3ωm √ log ( m ) ) ‖u ( k ) − y‖2 . ( 27 ) Step 2 . Control on ‖u ( k + 1 ) −u ( k ) ‖22 . Recall that by ( 25 ) , x ( k+1 ) L , i −x ( k ) L , i can be written as follows : x ( k+1 ) L , i − x ( k ) L , i = L∑ ` =1 D̃ ( k ) ` , i ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i = −η · L∑ ` =1 D̃ ( k ) ` , i ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i . It is worth noting that , ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥2 2 = ∥∥∥∥∥∥ n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] ∥∥∥∥∥∥ 2 2 ≤ n∑ j=1 ( uj ( k ) − yj ) 2 n∑ j=1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥22 ≤ O ( nmω ) ‖u ( k ) − y‖22 . ( 28 ) Also , observe that | ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) jj | ≤ 1 for all j ∈ [ m ] , so by Lemma A.3 of Zou et al . [ 2020 ] , we know ‖D̃ ( k ) ` , i ‖2 ≤ O ( √ L ) . Combining all the facts , we can conclude : ‖u ( k + 1 ) − u ( k ) ‖22 = m · n∑ i=1 ( vTx ( k+1 ) L , i − v Tx ( k ) L , i ) 2 ≤ m · ‖v‖22 n∑ i=1 ∥∥∥x ( k+1 ) L , i − x ( k ) L , i∥∥∥2 2 ≤ η2m · ‖v‖22 n∑ i=1 [ L∑ ` =1 ∥∥∥D̃ ( k ) ` , i ∥∥∥2 2 · ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥2 2 · ∥∥∥x ( k ) ` −1 , i∥∥∥2 2 ] ≤ O ( η2n2L2m2ω2 ) ‖u ( k ) − y‖22 ≤ O ( η2n2L2m2 ) ‖u ( k ) − y‖22 , ( 29 ) where in the third inequality , we additionally used the fact ‖v‖22 = O ( ω ) with probability at least 1− exp ( −Ω ( m ) ) , and the inequality ( 28 ) . In the last inequality , we used the assumption ω ≤ 1 . Step 3. λmin ( H ( k ) ) ≥ λ02 with sufficiently large m. Denote ρ ( A ) as a sprectral radius of a matrix A . Then , we have ‖H ( k ) −H∞L ‖2 ≥ ρ ( H ( k ) −H∞L ) ≥ −λmin ( H ( k ) −H∞L ) ≥ λmin ( H∞L ) − λmin ( H ( k ) ) ≥ λ0 − λmin ( H ( k ) ) , ( 30 ) where , in the second inequality , we used a triangle inequality , λmin ( H ( k ) −H∞L ) + λmin ( H∞L ) ≤ λmin ( H ( k ) ) . By Lemmas 4.10 and 4.11 , setting m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ ) and Õ ( λ 4/3 0 δ 1/3 n4/3L4 ) ≤ ω ≤ 1 , we have ‖H ( k ) −H∞L ‖2 ≤ ‖H ( k ) −H ( 0 ) ‖2 + ‖H ( 0 ) −H ∞ L ‖2 ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) +O ( ωn2L5/2 4 √ log ( nL/δ ) m ) ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) ≤ λ0 2 . ( 31 ) Thus , combining ( 30 ) and ( 31 ) yields that λmin ( H ( k ) ) ≥ λ02 . Step 4 . Concluding the proof . Recall that I ( k ) 1 = −mη · H ( k ) ( u ( k ) − y ) . Then observe that ( u ( k ) − y ) > I ( k ) 1 = −ηm · ( u ( k ) − y ) > H ( k ) ( u ( k ) − y ) ≤ −ηm · λmin ( H ( k ) ) ‖u ( k ) − y‖22 ≤ −ηm · λ0 2 ‖u ( k ) − y‖22 . ( 32 ) We set the step size η , radius of perturbation region τ and network width m as follows , η = Ω ( λ0 n2L2m ) , τ = O ( n √ ω√ mδλ0 ) , m ≥ Ω ( ω7n8L18 log3 ( m ) λ80δ ) . With the above settings , we can control the ‖u ( k + 1 ) − y‖22 by combining ( 27 ) , ( 29 ) and ( 32 ) as follows , ‖u ( k + 1 ) − y‖22 = ∥∥u ( k ) − y + ( u ( k + 1 ) − u ( k ) ) ∥∥2 2 = ‖u ( k ) − y‖22 − 2ηm · ( u ( k ) − y ) > H ( k ) ( u ( k ) − y ) − ( u ( k ) − y ) > I ( k ) 2 + ‖u ( k + 1 ) − u ( k ) ‖ 2 2 ≤ ( 1− ηmλ0 +O ( ηnL3τ1/3mω √ log ( m ) ) +O ( η2n2L2m2 ) ) ‖u ( k ) − y‖22 ≤ ( 1− ηmλ0 2 ) ‖u ( k ) − y‖22 . So far , we have shown from Step 1 to Step 4 that given the radius of perturbation region τ has the order O ( n√ω√ mδλ0 ) , then we can show the training error drops linearly to 0 with the discount factor ( 1 − ηmλ02 ) along with the proper choices of η and m. It remains us to prove the iterates W ( k ) ` for all ` ∈ [ L ] generated by GD algorithm indeed stay in the perturbation region B ( W ( 0 ) , τ ) over k ≥ 0 with τ = O ( n√ω√ mδλ0 ) . Step 5 . The order of the radius of perturbation region . We employ the induction process for the proof . The induction hypothesis is : ∀s ∈ [ k + 1 ] , ∥∥∥W ( s ) ` −W ( 0 ) ` ∥∥∥ 2 ≤ η · O ( √ nmω ) s−1∑ t=0 ( 1− ηmλ0 2 ) t 2 O ( √ n δ ) ≤ O ( n √ ω√ mδλ0 ) . ( 33 ) First , it is easy to see it holds for s = 0 . Now , suppose it holds for s = 0 , . . . , k , we consider s = k + 1.∥∥∥W ( k+1 ) ` −W ( k ) ` ∥∥∥ 2 = ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 = η · ∥∥∥∥∥∥ n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] ∥∥∥∥∥∥ 2 ≤ η · √√√√ n∑ j=1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥22 √√√√ n∑ j=1 ( uj ( k ) − yj ) 2 ≤ η · O ( √ nmω ) √ 2LS ( W ( k ) ) ≤ η · O ( √ nmω ) ( 1− ηmλ0 2 ) k 2 O ( √ n δ ) , ( 34 ) where in the second inequality , we used Lemmas 4.4 . Note that since it is assumed that W ( k ) ` ∈ B ( W ( 0 ) , τ ) , the Lemma is applicable with m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ ) . Simi- larly , since it is assumed that the induction hypothesis holds for s = 0 , . . . , k , we can see ‖u ( k ) − y‖22 ≤ ( 1 − ηmλ02 ) k‖u ( 0 ) − y‖22 . This inequality is plugged in the last inequality with Lemma 4.9 . By combining the inequalities ( 33 ) for s ∈ [ k ] and ( 34 ) , and triangle inequality , we conclude the proof : ∥∥∥W ( k+1 ) ` −W ( 0 ) ` ∥∥∥ 2 ≤ η · O ( √ nmω ) k∑ t=0 ( 1− ηmλ0 2 ) t 2 O ( √ n δ ) ≤ O ( n √ ω√ mδλ0 ) . Proposition 5.1 . For some δ ∈ [ 0 , 1 ] , if we set the width of the network as m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ 2 ) , and we set the step-size of gradient descent as η = O ( λ0 n2L2m ) . Then , with probability at least 1 − δ over the randomness of initialized parameters W ( 0 ) , we have for k = 0 , 1 , 2 , . . . , u ( k ) − y = ( I− ηmH∞L ) k ( u ( 0 ) − y ) + ξ ( k ) , where ‖ξ ( k ) ‖2 = k ( 1− ηmλ0 2 ) k−1 O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖y − u ( 0 ) ‖2 . Proof . Define ui ( k ) : = fW ( k ) ( xi ) , then we have u ( k + 1 ) − u ( k ) = −ηm ·H ( k ) ( u ( k ) − y ) + I ( k ) 1 = −ηm ·H∞L ( u ( k ) − y ) − ηm · ( H ( k ) −H∞L ) ( u ( k ) − y ) + I ( k ) 1 = −ηm ·H∞L ( u ( k ) − y ) + e ( k ) . By recursively applying the above equality , we can easily derive a following for any k ≥ 0 , u ( k ) − y = ( I− ηmH∞L ) k ( u ( 0 ) − y ) + k−1∑ t=0 ( I− ηmH∞L ) t e ( k − 1− t ) ︸ ︷︷ ︸ =ξ ( k ) . ( 35 ) Now , we want to show ξ ( k ) can be controlled in arbitrarily small number . First , e ( k ) needs to be bounded in an ` 2 norm : ‖e ( k ) ‖2 ≤ ηm ‖H ∞ L −H ( k ) ‖2 ‖u ( k ) − y‖2 + ∥∥∥I ( k ) 2 ∥∥∥ 2 ≤ ηm · O ( ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖u ( k ) − y‖2 , where , in the second inequality , τ = O ( n √ ω√ mδλ0 ) is plugged in ( 27 ) . Equipped with the bound on ‖e ( k ) ‖2 , we can easily bound the ‖ξ ( k ) ‖2 as follows : ∥∥∥∥∥ k−1∑ t=0 ( I− ηmH∞L ) t e ( k − 1− t ) ∥∥∥∥∥ 2 ≤ k−1∑ t=0 ‖I− ηmH∞L ‖ t 2 ‖e ( k − 1− t ) ‖2 ≤ k−1∑ t=0 ( 1− ηmλ0 ) t O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖u ( k − 1− t ) − y‖2 ≤ k−1∑ t=0 ( 1− ηmλ0 ) t O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ( 1− ηmλ0 2 ) k−1−t ‖u ( 0 ) − y‖2 = k ( 1− ηmλ0 2 ) k−1 O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ 2 ) ‖u ( 0 ) − y‖2 . ( 36 ) Note that in the third inequality , we used the result from Theorem 1 . F PROOF OF THEOREM 3.8 We begin the proof by decomposing the error f̂W ( k ) ( x ) − f∗ ( x ) for any fixed x ∈ Unif ( Sd−1 ) into two terms as follows : f̂W ( k ) ( x ) − f∗ ( x ) = ( f̂W ( k ) ( x ) − g∗ ( x ) ) ︸ ︷︷ ︸ ∆1 + ( g∗ ( x ) − f∗ ( x ) ) ︸ ︷︷ ︸ ∆2 . ( 37 ) Here , we denote the solution of kernel regression with kernel H∞L as g ∗ ( x ) , which is a minimum RKHS norm interpolant of the noise-free data set { xi , f ? ρ ( xi ) } ni=1 . To avoid the confusion of the notation , we write Ker ( x , X ) = ( H∞L ( x , x1 ) , . . . , H ∞ L ( x , xn ) ) n i=1 ∈ Rn and let y∗ = [ f ? ρ ( x1 ) , . . . , f ? ρ ( xn ) ] > . Then , we have a following closed form solution g∗ ( x ) as , g∗ ( x ) : = Ker ( x , X ) ( H∞L ) −1 y∗ . With the decomposition ( 37 ) , the proof sketch of Theorem 3.8 is as follows . 1 . Note that for any ` ∈ [ L ] , we have f̂W ( k ) ( x ) = 〈vec ( ∇W ` [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ) 〉 . We can write the term vec ( W ( k ) ` ) with respect to vec ( W ( 0 ) ` ) , H∞L and the residual term via recursive applications of GD update rule and the result from proposition 2.1 . Readers can refer ( 38 ) . Using the equality ( 38 ) , we can further decompose ∆1 into three terms . That is , ∆1 = ∆11 + ∆12 + ∆13 . Then , using the boundedness of ` 2-norm of network gradient and the fact that the size of ‖ξ ( k ) ‖2 can be controlled with wide enough network , we can control the size of ‖∆12‖2 and ‖∆13‖2 aribtarily small . 2 . In the term ∆2 , the g ? is an interpolant based on noiseless data . For large enough data points , g ? converges fastly to f ? at the rate OP ( 1√n ) . 3 . Lastly , the ∆11 is the only term that is involved with random error ε , and we show that ‖∆11‖2 is bounded away from 0 for small and large GD iteration index k. Step 1 . Control on ∆1 . For n data points ( x1 , . . . , xn ) and for the kth updated parameter W ( k ) , denote : ∇W ` [ fW ( k ) ( X ) ] = [ vec ( ∇W ` [ fW ( k ) ( x1 ) ] ) , · · · , vec ( ∇W ` [ fW ( k ) ( xn ) ] ) ] . Note that when ` = 1 , ∇W ` [ fW ( k ) ( X ) ] ∈ Rmd×n and when ` = 2 , . . . , L , ∇W ` [ fW ( k ) ( X ) ] ∈ Rm2×n . With this notation , we can rewrite the Gradient Descent update rule as vec ( W ( k+1 ) ` ) = vec ( W ( k ) ` ) − η∇W ` [ fW ( k ) ( X ) ] ( u ( k ) − y ) , k ≥ 0 . Applying Proposition 3.8 , we can get : vec ( W ( k ) ` ) − vec ( W ( 0 ) ` ) = k−1∑ j=0 ( vec ( W ( j+1 ) ` ) − vec ( W ( j ) ` ) ) = −η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ( u ( j ) − y ) = η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) − η · k−1∑ j=0 ∇W ` [ fW ( k ) ( X ) ] ξ ( j ) = η · k−1∑ j=0 ∇W ` [ fW ( 0 ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) − η · k−1∑ j=0 ∇W ` [ fW ( k ) ( X ) ] ξ ( j ) + η · k−1∑ j=0 ( [ ∇W ` [ fW ( j ) ( X ) ] −∇W ` [ fW ( 0 ) ( X ) ] ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) ) = η · k−1∑ j=0 ∇W ` [ fW ( 0 ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) + ξ ′ ( k ) . ( 38 ) First , we control ` 2-norm of the first term of ξ ′ ( k ) as follows : Note that ‖∇W ` [ fW ( j ) ( X ) ] ‖F ≤ O ( √ nmω ) by Lemma 4.4 for 0 ≤ j ≤ k − 1 . Then , we have∥∥∥∥∥∥η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ξ ( j ) ∥∥∥∥∥∥ 2 ≤ k−1∑ j=0 O ( η √ nmω ) O ( j ( 1− ηmλ0 2 ) j−1 ) O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖y − u ( 0 ) ‖2 ≤ O ( n11/6L3ω5/3 m2/3λ 7/3 0 δ 1/6 √ log ( m ) ) ‖y − u ( 0 ) ‖2 . ( 39 ) In the second inequality , ∑∞ j=1 j ( 1− ηmλ02 ) j = O ( 1 η2m2λ20 ) is used . Then , we control ` 2-norm of the second term of ξ ′ ( k ) as follows : ∥∥∥∥∥∥η · k−1∑ j=0 [ ∇W ` [ fW ( j ) ( X ) ] −∇W ` [ fW ( 0 ) ( X ) ] ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) ∥∥∥∥∥∥ 2 ≤ k−1∑ j=0 η ‖I− ηmH∞L ‖ j 2 ‖y − u ( 0 ) ‖2 √√√√ n∑ i=1 ∥∥∇W ` [ fW ( j ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥22 ≤ k−1∑ j=0 η ( 1− ηmλ0 ) jO ( n1/3m1/3L2ω2/3 λ 1/3 0 δ 1/6 √ log ( m ) ) O ( √ n ) ‖y − u ( 0 ) ‖2 ≤ O ( n5/6L2ω2/3 m2/3λ 4/3 0 δ 1/6 √ log ( m ) ) ‖y − u ( 0 ) ‖2 , ( 40 ) where in the second inequality , we used Lemmas 4.6 with τ = O ( n√ω√ mδλ0 ) . Now , we are ready to control ∆1 term . By using the equality ( 38 ) , we can decompose the term ∆1 as follows : Let us denote Gk = ∑k−1 j=0 ηm ( I − ηmH∞L ) j . Note that for any ` ∈ [ L ] , f̂W ( k ) ( x ) = 〈vec ( ∇W ` [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ) 〉 and recall that y = y∗ + ε . Then , for any fixed ` ′ ∈ [ L ] , we have : ∆1 = [ 〈 vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ′ ) 〉 −Ker ( x , X ) ( H∞L ) −1 y∗ ] + Ker ( x , X ) Gky −Ker ( x , X ) Gky = [ Ker ( x , X ) [ Gk − ( H∞L ) −1 ] y∗ + Ker ( x , X ) Gkε ] ︸ ︷︷ ︸ =∆11 + [ 1 m L∑ ` =1 vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] −Ker ( x , X ) ] Gky − 1 m ∑ ` : ` 6= ` ′ vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] Gky︸ ︷︷ ︸ =∆12 + [ 〈 vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( 0 ) ` ′ ) 〉 + vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ξ ′ ( k ) − 1 m vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ] .︸ ︷︷ ︸ =∆13 ( 41 ) Our goal in this step is to control ‖∆12‖2 and ‖∆13‖2 . Then , in the third step , we will show ‖∆11‖2 is the term , which governs the behavior of the prediction risk with respect to algorithm iteration k. First , we bound the ` 2 norm of the first term in ∆12 as : ∥∥∥∥∥ [ 1 m L∑ ` =1 vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] −Ker ( x , X ) ] Gky ∥∥∥∥∥ 2 ≤ 1 mL L∑ ` =1 ∥∥vec ( ∇W ` [ fW ( k ) ( x ) ] ) − vec ( ∇W ` [ fW ( 0 ) ( x ) ] ) ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) : Lemma 4.6 ∥∥∇W ` [ fW ( 0 ) ( X ) ] ∥∥F︸ ︷︷ ︸ ≤O ( √ ωnm ) : Lemma 4.4 ‖Gky‖2 + 1 L √√√√ n∑ i=1 ( 1 m L∑ ` =1 〈 ∇W ` [ fW ( 0 ) ( x ) ] , ∇W ` [ fW ( 0 ) ( xi ) ] 〉 Tr −Ker ( x , xi ) ) 2 ‖Gky‖2 ≤ { O ( n5/6L2ω7/6 m1/6δ1/6λ 1/3 0 √ log ( m ) ) +O ( ωn1/2L3/2 4 √ log ( nL/δ ) m ) } ‖Gk‖2 ‖y‖2 ≤ O ( n5/6L2ω7/6 m1/6δ1/6λ 4/3 0 √ log ( m ) · ‖y‖2 ) +O ( ωn1/2L3/2 λ0 4 √ log ( nL/δ ) m · ‖y‖2 ) , ( 42 ) where , in the second inequality , we plugged τ = O ( n√ω√ mδλ0 ) in the result of Lemma 4.6 and used Lemma 4.11 . In the last inequality , we used ‖Gk‖2 ≤ O ( 1 λ0 ) . Similarly , we can control the ` 2 norm of the second term in ∆12 as follows : ∥∥∥∥∥∥ 1m ∑ ` : ` 6= ` ′ vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] Gky ∥∥∥∥∥∥ 2 ≤ 1 m ∑ ` : ` 6= ` ′ ‖vec ( ∇W ` [ fW ( k ) ( x ) ] ) ‖2︸ ︷︷ ︸ ≤O ( √ ωm ) · ‖∇W ` [ fW ( 0 ) ( X ) ] ‖F︸ ︷︷ ︸ ≤O ( √ ωmn ) · ‖Gk‖2︸ ︷︷ ︸ ≤O ( 1 λ0 ) ‖y‖2 ≤ O ( ωL √ n λ0 ) · ‖y‖2 . ( 43 ) We turn our attention to controlling ‖∆13‖2 . The first term in ∆13 ; Recall that∥∥∥vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) ∥∥∥ 2 ≤ O ( √ mω ) by Lemma 4.4 . Then , the random variable vec ( ∇W ` [ fW ( k ) ( x ) ] ) > vec ( W ( 0 ) ` ) is simply a N ( 0 , O ( ω ) ) for 1 ≤ ` ≤ L. A straightforward application of Chernoff bound for normal random variable and taking union bound over the layer 1 ≤ ` ≤ L yield that : with probability at least 1− δ , ∣∣∣vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > vec ( W ( 0 ) ` ′ ) ∣∣∣ ≤ O ( √ω log ( L δ ) ) . ( 44 ) The ` 2 norm of the third term in ∆13 can be bounded as follows : ∥∥∥∥ 1mvec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ∥∥∥∥ 2 ≤ 1 m ∥∥∥vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) ∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ mω ) ∥∥∥∇W ` ′ [ fW ( 0 ) ( X ) ] ∥∥∥ F︸ ︷︷ ︸ ≤O ( √ ωmn ) ‖Gku ( 0 ) ‖2︸ ︷︷ ︸≤O ( √nωλ0δ ) ≤ O ( nω3/2 λ0δ ) . ( 45 ) In the last inequality , we used the Lemma 4.8 and ‖Gk‖2 ≤ O ( 1 λ0 ) . By combin- ing ( 39 ) , ( 40 ) , ( 44 ) , ( 45 ) with ∥∥∥∇W ` ′ [ fW ( 0 ) ( x ) ] ∥∥∥ F ≤ O ( √ mω ) , we have a following : ‖∆13‖2 ≤ ∥∥∥〈vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( 0 ) ` ′ ) 〉∥∥∥ 2 + ∥∥∥ ( vec ( ∇W ` ′ [ fW ( 0 ) ( x ) ] ) ) > ξ′ ( k ) ∥∥∥ 2 + ∥∥∥∥ 1m ( vec ( ∇W ` ′ [ fW ( 0 ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ∥∥∥∥ 2 ≤ O ( √ ω log ( L δ ) ) +O ( n11/6L3ω13/6 ‖y − u ( 0 ) ‖2 m1/6λ 4/3 0 δ 1/6 √ log ( m ) ) +O ( n5/6L2ω7/6 ‖y − u ( 0 ) ‖2 m1/6λ 7/3 0 δ 1/6 √ log ( m ) ) +O ( nω3/2 λ0δ ) = O ( n11/6L3ω13/6 ‖y − u ( 0 ) ‖2 m1/6λ 4/3 0 δ 1/6 √ log ( m ) ) +O ( n5/6L2ω7/6 ‖y − u ( 0 ) ‖2 m1/6λ 7/3 0 δ 1/6 √ log ( m ) ) +O ( nω3/2 λ0δ ) . ( 46 ) Step 2 . Control on ∆2 . First , note that there is a recent finding that the reproducing kernel Hilbert spaces of NTKs with any number of layers ( i.e. , L ≥ 1 ) have the same set of functions , if kernels are defined on Sd−1 . See Chen & Xu [ 2020 ] . Along with this result , we can apply the proof used in Lemma.D.2 . in Hu et al . [ 2021 ] for proving a following : ‖∆2‖2 = OP ( 1√ n ) . ( 47 ) Step 3 . The behavior of L2 risk is characterized by the term ∆11 . Recall the decompositions ( 37 ) and ( 41 ) , then we have : f̂W ( k ) ( x ) − f∗ ( x ) = ∆11 + ( ∆12 + ∆13 + ∆2 ) : = ∆11 + Θ . ( 48 ) Our goal in this step is mainly two-folded : ( i ) Control Eε ‖Θ‖22 arbitrarily small with proper choices of step-size of GD η and width of the network m. ( ii ) Show that how Eε ‖∆11‖22 affect the behavior of prediction risk over the GD iterations k. First , note that we have Eε ‖y‖22 = Eε ‖y ∗ + ε‖22 ≤ 2 ( y∗ ) > y∗ + 2Eε ‖ε‖22 = O ( n ) . ( 49 ) Second , recall Lemma 4.9 and note that over the random initialization , with probability at least 1−δ , Eε ‖y − u ( 0 ) ‖22 ≤ O ( n δ ) . ( 50 ) Now , by combining the bounds ( 42 ) , ( 46 ) and ( 47 ) , we have Eε ‖Θ‖22 ≤ 3Eε ( ‖∆12‖22 + ‖∆13‖ 2 2 + ‖∆2‖ 2 2 ) ≤ Eε [ O ( n5/3L4ω7/3 m1/3λ 8/3 0 δ 1/3 log ( m ) · ‖y‖22 ) +O ( ω2nL3 λ20 √ log ( nL/δ ) m · ‖y‖22 ) +O ( n11/3L6ω13/3 ‖y − u ( 0 ) ‖22 m1/3λ 8/3 0 δ 1/3 log ( m ) ) +O ( n5/3L4ω7/3 ‖y − u ( 0 ) ‖22 m1/3λ 14/3 0 δ 1/3 log ( m ) ) ] +O ( n2ω3 λ20δ 2 ) +O ( 1 n ) ≤ O ( ω2n2L3 λ20 √ log ( nL/δ ) m ) +O ( n14/3L6ω13/3 m1/3λ 8/3 0 δ 4/3 log ( m ) ) +O ( n8/3L4ω7/3 m1/3λ 14/3 0 δ 4/3 log ( m ) ) +O ( n2ω3 λ20δ 2 ) +O ( 1 n ) , ( 51 ) where in the third inequality , we used ( 49 ) and ( 50 ) . Case 1 . When k is large , the L2 risk is bounded away from zero by some constant . Now we control Eε ‖∆11‖22 . Recall the definitions ‖f‖ 2 2 : = ∫ x∈Sd−1 |f ( x ) | 2dx and Gk = ∑k−1 j=0 ηm ( I− ηmH∞L ) j . Let us denote S = y∗y∗ > . Then , we have Eε ‖∆11‖22 = ∫ x∈Sd−1 Ker ( x , X ) [ ( Gk − ( H∞L ) −1 ) y∗y∗ > ( Gk − ( H∞L ) −1 ) +G2k ] Ker ( X , x ) dx = ∫ x∈Sd−1 Ker ( x , X ) ( H∞L ) −1 Mk ( H∞L ) −1Ker ( X , x ) dx where Mk = ( I− ηmH∞L ) k S ( I− ηmH∞L ) k + ( I− ( I− ηmH∞L ) k ) 2 = [ ( I− ηmH∞L ) k − ( S + I ) −1 ] ( S + I ) [ ( I− ηmH∞L ) k − ( S + I ) −1 ] + I − ( S + I ) −1 . For the algorithm iterations k ≥ ( log ( n ) ηmλ0 ) C0 with some constant C0 > 1 , we have ( I − ηmH∞L ) k ( 1− ηmλ0 ) k · I exp ( −ηmλ0k ) · I exp ( −C0 log ( n ) ) = 1 nC0 · I . Since 1 + ‖y‖22 ≤ C1n for some constant C1 , we have λmax ( 1 nC0 · ( S + I ) ) = 1 + ‖y‖22 nC0 ≤ C1 nC0−1 < 1 . ( 52 ) Using the first item of Lemma ( 4.12 ) with the inequality ( 52 ) , we have ( I − ηmH∞L ) k 1 nC0 · I ≺ ( S + I ) −1 . ( 53 ) The above inequality ( 53 ) lead to a following result : ( S + I ) −1 − ( I − ηmH∞L ) k ( S + I ) −1 − 1nC0 · I . ( 54 ) It is obvious that both ( S + I ) −1 − ( I − ηmH∞L ) k and ( S + I ) −1 − 1nC0 · I are positive definite matrices due to ( 54 ) , and it is also easy to see that they are exchangeable . By using the second item of Lemma ( 4.12 ) , we have Mk = [ ( I− ηmH∞L ) k − ( S + I ) −1 ] ( S + I ) [ ( I− ηmH∞L ) k − ( S + I ) −1 ] + I − ( S + I ) −1 [ ( S + I ) −1 − 1 nC0 · I ] ( S + I ) [ ( S + I ) −1 − 1 nC0 · I ] + I − ( S + I ) −1 = 1 n2C0 S + ( 1− 1 nC0 ) 2 · I . Then , we have Eε ‖∆11‖22 1 n2C0 A+ ( 1− 1 nC0 ) 2 B c0B , where c0 ∈ ( 0 , 1 ) is a constant and A = ∫ x∈Sd−1 [ Ker ( x , X ) ( H∞L ) −1 y∗ ] 2 dx , and B = ∫ x∈Sd−1 [ Ker ( x , X ) ( H∞L ) −1 ] 2 dx . ( 55 ) By triangle inequality with the decomposition ( 48 ) and the bound on Eε ‖Θ‖22 in ( 51 ) , we have : Eε ∥∥∥f̂W ( k ) − f∗∥∥∥2 2 = Eε ‖∆11 + Θ‖22 ≥ 1 2 Eε ‖∆11‖2 − Eε ‖Θ‖ 2 2 ≥ c0 2 B −O ( 1 n ) −O ( n2ω3 λ20δ 2 ) − Õ ( 1 m1/3 poly ( ω , n , L , 1 λ0 , 1 δ ) ) . ( 56 ) For the third term in ( 56 ) , we can choose ω ≤ C2 ( λ0δ n ) 2/3 for some constant C2 > 0 such that the term can be bounded by c08 ∥∥∥Ker ( · , X ) ( H∞L ) −1∥∥∥2 2 . Similarly , the width m can be chosen large enough such that the fourth term in ( 56 ) is upper-bounded by c08 ‖Ker ( · , X ) ( H∞L ) −1‖22 . Using the above choices of k , ω , and m , we can further bound ( 56 ) : Eε ∥∥fW ( k ) − f∗∥∥22 ≥ c04 ∥∥∥Ker ( · , X ) ( H∞L ) −1∥∥∥22 −O ( 1 n ) . ( 57 ) Note that Eε‖f̂∞ − g∗‖22 = ‖Ker ( · , X ) ( H∞L ) −1‖22 where g∗ : = 0 and f̂∞ denotes the noise interpolator . Then , by Theorem 4.2. of Hu et al . [ 2021 ] , we know that Eε‖f̂∞−g∗‖22 ≥ c1 for some constant c1 > 0 . Then , we can take n large enough such that the term O ( 1 n ) is upper-bounded by c0c1 8 , and finish the proof . Case 2 . When k is small , the L2 risk is bounded away from zero by some constant . Recall the definition of ∆11 in the decomposition ( 41 ) , ∆11 : = Ker ( x , X ) Gk [ y∗ + ε ] −Ker ( x , X ) H∞L y∗ : = ∆∗11 −Ker ( x , X ) H∞L y∗ . ( 58 ) We denote the eigen-decomposition of the matrix H∞L : = ∑n i=1 λivivi > , then we can easily see a following : Gk = ηm k−1∑ j=0 ( n∑ i=1 ( 1− ηmλi ) jvivi > ) ηm k−1∑ j=0 n∑ i=1 vivi > ηmk · I . By using the above inequality , we have Eε ‖∆∗11‖ 2 2 = ∫ x∈Sd−1 Ker ( x , X ) Gk ( S + I ) GkKer ( X , x ) dx ≤ η2m2k2 ( ∫ x∈Sd−1 [ Ker ( x , X ) y∗ ] 2 dx+ ‖Ker ( · , X ) ‖22 ) = O ( η2m2k2ω2n2L2 ) . Recall the decompositions ( 37 ) and ( 41 ) , then we have : Eε ∥∥∥f̂W ( k ) − f∗∥∥∥2 2 = Eε ‖∆∗11 + Θ−Ker ( · , X ) H∞L y∗‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 − Eε ‖∆ ∗ 11 + Θ‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 − 2Eε ‖∆ ∗ 11‖ 2 2 − 2Eε ‖Θ‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 −O ( η2m2k2ω2n2L2 ) −O ( 1 n ) −O ( n2ω3 λ20δ 2 ) − Õ ( 1 m1/3 poly ( ω , n , L , 1 λ0 , 1 δ ) ) . ( 59 ) For some constant C ′ 1 > 0 , let k ≤ C ′ 1 ( 1 ηmnωL ) such that the second term in the bound ( 59 ) can be bounded by 18‖Ker ( · , X ) ( H∞L ) −1 y∗‖22 . For the fourth term in ( 59 ) , we can choose ω ≤ C ′ 2 ( λ0δ n ) 2/3 for some constant C ′ 2 > 0 such that the term can be bounded by 1 8 ∥∥∥Ker ( · , X ) ( H∞L ) −1y∗∥∥∥2 2 . Sim- ilarly , the width m can be chosen large enough such that the fifth term in ( 59 ) is upper-bounded by 1 8‖Ker ( · , X ) ( H∞L ) −1 y∗‖22 . Using the above choices of k , ω , and m , we can further bound ( 59 ) : Eε ∥∥fW ( k ) − f∗∥∥22 ≥ 14 ∥∥∥Ker ( · , X ) ( H∞L ) −1y∗∥∥∥22 −O ( 1 n ) ≥ C ′ 3 ∥∥f ? ρ∥∥22 −O ( 1n ) . ( 60 ) In the second inequality , we used ( 47 ) with triangle inequality . In the third inequality , we can take n large enough such that the term O ( 1 n ) is upper-bounded by C ′ 3 2 ∥∥f ? ρ∥∥22 . Lastly , by using the assumption that f ? ρ is a square-integrable function , we finish the proof . G PROOF OF THEOREM 3.11-TRAINING ERROR For the convenience of notation , we denote uD , i ( k ) = fW ( k ) D ( xi ) and let uD ( k ) = [ u1 , D ( k ) , . . . , un , D ( k ) ] > . In order to analyze the training error of ` 2-regularized estimator , ‖uD ( k ) − y‖22 , we decompose the term as follows : ‖uD ( k + 1 ) − y‖22 = ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖ 2 2 + ‖ ( 1− η2µL ) uD ( k ) − y‖ 2 2 − 2 ( y − ( 1− η2µL ) uD ( k ) ) > ( uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ) ( 61 ) Equipped with this decomposition , the proof consists of the following steps : 1 . We decompose the decayed prediction difference uD ( k+ 1 ) − ( 1− η2µL ) uD ( k ) into two terms . We note that the first term is related with a gram matrix HD ( k ) and denote a second term as I ( k ) D . 2 . The term I ( k ) D can be further decomposed into three terms , where we denote them as I ( k ) 2 , D , I ( k ) 3 , D and I ( k ) 5 , D . The crux for controlling the ` 2-norm of the above three terms is to utilize the results from the Appendix A.4 . The applications of Lemmas in the Appendix A.4 is possible , since we can inductively guarantee that ‖W ( k ) D , ` − W ( 0 ) D , ` ‖2 is sufficiently small enough for large enough m. 3 . Given the decomposition ( 61 ) , we further decompose it into four terms as follows : ( 61 ) = ‖ ( 1− η2µL ) uD ( k ) − y‖22︸ ︷︷ ︸ : =T1 + ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖22︸ ︷︷ ︸ : =T2 + 2mη1 ( y − ( 1− η2µL ) uD ( k ) ) > HD ( k ) ( uD ( k ) − y ) ︸ ︷︷ ︸ : =T3 −2 ( y − ( 1− η2µL ) uD ( k ) ) > I ( k ) D︸ ︷︷ ︸ : =T4 . ( 62 ) In this step , we obtain the upper-bound of ‖Ti‖2 for i = 1 , 2 , 3 , 4 obtained in Step 4 . 4 . We combine the upper-bounds of ‖Ti‖2 for i = 1 , 2 , 3 , 4 in step 3 and obtain the bound on ‖uD ( k + 1 ) − y‖22 with respect to ‖uD ( k ) − y‖22 and ‖y‖2 . 5 . Lastly , we inductively show that the weights generated from regularized gradient descent stay within a perturbation region B ( W ( 0 ) , τ ) , irrespective with the number of iterations of algorithm . We start the proof by analyzing the term u ( k + 1 ) − ( 1− η2µL ) u ( k ) . Step 1 . Dynamics of uD ( k + 1 ) − ( 1−η2µL ) ·uD ( k ) . Recall ( Σ ( k ) D , ` , i ) jj = 1 ( 〈w ( k ) D , ` , j , x ( k ) D , ` −1 , i〉 ≥ 0 ) and we introduce a diagonal matrix Σ̃ ( k ) D , ` , i , whose jth entry is defined as follows : ( Σ̃ ( k ) D , ` , i ) jj = ( Σ ( k+1 ) D , ` , i −Σ ( k ) D , ` , i ) jj · 〈w ( k+1 ) D , ` , j , x ( k+1 ) D , ` −1 , i〉 〈w ( k+1 ) D , ` , j , x ( k+1 ) D , ` −1 , i〉 − 〈w ( k ) D , ` , j , x ( k ) D , ` −1 , i〉 . With this notation , the difference x ( k+1 ) D , L , i − x ( k ) D , L , i can be rewritten via the recursive applications of Σ̃ ( k ) D , ` , i : Then , we introduce following notations : D ( k ) D , ` , i = ( L∏ r= ` +1 Σ ( k ) D , r , iW ( k ) D , r ) Σ ( k ) D , ` , i , D̃ ( k ) D , ` , i = ( L∏ r= ` +1 ( Σ ( k ) D , r , i + Σ̃ ( k ) D , r , i ) W ( k+1 ) D , r ) ( Σ ( k ) D , ` , i + Σ̃ ( k ) D , ` , i ) . Now , we can write uD , i ( k + 1 ) − uD , i ( k ) by noting that uD , i ( k ) = √ m · vTx ( k ) D , L , i : uD , i ( k + 1 ) − uD , i ( k ) = √ m · vT ( x ( k+1 ) D , L , i − x ( k ) D , L , i ) = √ m · vT L∑ ` =1 D̃ ( k ) D , ` , i ( W ( k+1 ) D , ` −W ( k ) D , ` ) x ( k ) D , ` −1 , i = √ m · vT L∑ ` =1 D̃ ( k ) D , ` , i ( − η1∇W ` [ LS ( W ( k ) D ) ] − η2µW ( k ) D , ` + η2µW ( 0 ) D , ` ) x ( k ) D , ` −1 , i = −η1 √ m · vT L∑ ` =1 D ( k ) D , ` , i∇W ` [ LS ( W ( k ) D ) ] x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 1 , D , i −η1 √ m · vT L∑ ` =1 ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ∇W ` [ LS ( W ( k ) D ) ] x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 2 , D , i −η2µ √ m · vT L∑ ` =1 ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ( W ( k ) D , ` −W ( 0 ) D , ` ) x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 3 , D , i −η2µ √ m · vT L∑ ` =1 D ( k ) D , ` , iW ( k ) D , ` x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 4 , D , i +η2µ √ m · vT L∑ ` =1 D ( k ) D , ` , iW ( 0 ) D , ` x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 5 , D , i ( 63 ) where in the second equality , we used the recursive relation ( 24 ) , and in the third equality , modified GD update rule ( 6 ) is applied . Furthermore , I ( k ) 1 , D , i can be rewritten as follows : I ( k ) 1 , D , i = −η1 √ m · vT L∑ ` =1 D ( k ) D , ` , i n∑ j=1 ( uD , j ( k ) − yj ) ∇W ` [ f W ( k ) D ( xj ) ] x ( k ) D , ` −1 , i = −η1 · n∑ j=1 ( uD , j ( k ) − yj ) · ( √ m L∑ ` =1 vTD ( k ) D , ` , i∇W ` [ f W ( k ) D ( xj ) ] x ( k ) D , ` −1 , i ) = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) · 1 m L∑ ` =1 〈 ∇W ` [ f W ( k ) D ( xi ) ] , ∇W ` [ f W ( k ) D ( xj ) ] 〉 Tr = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) ·HD , i , j ( k ) . ( 64 ) With I ( k ) 4 , i = ( −η2µL ) · uD , i ( k ) and ( 64 ) , we can rewrite ( 63 ) as follows : uD , i ( k + 1 ) − ( 1− η2µL ) uD , i ( k ) = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) ·HD , i , j ( k ) + I ( k ) 2 , D , i + I ( k ) 3 , D , i + I ( k ) 5 , D , i . ( 65 ) Step 2 . Control of the size ∥∥∥I ( k ) D ∥∥∥ 2 . Let I ( k ) D = [ I ( k ) 2 , D,1 + I ( k ) 3 , D,1 + I ( k ) 5 , D,1 , . . . , I ( k ) 2 , D , n + I ( k ) 3 , D , n + I ( k ) 5 , D,1 ] > . Now , we control the bound on the ∥∥∥I ( k ) D ∥∥∥2 2 . Recall that in Eq . ( 27 ) , we have ∥∥∥I ( k ) 2 , D∥∥∥ 2 ≤ O ( η1nL 3τ1/3ωm √ log ( m ) ) ‖uD ( k ) − y‖2 . ( 66 ) Similarly , ∥∥∥I ( k ) 3 , D∥∥∥ 2 can be bounded : ∥∥∥I ( k ) 3 , D∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 3 , D , i∣∣∣ ≤ η2µ√m · n∑ i=1 [ L∑ ` =1 ∥∥∥∥v > ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( L2τ1/3 √ ω log ( m ) ) · ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2︸ ︷︷ ︸ ≤τ · ∥∥∥x ( k ) D , ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) ] ≤ O ( η2µnL 3τ4/3 √ ωm log ( m ) ) . ( 67 ) Lastly ∥∥∥I ( k ) 5 , D∥∥∥ 2 can be bounded : ∥∥∥I ( k ) 5 , D∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 5 , D , i∣∣∣ ≤ n∑ i=1 ∣∣∣∣∣η2µ√m · vT L∑ ` =1 D ( k ) D , ` , iW ( k ) D , ` x ( k ) D , ` −1 , i ∣∣∣∣∣+ n∑ i=1 ∣∣∣∣∣η2µ√m · vT L∑ ` =1 D ( k ) D , ` , i ( W ( k ) D , ` −W ( 0 ) D , ` ) x ( k ) D , ` −1 , i ∣∣∣∣∣ ≤ η2µL · n∑ i=1 |ui , D ( k ) |+ η2µ √ m · n∑ i=1 [ L∑ ` =1 ‖v‖2︸ ︷︷ ︸ ≤O ( √ ω ) · ∥∥∥D ( k ) D , ` , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ L ) · ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2︸ ︷︷ ︸ ≤τ · ∥∥∥x ( k ) D , ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) ] ≤ O ( η2µnL √ ω log ( L/δ ) ) +O ( η2µnL 3/2τ √ mω ) , ( 68 ) where in the last inequality , we employed the same logic used in ( 44 ) with the Lemma 4.2 to obtain the upper-bound on the |ui , D ( k ) | . We set the orders of the parameters µ , η1 , η2 , τ , and ω as follows : µ = Θ ( n d−1 2d−1 ) , η1 = Θ ( 1 m n− 3d−2 2d−1 ) , η2 = Θ ( 1 L n− 3d−2 2d−1 ) , τ = O ( L √ ω√ mδ n d 2d−1 ) , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) . ( 69 ) Plugging the choices of parameters ( 69 ) with sufficiently large m in ( 66 ) , ( 67 ) and ( 68 ) yields ∥∥∥I ( k ) D ∥∥∥ 2 ≤ O ( L37/12n− 9d−8 12d−6 √ log ( m ) m1/6δ1/3 ) · ‖uD ( k ) − y‖2 +OP ( 1 n2 ) . ( 70 ) Step 3 . Upper-bound of ‖Ti‖2 on i = 1 , 2 , 3 , 4 . First , we work on getting the upper-bound on λmax ( HD ( k ) ) . By the Gershgorin ’ s circle theorem [ Varga , 2004 ] , we know the maximum eigenvalue of symmetric positive semi-definite matrix is upper-bounded by the maximum absolute column sum of the matrix . Using this fact , we can bound the λmax ( HD ( k ) ) as : λmax ( HD ( k ) ) ≤ max i=1 , ... , n n∑ j=1 |HD , i , j ( k ) | ≤ max i=1 , ... , n n∑ j=1 ∣∣∣∣∣ 1m L∑ ` =1 〈 ∇W ` [ f W ( k ) D ( xi ) ] , ∇W ` [ f W ( k ) D ( xj ) ] 〉 Tr ∣∣∣∣∣ ≤ max i=1 , ... , n n∑ j=1 1 m L∑ ` =1 ∥∥∥∇W ` [ fW ( k ) D ( xi ) ] ∥∥∥F︸ ︷︷ ︸ ≤O ( √ mω ) ∥∥∥∇W ` [ fW ( k ) D ( xj ) ] ∥∥∥F︸ ︷︷ ︸ ≤O ( √ mω ) ≤ O ( nLω ) . ( 71 ) Recall the decomposition ( 62 ) . Our goal is to obtain the upper-bound on Ti for i = 1 , 2 , 3 , 4 . Control on T1 . By using the inequality 2η2µL ( 1−η2µL ) y > ( y−uD ( k ) ) ≤ η2µL ‖y‖22+η2µL ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 , we have ‖y − ( 1− η2µL ) uD ( k ) ‖22 = ∥∥ ( 1− η2µL ) ( y − uD ( k ) ) + η2µLy∥∥22 = ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 + η 2 2µ 2L2 ‖y‖22 + 2η2µL ( 1− η2µL ) y > ( y − uD ( k ) ) ≤ ( η2µL+ η22µ2L2 ) ‖y‖ 2 2 + ( 1 + η2µL ) ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 . ( 72 ) Control on T2 . Recall the equality ( 65 ) . Then , through applications of the Young ’ s inequality ‖a+ b‖22 ≤ 2 ‖a‖ 2 2 + 2 ‖b‖ 2 2 for a , b ∈ Rn , we have ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖22 = ∥∥∥−mη1 ·HD ( k ) ( uD ( k ) − y ) + I ( k ) D ∥∥∥2 2 ≤ 2m2η21λmax ( HD ( k ) ) 2 ‖y − uD ( k ) ‖22 + 2 ∥∥∥I ( k ) D ∥∥∥2 2 . ( 73 ) Similarly with T1 and T2 , we can control T3 and T4 as follows : Control on T3 . Recall HD ( k ) is a Gram matrix by definition . Then , by using the fact λmin ( HD ( k ) ) ≥ 0 and Cauchy-Schwarz inequality , we have 2mη1 ( y − ( 1− η2µL ) uD ( k ) ) > HD ( k ) ( uD ( k ) − y ) = −2mη1 ( 1− η2µL ) ( y − uD ( k ) ) > HD ( k ) ( y − uD ( k ) ) + ( 2mη1η2µL ) · y > HD ( k ) ( uD ( k ) − y ) ≤ ( 2mη1η2µL ) · λmax ( HD ( k ) ) ‖y − uD ( k ) ‖22 + ( 2mη1η2µL ) · ( λmax ( HD ( k ) ) ‖y‖2 ‖y − uD ( k ) ‖2 ) − 2mη1λmin ( HD ( k ) ) ‖y − uD ( k ) ‖22 = ( 4mη1η2µL ) · λmax ( HD ( k ) ) ‖y − uD ( k ) ‖22 + ( 4mη1η2µL ) · λmax ( HD ( k ) ) ‖y‖22 . ( 74 ) Control on T4 . By a simple Cauchy-Schwarz and Young ’ s inequality , we have − 2 ( y − ( 1− η2µL ) uD ( k ) ) > I ( k ) D = −2 ( 1− η2µL ) ( y − uD ( k ) ) > I ( k ) D + 2η2µL · y > ID ( k ) ≤ 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ‖y‖22 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ( 75 ) Step 4 . Upper-bound of the decomposition on training error ( 62 ) . Before getting the upper bound of the decomposition ( 62 ) , we first work on obtaining the bound of ( 76 ) . Set κ = O ( 1 n2 ) and notice η2µL = O ( 1 n ) by ( 69 ) , then we have 2 ∥∥∥I ( k ) D ∥∥∥2 2 + 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ( 76 ) = ( 2 + η2µL ) ∥∥∥I ( k ) D ∥∥∥2 2 + 2κ ( 1− η2µL ) ‖y − uD ( k ) ‖2 · 1 κ ∥∥∥I ( k ) D ∥∥∥ 2 ≤ ( 2 + η2µL+ 1 κ2 ) ∥∥∥I ( k ) D ∥∥∥2 2 + κ2 ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 = 1 κ2 · ∥∥∥I ( k ) D ∥∥∥2 2 + κ2 ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 ≤ { 1 κ2 · O ( L37/6n− 9d−8 6d−3 log ( m ) m1/3δ2/3 ) + κ2 ( 1− η2µL ) 2 } · ‖y − uD ( k ) ‖22 + 1κ2 · OP ( 1 n4 ) ≤ ( η2µL ) 4 ( 1− η2µL ) 2 · ‖y − uD ( k ) ‖22 + η2µL · ‖y‖22 , ( 77 ) where in the second inequality , the Eq . ( 70 ) is used with ( a + b ) 2 ≤ 2a2 + 2b2 for a , b ∈ R , and in the last inequality , we used ‖y‖22 = O ( n ) and the sufficiently large m to control the order of the coefficient terms of ‖y − uD ( k ) ‖22 . Specifically , we choose m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Now , by combining the inequalities ( 72 ) , ( 73 ) , ( 74 ) , ( 75 ) , ( 71 ) and ( 77 ) , we obtain the upper-bound on the decomposition ( 62 ) ; ‖uD ( k + 1 ) − y‖22 ≤ ( 2η2µL+ η 2 2µ 2L2 + 4mη1η2µL · λmax ( HD ( k ) ) ) · ‖y‖22 + ( ( 1 + η2µL ) ( 1− η2µL ) 2 + 2m2η21λmax ( HD ( k ) ) 2 + 4mη1η2µL · λmax ( HD ( k ) ) ) · ‖y − uD ( k ) ‖22 + ( 2 ∥∥∥I ( k ) D ∥∥∥2 2 + 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ) ≤ { 3η2µL+ η 2 2µ 2L2 +O ( ωmnη1η2µL 2 ) } · ‖y‖22 + { ( 1 + η2µL+ η 4 2µ 4L4 ) ( 1− η2µL ) 2 +O ( ω2m2n2η21L 2 ) +O ( ωmnη1η2µL 2 ) } · ‖y − uD ( k ) ‖22 : = A · ‖y‖22 + ( 1− B ) · ‖y − uD ( k ) ‖ 2 2 . ( 78 ) With the order choices of µ , η1 and η2 as in ( 69 ) , it is easy to see the leading terms of both A and B are same as η2µL = o ( 1n ) . Then , by recursively applying the inequality ( 78 ) , we can get the upper-bound on the training error . ‖y − uD ( k + 1 ) ‖22 ≤ A · ‖y‖ 2 2 + ( 1− B ) · ‖y − uD ( k ) ‖ 2 2 ≤ A‖y‖22 · ( k∑ j=0 ( 1− B ) j ) + ( 1− B ) k+1 · ‖y − uD ( 0 ) ‖22 ≤ A B · ‖y‖22 + ( 1− B ) k+1 · ‖y − uD ( 0 ) ‖22 ≤ O ( n ) + ( 1− η2µL ) k+1 · ‖y − uD ( 0 ) ‖22 . ( 79 ) In the last inequality , we used AB = o ( 1 ) , B ≥ η2µL and ‖y‖ 2 2 = O ( n ) . Step 5 . The order of the radius of perturbation region . It remains us to prove the radius of perturbation region τ has the order OP ( L √ ω√ m n d 2d−1 ) . First , recall that the ` 2-regularized GD update rule is as : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] + η2µW ( 0 ) D , ` , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . ( 80 ) Similarly with the proof in the Theorem 3.5 , we employ the induction process for the proof . The induction hypothesis is∥∥∥W ( s ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 ≤ O ( η1n √ mω√ δη2µ ) , ∀s ∈ [ k + 1 ] . ( 81 ) It is easy to see it holds for s = 0 , and suppose it holds for s = 0 , 1 , . . . , k , we consider k+1 . Using the update rule ( 80 ) , we have∥∥∥W ( k+1 ) D , ` −W ( k ) D , ` ∥∥∥ 2 ≤ η2µ ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 + η1 ∥∥∥∇W ` [ LS ( W ( k ) D ) ] ∥∥∥ 2 = η2µ ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 + η1 ∥∥∥∥∥ n∑ i=1 ( yi − uD , i ( k ) ) ∇W ` [ fWD ( k ) ( xi ) ] ∥∥∥∥∥ 2 ≤ O ( η1n √ mω√ δ ) +O ( η1 √ nmω ) · ‖y − uD ( k ) ‖2 ≤ O ( η1n √ mω√ δ ) +O ( η1 √ nmω ) · { O ( √ n ) + ( 1− η2µL ) k 2O ( √ n δ ) } ≤ O ( η1n √ mω√ δη2µ ) . In the first inequality , we use the induction hypothesis for s = k , and Lemma 4.4 . In the second inequality , since the induction hypothesis holds for s = 0 , 1 , . . . , k , we employ ‖y − uD ( k ) ‖2 ≤ O ( √ n ) + ( 1− η2µL ) k 2 ‖y − uD ( 0 ) ‖2 with the Lemma 4.9 . In the last inequality , we use η2µ < 1 . By triangle inequality , the induction holds for s = k + 1 . Plugging the proper choices of η1 , η2 and µ as suggested in ( 69 ) to O ( η1n √ mω√ δη2µ ) yields ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ OP ( L √ ω√ m n d 2d−1 ) . H PROOF OF THEOREM 3.11-KERNEL RIDGE REGRESSOR APPROXIMATION We present a following proof sketch on the approximation of regularized DNN estimator to kernel ridge regressor . 1 . The key idea for proving the second result in Theorem 3.8 is to write the distance between ui , D ( k ) ( where D is to denote the prediction is obtained from regularized GD rule ) and kernel regressor B : = H∞L ( Cµ · I + H∞L ) −1 y in terms of NTK matrix H∞L , which is as follows : uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) . Above equality describes how the regularized estimator evolves to fit the kernel regressor as iteration of algorithm goes by . 2 . We can bound the ` 2-norm of residual term eD ( k ) as O ( 1/n ) , and show that the ` 2 norm of the first term on the RHS of equation ( 4.3 ) decays at the rateO ( √ n ( 1− η2µL ) k ) . Here the √ n comes from the bound ‖B‖2 ≤ O ( √ n ) , since we know ‖u ( 0 ) ‖2 hasO ( √ nω ) with small ω ≤ 1 . This yields the claim . Recall the equality ( 65 ) . Then , we have uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) = −mη1 ·HD ( k ) ( uD ( k ) − y ) + I ( k ) 2 , D + I ( k ) 3 , D + I ( k ) 5 , D = −mη1 ·H∞L ( uD ( k ) − y ) −mη1 · ( HD ( k ) −H∞L ) ( uD ( k ) − y ) + I ( k ) 2 , D + I ( k ) 3 , D + I ( k ) 5 , D = −mη1 ·H∞L ( uD ( k ) − y ) + ξD ( k ) . ( 82 ) With τ = O ( L √ ω√ mδ n d 2d−1 ) , similarly with Lemma 4.10 and a direct employment of the result from Lemma 4.11 , we can control the distance from HD ( k ) to H∞L under operator norm as follows : ‖HD ( k ) −H∞L ‖2 ≤ ‖HD ( k ) −H ( 0 ) ‖2 + ‖H ( 0 ) −H ∞ L ‖2 ≤ O ( ω7/6L10/3n 7d−3 6d−3 6 √ log3 ( m ) mδ2 ) +O ( ωL5/2n 4 √ log ( nL/δ ) m ) ≤ O ( L19/12n− 21d−8 12d−6 6 √ log3 ( m ) mδ2 ) +O ( Ln− 18d−6 12d−6 4 √ log ( nL/δ ) m ) ≤ O ( L19/12n− 21d−8 12d−6 6 √ log3 ( m ) mδ2 ) , ( 83 ) where in the third inequality , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) is plugged-in . The last inequality holds with d ≥ 2 with large enough n and the condition on width m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Then , the ` 2 norm of ξD ( k ) can be bounded as : ‖ξD ( k ) ‖2 ≤ mη1 · ‖H ∞ L −HD ( k ) ‖2 ‖uD ( k ) − y‖2 + ∥∥∥I ( k ) D ∥∥∥ 2 ≤ O ( L19/12n− 12d−5 6d−3 √ log ( m ) m1/6δ1/3 ) · ‖uD ( k ) − y‖2︸ ︷︷ ︸ ≤O ( √ n/δ ) +OP ( 1 n2 ) ≤ O ( L19/12n− 18d−7 12d−6 √ log ( m ) m1/6δ5/6 ) +OP ( 1 n2 ) = OP ( 1 n2 ) , ( 84 ) where in the second inequality , we used ( 83 ) with η1 = O ( 1 mn − 3d−22d−1 ) to control the first term and employed Eq . ( 70 ) to control the second term . In the last equality , we usedm ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Now , by setting B : = ( η2µL η1m I + H∞L ) −1 H∞L y , we can easily convert the equality ( 82 ) as follows : for k ≥ 1 , uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) ( uD ( k − 1 ) −B ) + ξD ( k − 1 ) . ( 85 ) The recursive applications of the equality ( 85 ) yields uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + k∑ j=0 ( ( 1− η2µL ) · I −mη1H∞L ) j ξD ( k − j − 1 ) = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) . ( 86 ) Now , we bound the ` 2 norm of eD ( k ) in ( 86 ) : ‖eD ( k ) ‖2 = ∥∥∥∥∥∥ k∑ j=0 ( ( 1− η2µL ) · I −mη1H∞L ) j ξD ( k − j − 1 ) ∥∥∥∥∥∥ 2 ≤ k∑ j=0 ∥∥ ( 1− η2µL ) · I −mη1H∞L ∥∥j2 ‖ξD ( k − j − 1 ) ‖2 ≤ k∑ j=0 ( 1− η2µL ) j ‖ξD ( k − j − 1 ) ‖2 = O ( 1n ) , ( 87 ) in the last inequality , we used η2µL = O ( 1 n ) and Eq . ( 84 ) . Now , we control the ` 2-norm of the first term in ( 86 ) as : ∥∥∥∥∥ ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) ∥∥∥∥∥ 2 ≤ ( 1− η2µL ) k ‖uD ( 0 ) −B‖2 ≤ O ( √ n ( 1− η2µL ) k ) , ( 88 ) where in the second inequality , we used ‖uD ( 0 ) ‖2 ≤ O ( √ nω/δ ) and the fact that ‖B‖2 ≤ ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 H∞L ∥∥∥∥∥ 2 · ‖y‖2 ≤ O ( √ n ) . By combining ( 87 ) and ( 88 ) and using a fact ( 1− η2µL ) k ≤ exp ( −η2µLk ) , we conclude that after k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) , the error ‖uD ( k ) −B‖2 decays at the rate O ( 1 n ) . I PROOF OF THEOREM 3.12 We begin the proof by decomposing the error f̂ W ( k ) D ( x ) − f∗ ( x ) for any fixed x ∈ Unif ( Sd−1 ) into two terms as follows : f̂ W ( k ) D ( x ) − f∗ ( x ) = ( f̂ W ( k ) D ( x ) − g∗µ ( x ) ) ︸ ︷︷ ︸ ∆D,1 + ( g∗µ ( x ) − f∗ ( x ) ) ︸ ︷︷ ︸ ∆D,2 . ( 89 ) Here , we devise a solution of kernel ridge regression g∗µ ( x ) in the decomposition ( 89 ) : g∗µ ( x ) : = Ker ( x , X ) ( Cµ · I + H∞L ) −1 y , for some constant C > 0 . Specifically , in the proof to follow , we choose η1 and η2 such that C = η2Lη1m for the theoretical convenience . Our goal is to show that all the terms ‖∆D,1‖ 2 2 , and ‖∆D,2‖22 have the order either equal to or smaller than O ( n− d 2d−1 ) with the proper choices on m , µ , η1 and η2 . Since the high-level proof idea is similar with that of Theorem 3.8 , we omit the step-by-step proof sketch of Theorem 3.12 . The most notable difference between the proof strategies of the two theorems is that the regularized DNN approximate the kernel ridge regressor of noisy data , whereas in Theorem 3.8 , unregularized DNN approximate the interpolant based on noiseless data . Step 1 . Control on ∆D,2 . First , note that there is a recent finding that the reproducing kernel Hilbert spaces induced from NTKs with any number of layers ( i.e. , L ≥ 1 ) have the same set of functions , if kernels are defined on Sd−1 . See Chen & Xu [ 2020 ] . Along with this result , under the choice of model parameters as suggested in ( 69 ) , we can apply exactly the same proof used in Theorem.3.2 in Hu et al . [ 2021 ] for proving a following : ‖∆D,2‖22 : = ∥∥g∗µ − f∗∥∥22 = OP ( n− d2d−1 ) , ∥∥g∗µ∥∥2H = OP ( 1 ) . ( 90 ) Step 2 . Control on ∆D,1 . For n data points ( x1 , . . . , xn ) and for the kth updated parameter W ( k ) D , denote : ∇W ` [ f W ( k ) D ( X ) ] = [ vec ( ∇W ` [ f W ( k ) D ( x1 ) ] ) , · · · , vec ( ∇W ` [ f W ( k ) D ( xn ) ] ) ] . Note that when ` = 1 , ∇W ` [ f W ( k ) D ( X ) ] ∈ Rmd×n and when ` = 2 , . . . , L , ∇W ` [ f W ( k ) D ( X ) ] ∈ Rm2×n . With this notation , we can write the vectorized version of the update rule ( 80 ) as : vec ( W ( k ) D , ` ) = vec ( W ( 0 ) D , ` ) − η1 k−1∑ j=0 ( 1− η2µ ) j∇W ` [ fWD ( k−j−1 ) ( X ) ] ( uD ( k − j − 1 ) − y ) , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . Using the equality , we can get the decomposition : vec ( W ( k ) D , ` ) = vec ( W ( 0 ) D , ` ) ︸ ︷︷ ︸ : =E1 −η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( uD ( k − j − 1 ) − y ) ︸ ︷︷ ︸ : =E2 −η1 k−1∑ j=0 ( 1− η2µ ) j [ ∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ] ( uD ( k − j − 1 ) − y ) .︸ ︷︷ ︸ : =E3 ( 91 ) Let zD , k ( x ) : = vec ( ∇W ` [ f W ( k ) D ( x ) ] ) , and note that f W ( k ) D ( x ) = 〈zD , k ( x ) , vec ( W ( k ) D , ` ) 〉 . Then , by the definition of ∆D,1 and the decomposition ( 91 ) , we have ∆D,1 = 1 L L∑ ` =1 〈zD , k ( x ) , E1 + E2 + E3〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y = 1 L L∑ ` =1 〈zD , k ( x ) , E1〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E3〉 + 1 L L∑ ` =1 〈zD , k ( x ) , E2〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y︸ ︷︷ ︸ : =C ( 92 ) First , we focus on controlling the ` 2 bound on the first two terms in ( 92 ) . Observe that the first term can be bounded as : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E1〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E1〉|2 . ( 93 ) Recall that ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) by Lemma 4.4 . Then , the random variable zD , k ( x ) > vec ( W ( 0 ) D , ` ) | zD , k ( x ) is simply a N ( 0 , O ( ω ) ) for 1 ≤ ` ≤ L. A straightforward application of Chernoff bound for normal random variable and taking union bound over the layer 1 ≤ ` ≤ L yield that : with probability at least 1− δ , 1 L L∑ ` =1 ∣∣∣zD , k ( x ) > vec ( W ( 0 ) D , ` ) ∣∣∣2 ≤ O ( ω log ( L δ ) ) . ( 94 ) The ` 2 norm of the second term in ( 92 ) can be similarly bounded as ( 93 ) in addition with the CauchySchwarz inequality : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E3〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E3〉|2 ≤ 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 . ( 95 ) The ‖E3‖2 is bounded as : ‖E3‖2 = ∥∥∥∥∥∥η1 k−1∑ j=0 ( 1− η2µ ) j [ ∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ] ( uD ( k − j − 1 ) − y ) ∥∥∥∥∥∥ 2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · ∥∥∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ∥∥2 ‖uD ( k − j − 1 ) − y‖2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · ∥∥∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ∥∥F ‖uD ( k − j − 1 ) − y‖2 = η1 k−1∑ j=0 ( 1− η2µ ) j · √√√√ n∑ i=1 ∥∥∇W ` [ fWD ( k−j−1 ) ( xi ) ] −∇W ` [ fWD ( 0 ) ( xi ) ] ∥∥2F ‖uD ( k − j − 1 ) − y‖2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · √√√√2 n∑ i=1 ∥∥∇W ` [ fWD ( k−j−1 ) ( xi ) ] −∇W ` [ fWD ( 0 ) ( xi ) ] ∥∥22 ‖uD ( k − j − 1 ) − y‖2 ≤ η1 η2µ · O ( τ1/3L2 √ ωmn log ( m ) ) · O ( √ n ) ≤ O ( L10/3ω1/6 m2/3δ1/3 n 4d 6d−3 √ log ( m ) ) . ( 96 ) In the first , second and third inequalities , we used a simple fact that for the matrix A ∈ Rd1×d2 with rank r , then ‖A‖2 ≤ ‖A‖F ≤ √ r‖A‖2 . Recall that the rank of the matrix∇W ` [ fWD ( k−j−1 ) ( x ) ] − ∇W ` [ fWD ( 0 ) ( x ) ] is at most 2 . In the second to the last inequality , we use the result of Lemma 4.6 and the ‖uD ( i ) − y‖2 ≤ O ( √ n ) for any i ≥ 1 . In the last inequality , we plug the correct orders as set in ( 69 ) to τ , η1 , η2 and µ . Back to the inequality ( 95 ) , using the ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) and ( 96 ) , we can get 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 ≤ OP ( L20/3ω4/3 m1/3 n 8d 6d−3 log ( m ) ) . ( 97 ) Before controlling the ` 2 norm of C in ( 92 ) , recall that we set B : = ( η2µL η1m I + H∞L ) −1 H∞L y and the dynamics of uD ( k ) −B can be expressed in terms of H∞L as follows : For any k ≥ 1 , uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) , ( 98 ) with ‖eD ( k ) ‖2 ≤ O ( 1 n ) . Using ( 98 ) , we can further decompose the term E2 in ( 91 ) as : E2 : = −η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( uD ( k − j − 1 ) − y ) = η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 B − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 uD ( 0 ) − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) jeD ( k − j − 1 ) − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( B− y ) = E2,1 + E2,2 + E2,3 + E2,4 . ( 99 ) Then , we can re-write the error term C in ( 92 ) as : C = 1 L L∑ ` =1 〈zD , k ( x ) , E2,1〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E2,2〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E2,3〉 + { 1 L L∑ ` =1 〈zD , k ( x ) , E2,4〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y } ︸ ︷︷ ︸ : =D . ( 100 ) Our goal is to control the ` 2 norm of each summand in the equality ( 100 ) . For the first three terms in ( 100 ) , a simple Cauchy-Schwarz inequality can be applied : for i = 1 , 2 , 3 : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E2 , i〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E2 , i〉|2 ≤ 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 · ‖E2 , i‖ 2 2 . We work on obtaining the bound of ∑L ` =1 ‖E2,1‖ 2 2 . Let Tk be defined as Tk : = k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 . Then , we have L∑ ` =1 ‖E2,1‖22 = η 2 1 L∑ ` =1 ( B > T > k ∇W ` [ fWD ( 0 ) ( X ) ] > ∇W ` [ fWD ( 0 ) ( X ) ] TkB ) = mη21B > T > k H ( 0 ) TkB = mη21B > T > k ( H ( 0 ) −H∞L ) TkB +mη21B > T > k H∞L TkB ≤ mη21 ‖H ( 0 ) −H∞L ‖2 ·B > T 2k B +mη21B > T > k H∞L TkB . ( 101 ) To obtain the upper-bound on ( 101 ) , we need to control the terms T > k H∞L Tk and B > T 2k B . Let us denote H∞L = ∑n i=1 λiviv > i be the eigen-decomposition of H ∞ L . Using 1− η2µL ≤ 1− η2µ , note that Tk = k−1∑ j=0 ( 1− η2µ ) j ( 1− η2µL ) k−j−1 ( I − mη1 1− η2µL H∞L ) k−j−1 ( 1− η2µ ) k−1 k−1∑ i=0 ( I − mη1 1− η2µ H∞L ) i = ( 1− η2µ ) k−1 n∑ j=0 ( 1− ( 1− mη11−η2µλj ) k mη1 1−η2µλj ) vjv > j ( 1− η2µ ) k mη1λ0 · I . ( 102 ) A similar logic can be applied to bound T > k H∞L Tk : T > k H∞L Tk ( 1− η2µ ) k−1 n∑ j=0 ( 1− ( 1− mη11−η2µλj ) k mη1 1−η2µλj ) 2 λjvjv > j ( 1− η2µ ) 2k m2η21 · ( H∞L ) −1 . ( 103 ) Recall the definition of the notation B : = H∞L ( η2µL η1m I + H∞L ) −1 y . Then , we can bound the term B > T > k H∞L TkB : B > T > k H∞L TkB ≤ ( 1− η2µ ) 2k m2η21 ·B > ( H∞L ) −1 B = ( 1− η2µ ) 2k m2η21 · y > ( η2µL η1m I + H∞L ) −1 H∞L ( η2µL η1m I + H∞L ) −1 y = O ( ( 1− η2µ ) 2k m2η21 ) , ( 104 ) where in the last equality , we used ∥∥g∗µ∥∥2H = OP ( 1 ) in ( 90 ) . Now we turn our attention to bound the term B > T 2k B , B > T 2k B ≤ ( 1− η2µ ) 2k m2η21λ 2 0 y > ( η2µL η1m I + H∞L ) −1 ( H∞L ) 2 ( η2µL η1m I + H∞L ) −1 y = O ( ( 1− η2µ ) 2k n m2η21λ 2 0 ) , ( 105 ) where we used ‖y‖22 = O ( n ) in the last inequality . Combining the bounds ( 104 ) , ( 105 ) and the result from Lemma 4.11 , we can further bound ( 101 ) and have : L∑ ` =1 ‖E2,1‖22 ≤ O ( ω ( 1− η2µ ) 2k mλ20 n2L5/2 4 √ log ( nL/δ ) m + ( 1− η2µ ) 2k m ) ≤ O ( ( 1− η2µ ) 2k m ) , ( 106 ) where in the second inequality , we used m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Similarly , we can bound∑L ` =1 ‖E2,2‖ 2 2 : L∑ ` =1 ‖E2,2‖22 = η 2 1 L∑ ` =1 ( uD ( 0 ) > T > k ∇W ` [ fWD ( 0 ) ( X ) ] > ∇W ` [ fWD ( 0 ) ( X ) ] TkuD ( 0 ) ) = mη21uD ( 0 ) > T > k H ( 0 ) TkuD ( 0 ) = mη21uD ( 0 ) > T > k ( H ( 0 ) −H∞L ) TkuD ( 0 ) +mη21uD ( 0 ) > T > k H∞L TkuD ( 0 ) ≤ mη21 ‖H ( 0 ) −H∞L ‖2 · uD ( 0 ) > T 2k uD ( 0 ) +mη21uD ( 0 ) > T > k H∞L TkuD ( 0 ) ≤ mη21 ( 1− η2µ ) 2k m2η21λ 2 0 O ( ωnL5/2 4 √ log ( nL/δ ) m ) ‖uD ( 0 ) ‖22 +mη21 ( 1− η2µ ) 2k m2η21 uD ( 0 ) > ( H∞L ) −1uD ( 0 ) ≤ O ( ( 1− η2µ ) 2k n2ω2L5/2 mλ20δ 2 4 √ log ( nL/δ ) m + nω ( 1− η2µ ) 2k mλ0δ2 ) = OP ( nω ( 1− η2µ ) 2k mλ0 ) . ( 107 ) Here , in the second inequality , we used the inequalities ( 102 ) and ( 103 ) and Lemma 4.11 . In the third inequality , we used the Lemma 4.8 , ‖u ( 0 ) ‖2 = O ( √nω δ ) with probability at least 1− δ . In the last equality , we used m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Next , we bound ∑L ` =1 ‖E2,3‖ 2 2 as : L∑ ` =1 ‖E2,3‖22 = mη 2 1 · ( k−1∑ j=0 ( 1− η2µ ) j ( ek−j−1 ) ) > HD ( 0 ) ( k−1∑ j=0 ( 1− η2µ ) j ( ek−j−1 ) ) ≤ mη 2 1 η22µ 2 · λmax ( HD ( k ) ) · ‖ek−j−1‖22 ≤ mη21 η22µ 2 · O ( ωnL ) · O ( 1 n2 ) = O ( L3 m ω · n− 4d−3 2d−1 ) . ( 108 ) Now , we focus on obtaining the ` 2 norm bound on D in ( 100 ) . Recall the definition of the notation B : = H∞L ( η2µL η1m I + H∞L ) −1 y . A simple calculation yields that B− y = H∞L ( η2µL η1m I + H∞L ) −1 y − y = −η2µL mη1 ( η2µL η1m I + H∞L ) −1 y . Then , we can re-write the expression of the D as : D : = ( η2µL mη1 ) · η1 1 L L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 k−1∑ j=0 ( 1− η2µ ) j ( η2µL η1m I + H∞L ) −1 y −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y = ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y − ( 1− η2µ ) k 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y = ( 1 m L∑ ` =1 〈 z0 , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y − ( 1− η2µ ) k 1 m L∑ ` =1 〈 z0 , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y + ( 1− ( 1− η2µ ) k ) ( 1 m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ) ( η2µL η1m I + H∞L ) −1 y , ( 109 ) where in the second equality , ∑k−1 j=0 ( 1− η2µ ) j = 1− ( 1−η2µ ) k η2µ is used . The ` 2 norm of first term in the ( 109 ) can be bounded as : ∥∥∥∥∥ ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ∥∥∥∥∥ ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 = √√√√ n∑ i=1 ( 1 m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( xi ) ] 〉 −Ker ( x , xi ) ) 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ O ( ω √ nL5/2 4 √ log ( nL/δ ) m ) · O ( η1m η2µL √ n ) = O ( ωη1mn η2µL L5/2 4 √ log ( nL/δ ) m ) = O ( ωL5/2n d 2d−1 4 √ log ( nL/δ ) m ) , ( 110 ) where , in the second inequality , we used Lemma 4.11 , and also we used∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ √ y > ( η2µL η1m I + H∞L ) −2 y ≤ √ η21m 2 η22µ 2L2 · ‖y‖22 = O ( η1m η2µL √ n ) . ( 111 ) The ` 2 norm of the second term in ( 109 ) can be easily bounded as : ∥∥∥∥∥ ( 1− η2µ ) k 1m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ∥∥∥∥∥ ( 1− η2µ ) k ( 1 m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 + ∥∥∥∥∥ ( 1− η2µ ) kKer ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− η2µ ) k ∥∥∥∥∥ 1m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 + ( 1− η2µ ) k ∥∥∥∥∥Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− η2µ ) k · O ( ω√nL3/2 4√ log ( nL/δ ) m ) · O ( η1mη2µL√n ) +O ( ( 1− η2µ ) k ) ≤ ( 1− η2µ ) k · O ( ωL3/2n d2d−1 4√ log ( nL/δ ) m ) +O ( ( 1− η2µ ) k ) . ( 112 ) Lastly , the ` 2 norm of the third term in ( 109 ) is bounded as : ∥∥∥∥∥ ( 1− ( 1− η2µ ) k ) ( 1 m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ∥∥∥∥∥ 1m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ( 1 m L∑ ` =1 ‖zD , k ( x ) − zD,0 ( x ) ‖F ∥∥∇W ` [ fWD ( 0 ) ( X ) ] ∥∥F ) · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ( L m O ( τ1/3L2 √ ωm log ( m ) ) · O ( √ ωmn ) ) · O ( η1m η2µL √ n ) ≤ ( 1− ( 1− η2µ ) k ) · O ( ω7/6L10/3n 4d6d−3 √log ( m ) m1/6δ1/3 ) ≤ O ( ω7/6L10/3n 4d 6d−3 √ log ( m ) m1/6δ1/3 ) , ( 113 ) where in the fourth inequality , τ = OP ( L √ ω√ m n d 2d−1 ) is plugged in . Combining the inequali- ties ( 110 ) , ( 112 ) and ( 113 ) , we get the bound on ‖D‖2 in ( 109 ) : ‖D‖22 ≤ O ( ω2L5n 2d 2d−1 √ log ( nL/δ ) m ) + ( 1− η2µ ) 2kO ( ω2L3n 2d2d−1√ log ( nL/δ ) m ) +O ( ( 1− η2µ ) 2k ) +O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) ≤ O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) +O ( ( 1− η2µ ) 2k ) . ( 114 ) Step 3 . Combining all pieces . Recall ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) . With this fact , combining the bounds ( 94 ) , ( 97 ) , ( 106 ) , ( 107 ) , ( 108 ) and ( 114 ) , we can bound the ‖∆D,1‖22 via the decomposition ( 92 ) as follows : ‖∆D,1‖22 ≤ 1 L L∑ ` =1 ∣∣∣zD , k ( x ) > vec ( W ( 0 ) D , ` ) ∣∣∣2 + 1L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2,1‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2,2‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2.3‖ 2 2 + ∥∥∥∥∥ 1L L∑ ` =1 〈zD , k ( x ) , E2,4〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 2 ≤ O ( ω log ( L δ ) ) +OP ( L20/3ω4/3 m1/3 n 8d 6d−3 log ( m ) ) +OP ( ω ( 1− η2µ ) 2k L ) +OP ( nω2 ( 1− η2µ ) 2k Lλ0 ) +OP ( L2 m ω · n− 4d−3 2d−1 ) +O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) +OP ( ( 1− η2µ ) 2k ) ≤ OP ( n− d 2d−1 ) . | The paper proves a bound on the excess risk in a nonparametric regression setting, where the estimator is a deep ReLU network trained by GD with L2 regularization, and the regression function belongs to the RKHS induced by the Neural Tangent Kernel (NTK) related to a deep network. In particular, this work considers presence of label noise and argues that without an explicit regularization such learning procedure is not consistent in general as the network reaches interpolating regime. The paper proposes to use a relative L2 regularization (relative to the initialization) and shows that with an appropriate tuning of parameters the procedure is consistent and yields an optimal rate for regression on RKHS. The proof is heavily based on the NTK machinery, and in particular on the "coupling" between predictions of the network and a ridge regressor given the NTK kernel matrix. | SP:232f05c3b3a9b5ca1cea27b2aea1293a7da190c8 |
Generalization of Overparametrized Deep Neural Network Under Noisy Observations | 1 INTRODUCTION Over the past few years , Neural Tangent Kernel ( NTK ) [ Arora et al. , 2019b ; Jacot et al. , 2018 ; Lee et al. , 2018 ; Chizat & Bach , 2018 ] has been one of the most seminal discoveries in the theory of neural network . The underpinning idea of the NTK-type theory comes from the observation that in a wide-enough neural net , model parameters updated by gradient descent ( GD ) stay close to their initializations during the training , so that the dynamics of the networks can be approximated by the first-order Taylor expansion with respect to its parameters at initialization . The linearization of learning dynamics on neural networks has been helpful in showing the linear convergence of the training error on both overparametrized shallow [ Li & Liang , 2018 ; Du et al. , 2018 ] and deep neural networks [ Allen-Zhu et al. , 2018 ; Zou et al. , 2018 ; 2020 ] , as well as the characterizations of generalization error on both models [ Arora et al. , 2019a ; Cao & Gu , 2019 ] . These findings clearly lead to the equivalence between learning dynamics of neural networks and the kernel methods in an reproducing kernel Hilbert spaces ( RKHS ) associated with NTK . 1 Specifically , Arora et al . [ 2019a ] provided the O ( n−1/2 ) generalization bound of shallow neural network , where n denotes the training sample size . Recently , in the context of nonparametric regression , two papers Nitanda & Suzuki [ 2020 ] and Hu et al . [ 2021 ] showed that neural network can obtain the convergence rate faster than O ( n−1/2 ) by specifying the complexities of target function and hypothesis space . Specifically , Nitanda & Suzuki [ 2020 ] showed that the shallow neural network with smoothly approximated ReLU ( swish , see Ramachandran et al . [ 2017 ] ) activation trained via ` 2-regularized averaged stochastic gradient descent ( SGD ) can recover the target function from RKHSs induced from NTK with swish activation . Similarly , Hu et al . [ 2021 ] showed that a shallow neural network with ReLU activation trained via ` 2-regularized GD can generalize well , when the target function ( i.e. , f ? ρ ) is fromHNTK1 . Notably , the rate that the papers Nitanda & Suzuki [ 2020 ] and Hu et al . [ 2021 ] obtained is minimax optimal , meaning that no estimators perform substantially better than the ` 2-regularized GD or averaged SGD algorithms for recovering functions from respective function spaces . Nevertheless , 1Henceforth , we denoteHNTK1 andHNTKL as RKHSs induced from NTK of shallow L = 1 and deep neural networks L ≥ 2 with ReLU activations , respecitvely . these results are restricted to shallow neural networks , and can not explain the generalization abilities of deep neural network ( DNN ) . Similarly with Arora et al . [ 2019a ] , Cao & Gu [ 2019 ] obtained the O ( n−1/2 ) generalization bound , showing that the SGD generalize well for f ? ρ ∈ HNTKL , when f ? ρ has a bounded RKHS norm . However , the rate they obtained is slower than the minimax rate we can actually achieve . Furthermore , their results become vacuous under the presence of additive noises on the data set . Motivated from these observations , the fundamental question in this study is as follows : When the noisy dataset is generated from a function fromHNTKL , does the overparametrized DNN obtained via ( ` 2-regularized ) GD provably generalize well the unseen data ? We consider a neural network that has L ≥ 2 hidden layers with width m n . ( i.e. , overparametrized deep neural network . ) We focus on the least-squares loss and assume that the activation function is ReLU . A positivity assumption of NTK from ReLU DNN is imposed , meaning that λ0 > 0 , where λ0 denotes the minimum eigenvalue of the NTK . We give a more formal mathematical definition of ReLU DNN in the following Subsection 2.2 . Under these settings , we provide an affirmative answer to the above question by investigating the behavior of L2-prediction error of the obtained neural network with respect to GD iterations . 1.1 CONTRIBUTIONS Our derivations of algorithm-dependent prediction risk bound require the analysis on training dynamics of the estimated neural network through ( regularized ) GD algorithm . We include these results as the contributions of our paper , which can be of independent interests as well . • In an unregulaized case , under the assumption λ0 > 0 , we show randomly initialized GD converges to 0 training loss at a linear rate . As will be detailed in subsection 3.3 , this is the different result from the seminal work of Allen-Zhu et al . [ 2018 ] , where they also prove a linear convergence of training loss of ReLU DNN , but under different data distribution assumption . • We show that the DNN updated via vanilla GD does not recover the ground truth function f ? ρ ∈ HNTKL under noisy observations , if the DNN is trained for either too short or too long : that is , the prediction error is bounded away from 0 by some constant as n goes to infinity . • In regularized case , we prove the mean-squared error ( MSE ) of DNN is upper bounded by some positive constant . Additionally , we proved the dynamics of the estimated neural network get close to the solution of kernel ridge regression associated with NTK from ReLU DNN . • We show that the ` 2-regularization can be helpful in achieving the minimax optimal rate of the prediction risk for recovering f ? ρ ∈ HNTKL under the noisy data . Specifically , it is shown that after some iterations of ` 2-regularized GD , the minimax optimal rate ( which is O ( n− d 2d−1 ) , where d is a feature dimension . ) can be achieved . Note that our paper is an extension of Hu et al . [ 2021 ] to DNN model , showing that the ` 2-regularized DNN can achieve a minimax optimal rate of prediction error for recovering f ? ρ ∈ HNTKL . However , we would like to emphasize that our work is not a trivial application of their work from at least two technical aspects . These aspects are more detailed in the following subsection . 1.2 TECHNICAL COMPARISONS WITH HU ET AL . [ 2021 ] Firstly , in the analysis of training loss of regularized shallow neural-net , Hu et al . [ 2021 ] begin the proof by decomposing the difference between two individual predictions into two terms : one that is related with the gram matrix evaluated at each iteration of the algorithm and the perturbation term . Henceforth , we name this decompostion as “ Gram+Pert ” decomposition . This decomposition can be checked with the equality ( E.2 ) in the supplementary PDF of Hu et al . [ 2021 ] . The key ingredients for the decomposition are ( i ) the simple gradient structure of the shallow neural net , and ( ii ) the partitioning of the nodes in the hidden-layer into two sets : a set of nodes whose activation patterns change from their initializations during training , and the complement of the set . This construction of the sets peels off the ReLU activation in the difference so that the GD algorithm can be involved in the analysis . However , because of the compositional structure of the network , the same nodes partitioning technique can not be applied for obtaining the decomposition in the DNN setting with ReLU activation . To avoid this difficulty , we employ a specially designed diagonal matrix Σ̃ and this matrix can peel off the ReLU function for each layer of the network . ( See the definition of Σ̃ in the proof of Theorem 3.5 in the Appendix . ) Recursive applications of this diagonal matrix across the entire hidden layers enable the Gram+Pert decomposition in our setting . It should be noted that the diagnoal matrix Σ̃ had been employed in Zou et al . [ 2020 ] , which analyzed the behavior of training loss of classification problem via ReLU DNN under logistic loss . However , since their result is dependent on different data distribution assumption under the different loss function from ours , they didn ’ t employ the Gram+Pert decomposition . Thus their technical approaches are different from ours . Secondly , Hu et al . [ 2021 ] directly penalized the weight parameter W by adding ‖W‖2F to the objective function . The ` 2-regularization solely on the W has an effect of pushing the weight towards the origin . This makes ‖W ( k ) −W ( 0 ) ‖2 ≤ O ( 1 ) 2 , allowing most activation patterns of the nodes in the hidden layer can change during the training , even in overparametrized setting . Here , W ( k ) denotes the updated weight parameter at kth itertaion of algorithm , and ‖ · ‖2 denotes the spectral norm of the matrix . Nonetheless , this doesn ’ t affect the analysis on obtaining the upper-bound of MSE in shallow neural net , since the network has only a single hidden layer . In contrast , in the DNN setting , we allow the non-convex interactions of parameters across the hidden layers . To the best of our knowledge , a technique for controlling the size of ` 2-norm of network gradient has not been developed under this setting , yet . We circumvent this difficulty by regularizing the distance between the updated and the initialized parameter , instead by directly regularizing the updated parameter . This ensures that the updated parameter by ` 2-regularized GD stays in a close neighborhood to its initialization , so that with heavy over-parametrization , the dynamics of network becomes linearized in parameter and we can ignore the non-convex interactions of parameters across the hidden layers . Specifically , under suitable model parameter setting , we prove that ‖W ( k ) ` −W ( 0 ) ` ‖2 ≤ ÕP ( 1√ m ) 3 over all ` ∈ { 1 , . . . , L } . Here , ÕP ( · ) hides the dependencies on the model parameters ; L , ω , and n. This result allows us to adopt the so-called “ Forward Stability ” argument developed by Allen-Zhu et al . [ 2018 ] , and eventually leads to the control of network gradient under ` 2 sense . 1.3 ADDITIONAL RELATED WORKS There has been another line of works trying to characterize the generalizabilities of DNN under noisy observation settings . Specifically , it has been shown that the neural network model can achieve minimax style optimal convergence rates of L2-prediction risk both in regression [ Bauer & Kohler , 2019 ; Liu et al. , 2019 ; Schmidt-Hieber , 2020 ] and classification [ Kim et al. , 2021 ] problems . Nonetheless , a limitation of the aforementioned papers is that they assume an adequate minimizer of the empirical risk can be obtained . In other words , the mathematical proofs of their theorems do not correspond to implementable algortihms . Recently , several papers , which study the generalization properties of neural network with algorithmic guarantees , appear online . Specifically , Kohler & Krzyzak [ 2019 ] showed that the data interpolants obtained through DNN by vanilla GD is inconsistent . This result is consistent with our result , but they consider the overparametrized DNN that is a linear combination of Ω ( n10d 2 ) smaller neural network , and the activation function they consider is sigmoid function , which is smooth and differentiable . Along this line of research , Kuzborskij & Szepesvári [ 2021 ] ( regression ) and Ji et al . [ 2021 ] ( classification ) showed that when training overparametrized shallow neural network , early stopping of vanilla GD enables us to obtain consistent estimators . Notation . We use the following notation for asymptotics : For sufficiently large n , we write f ( n ) = O ( g ( n ) ) , if there exists a constant K > 0 such that f ( n ) ≤ Kg ( n ) , and f ( n ) = Ω ( g ( n ) ) if f ( n ) ≥ K ′g ( n ) for some constant K ′ > 0 . The notation f ( n ) = Θ ( g ( n ) ) means that f ( n ) = O ( g ( n ) ) and 2This was empirically shown to be true in paper Wei et al . [ 2019 ] . See Figure 3 in their paper . We provide a brief mathematical explanation on why this result is hard to be shown in Appendix C. 3Readers can find the proof of this result in Appendix G. f ( n ) = Ω ( g ( n ) ) . Let 〈A , B〉Tr : = Tr ( A > B ) for the two matrices A , B ∈ Rd1×d2 . We adopt the shorthand notation denoting [ n ] : = { 1 , 2 , . . . , n } for n ∈ N. 2 PROBLEM FORMULATION 2.1 NON-PARAMETRIC REGRESSION Let X ⊂ Rd and Y ⊂ R be the measureable feature space and output space . We denote ρ as a joint probability measure on the product space X × Y , and let ρX be the marginal distribution of the feature space X . We assume that the noisy data-set D : = { ( xi , yi ) } ni=1 are generated from the non-parametric regression model yi = f ? ρ ( xi ) + εi , where εi i.i.d.∼ N ( 0 , 12 ) for i = 1 , . . . , n. Let f̂W ( k ) ( · ) be the value of neural network evaluated with the parameters W at the k-th iterations of GD update rule . At k = 0 , we randomly initialize the weight parameters in the model following He initialization [ He et al. , 2015 ] with a slight modification . Then , the L2 prediction risk is defined as the difference between two expected risks ( i.e. , excess risk ) R ( f̂W ( k ) ) : = Eρ∼ ( x , y ) [ ( y − f̂W ( k ) ( x ) ) 2 ] and R ( f ? ρ ) : = Eρ∼ ( x , y ) [ ( y − f ? ρ ( x ) ) 2 ] , where f ? ρ ( x ) : = E [ y|x ] . Then , we can easily show the prediction risk has a following form : R ( f̂k , f ? ρ ) : = R ( f̂W ( k ) ) −R ( f ? ) = Eρx , ε [ ( f̂W ( k ) ( x ) − f ? ρ ( x ) ) 2 ] . ( 1 ) Note that the expectation is taken over the marginal probability measure of feature space , ρx , and the noise of the data , ε . However , the ( 1 ) is still a random quantity due to the randomness of the initialized parameters ( W ( 0 ) ` ) ` =1 , ... , L . 2.2 DEEP NEURAL NETWORK WITH RELU ACTIVATION Following the setting introduced in Allen-Zhu et al . [ 2018 ] , we consider a fully-connected deep neural networks with L hidden layers and m network width . For L ≥ 2 , the output of the network fW ( · ) ∈ R with input data x ∈ X can be formally written as follows : fW ( x ) = √ m · vTσ ( WLσ ( WL−1 · · ·σ ( W1x ) · · · ) ) , ( 2 ) where Sd−1 is a unit sphere in d-dimensional euclidean space , σ ( · ) is an entry-wise activation function , W1 ∈ Rm×d , W2 , . . . , WL ∈ Rm×m denote the weight matrices for hidden layers and v ∈ Rm×1 denote the weight vector for the output layer . Following the existing literature , we will consider ReLU activation function σ ( x ) = max ( x , 0 ) , which is the most commonly used activation function by practitioners . Random Initialization . Each entries of weight matrices in hidden layers are assumed to be generated from ( Wi , j ) ` =1 , ... , L ∼ N ( 0 , 2m ) , and entries of the output layer are drawn from vj ∼ N ( 0 , ωm ) . This initialization scheme helps the forward propagation neither explode nor vanish at the initialization , seeing Allen-Zhu et al . [ 2018 ] ; Zou et al . [ 2018 ; 2020 ] . Note that we initialize the parameters in the last layer with variance ωm , where ω ≤ 1 is a model parameter to be chosen later for technical convenience . Unregularized GD update rule . We solve a following ` 2-loss function with the given dataset D : LS ( W ) = 1 2 n∑ i=1 ( yi − fW ( xi ) ) 2 . ( 3 ) Let W ( 0 ) 1 , . . . , W ( 0 ) L be the initialized weight matrices introduced above , and we consider a following gradient descent update rule : W ( k ) ` = W ( k−1 ) ` − η∇W ` ( LS ( W ( k−1 ) ` ) ) , ` ∈ [ L ] , k ≥ 1 , ( 4 ) where ∇W ` ( LS ( · ) ) is a partial gradient of the loss function LS ( · ) with respect to the ` -th layer parameters W ` , and η > 0 is the learning rate of the gradient descent . ` 2-regularized GD update rule . The estimator is obtained by minimizing a ` 2-regularized function ; ΦD ( W ) : = LS ( WD ) + µ 2 L∑ ` =1 ∥∥∥WD , ` −W ( 0 ) D , ` ∥∥∥2 F . ( 5 ) Naturally , we update the model parameters { WD , ` } ` =1 , ... , L via modified GD update rule : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] + η2µW ( 0 ) D , ` , ∀ ` ∈ [ L ] , ∀k ≥ 1 . ( 6 ) The notations η1 , η2 are step sizes , and µ > 0 is a tuning parameter on regularization . We adopt the different step sizes for the partial gradient and regularized term for the theoretical conveniences . Furthermore , we add the additional subscript D to the update rule ( 6 ) to denote the variables are under the regularized GD update rule . Recall that the W ( 0 ) D , ` are initialized parameters same with the unregularized case . For simplicity , we fix the output layer , and train L hidden layers for both unregularized and regularized cases . 3 MAIN THEORY First , we describe the neural tangent kernel ( NTK ) matrix of ( 2 ) , which is first proposed by Jacot et al . [ 2018 ] and further studied by Arora et al . [ 2019b ] ; Du et al . [ 2019 ] ; Lee et al . [ 2018 ] ; Yang [ 2019 ] . NTK matrix of DNN is a L-times recursively defined n × n kernel matrix , whose entries are the infinite-width limit of the gram matrix . Let ∇W ` [ fW ( 0 ) ( · ) ] be the gradient of the ReLU DNN ( 2 ) with respect to the weight matrix in ` th hidden layer at random initialization . Note that when ` = 1 , ∇W ` [ fW ( 0 ) ( · ) ] ∈ Rm×d and when ` ∈ { 2 , . . . , L } , ∇W ` [ fW ( 0 ) ( · ) ] ∈ Rm×m . Then , as m→∞ , H ( 0 ) : = ( 1 m L∑ ` =1 〈 ∇W ` [ fW ( 0 ) ( xi ) ] , ∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ) n×n → H∞L , ( 7 ) where H∞L : = { Ker ( xi , xj ) } n i , j=1 . Here , Ker ( · , · ) denotes a NTK function of ( 2 ) to be defined as follows : Definition 3.1 . ( NTK function of ( 2 ) ) . For any x , x′ ∈ X and ` ∈ [ L ] , define Φ ( 0 ) ( x , x′ ) = 〈x , x′〉 , Θ ( ` ) ( x , x′ ) = ( Φ ( ` −1 ) ( x , x ) Φ ( ` −1 ) ( x , x′ ) Φ ( ` −1 ) ( x′ , x ) Φ ( ` −1 ) ( x′ , x′ ) ) ∈ R2×2 , Φ ( ` ) ( x , x′ ) = 2 · E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ ( u ) · σ ( v ) ] , and Φ̇ ( ` ) ( x , x′ ) = 2 · E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ̇ ( u ) · σ̇ ( v ) ] , where σ̇ ( u ) = 1 ( u ≥ 0 ) . Then , we can derive the final expression of NTK function of ( 2 ) as follows : Ker ( x , x′ ) = ω 2 · L∑ ` =1 ( Φ ( ` −1 ) ( x , x′ ) · L∏ ` ′= ` Φ̇ ( ` ′ ) ( x , x′ ) ) . ( 8 ) The expression in ( 8 ) is adapted from Cao & Gu [ 2019 ] . As remarked in Cao & Gu [ 2019 ] , a coefficient 2 in Φ ( ` ) and Φ̇ ( ` ) remove the exponential dependence on the network depth L in the NTK function . However , when compared with the NTK formula in Cao & Gu [ 2019 ] , ( 8 ) is different from two aspects : ( i ) An additional factor ω in ( 8 ) ) comes from the difference in initialization settings of the output layer , in which Cao & Gu [ 2019 ] considers vj ∼ N ( 0 , 1m ) , whereas we consider vj ∼ N ( 0 , ωm ) . ( ii ) Φ ( L ) is not added in the final expression of ( 8 ) ) , whereas it is added in the definition provided in Cao & Gu [ 2019 ] . This is because we only train the L hidden layers but fix the output layer , while Cao & Gu [ 2019 ] train the entire layers of the network including the output layer . As already been pointed by several papers Cho & Saul [ 2009 ] ; Jacot et al . [ 2018 ] , it can be proved that the NTK function ( 8 ) is a positive semi-definite kernel function . Furthermore , Cho & Saul [ 2009 ] prove that the expectations in Φ and Φ̇ have closed form solutions , when the covariance matrices have the form ( 1 tt 1 ) with |t| ≤ 1 : E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ ( u ) · σ ( v ) ] = 1 2π ( t · ( π − arccos ( t ) ) + √ 1− t2 ) , E ( u , v ) ∼N ( 0 , Θ ( ` ) ) [ σ̇ ( u ) · σ̇ ( v ) ] = 1 2π ( π − arccos ( t ) ) . ( 9 ) Clearly , ( 8 ) is symmetric and continuous on the product space X ×X , from which it can be implied that Ker ( · , · ) is a Mercer kernel inducing an unique RKHS . Following Ghorbani et al . [ 2020 ] , we define the RKHS induced by ( 8 ) as : Definition 3.2 . ( NTK induced RKHS ) . For some integer p ∈ N , set of points { x̃j } pj=1 ⊂ X , and weight vector α : = { α1 , . . . , αp } ∈ Rp , define a complete vector space of functions , f : X → R , HNTKL : = cl ( { f ( · ) = p∑ j=1 αjKer ( · , x̃j ) } ) , ( 10 ) where cl ( · ) denotes closure . In the remaining of our work , we assume the regression function f ? ρ ( x ) : = E [ y|x ] belongs toHNTKL . 3.1 ASSUMPTIONS . In this subsection , we state the assumptions imposed on the data distribution with some remarks . ( A1 ) ρX is an uniform distribution on Sd−1 : = { x ∈ Rd | ‖x‖2 = 1 } , and noisy observations are assumed to be bounded . ( i.e. , ρx ∼ Unif ( Sd−1 ) , yi = O ( 1 ) , ∀i ∈ [ n ] . ) ( A2 ) Draw n i.i.d . samples { xi , f ? ρ ( xi ) } ni=1 from the joint measure ρ . Then , with probability at least 1− δ , we have λmin ( H∞L ) = λ0 > 0 . Remark 3.3 . • When the feature space is restricted on the unit sphere , the NTK function in ( 8 ) becomes rotationally invariant zonal kernel . This setting allows to adopt the results of spectral decay of ( 8 ) in the basis of spherical harmonic polynomials for measuring the complexity of hypothesis space , HNTKL . See the Appendix 3.2 and references therein . • Assumption ( A2 ) is commonly employed in NTK related literature for proving global convergence of training error and generalization error of both deep and shallow neural network , Du et al . [ 2018 ; 2019 ] ; Arora et al . [ 2019a ] . Note that the ( A2 ) holds as long as no two xi and xj are parallel to each other , which is true for most of the real-world distributions . See the proof of this claim in Du et al . [ 2019 ] . 3.2 MINIMAX RATE FOR RECOVERING f ? ρ ∈ HNTKL The obtainable minimax rate of L2-prediction error is directly related with the complexity of function space of interest . In our setting , the complexity of RKHS HNTKL can be characterized by the eigen-decay rate of the NTK function . Since Ker ( x , x′ ) is defined on the sphere , the decomposition can be given in the basis of spherical harmonics as follows : Ker ( x , x′ ) = ∞∑ k=0 µk N ( d , k ) ∑ j=1 Yk , j ( x ) Yk , j ( x ′ ) , where Yk , j , j = 1 , . . . , N ( d , k ) are spherical harmonic polynomials of degree k and { µk } ∞k=0 are non-negative eigenvalues . Recently , several researchers , both empirically [ Basri et al. , 2020 ] and theoretically [ Chen & Xu , 2020 ; Geifman et al. , 2020 ; Bietti & Bach , 2021 ] , showed that , for large enough harmonic function frequency k , the decay rate of the eigenvalues µk is in the order of Θ ( k−d ) 4 . Given this result and the fact N ( d , k ) = 2k+d−3k ( k+d−3 d−2 ) grows as kd−2 for large k , it can be easily shown λj = Θ ( j− d d−1 ) , when Ker ( x , x′ ) = ∑∞ j=1 λjφj ( x ) φj ( x ′ ) , for eigen-values λ1 ≥ λ2 ≥ · · · ≥ 0 and orthonormal basis { φj } ∞j=1 . Furthermore , it is a well known fact that if the eigenvalues decay at the rate λj = Θ ( j−2ν ) , then the corresponding minimax rate for estimating function in RKHS is O ( n− 2ν 2ν+1 ) , [ Raskutti et al. , 2014 ; Yuan & Zhou , 2016 ; Hu et al. , 2021 ] . By setting 2ν = dd−1 , we can see the minimax rate for recovering f ? ρ ∈ HNTKL isO ( n− d 2d−1 ) . Remark 3.4 . We defer all the technical proofs of the Theorems in subsections 3.3 and 3.4 in the Appendix for conciseness of the paper . At the beginning of each proof , a high-level proof idea is also provided . We also provide numerical experiments which can corroborate our theoretical findings in the Appendix A . 3.3 ANALYSIS OF UNREGULARIZED DNN In this subsection , we provide the results on the training loss of DNN estimator obtained via minimizing unregularized ` 2-loss ( 3 ) and on the corresponding estimator ’ s L2-prediction riskR ( f̂k , f ? ρ ) . Theorem 3.5 . ( Optimization ) For some δ ∈ [ 0 , 1 ] , if we set the width of the network as m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) , and we set the step-size of gradient descent as η = O ( λ0 n2L2m ) . Then , with probability at least 1−δ over the randomness of initialized parametersW ( 0 ) : = { W ( 0 ) ` } L+1 ` =1 with W ( 0 ) L+1 = v , we have for k = 0 , 1 , 2 , . . . , LS ( W ( k ) ) ≤ ( 1− ηmλ0 2 ) k LS ( W ( 0 ) ) . ( 11 ) In other words , the training loss drops to 0 at a linear rate . We acknowledge a series of past works Allen-Zhu et al . [ 2018 ] ; Du et al . [ 2019 ] have similar spirits with those in Theorem 3.5 . However , it is worth noting that their results are not applicable in our problem settings and data assumptions . Specifically , the result of Du et al . [ 2019 ] is based on the smooth and differentiable activation function , whereas the Theorem 3.5 is about the training error of ReLU activation function , which is not differentiable at 0 . Furthermore , the result of Allen-Zhu et al . [ 2018 ] relies on φ-separateness assumption stating that the every pair of feature vectors { xi , xj } n i 6=j is apart from each other by some constant φ > 0 in a Euclidean norm . In our work , the positivity assumption on the minimum eigenvalue of the NTK is imposed ( i.e. , λ0 > 0 ) . Remark 3.6 . Reducing the order of network width is definitely another line of interesting research direction . We are aware of some works in literature , but we chose not to adopt the techniques since this can make the analysis overly complicated . To the best of our knowledge , the paper that most neatly summarizes this line of literature is Zou & Gu [ 2019 ] . See the table in page 3 in their paper . The order of width they obtained is Ω ( n8L12 φ8 ) , where they impose φ-separateness assumption . Remark 3.7 . There has been an attempt to make a connection between the positivity and φseparateness assumptions . Recently , Zou & Gu [ 2019 ] proved the relation λ0 = Ω ( φn−2 ) 5 in a shallow-neural net setting . See Proposition 3.6. of their work . However , it is still an open question on whether this relation holds in DNN setting as well . The results in Theorem 3.5 suggest a positive conjecture on this question . Indeed , plugging the relation λ0 = Ω ( φn−2 ) in ( 11 ) and in the η = O ( λ0 n2L2m ) yield the discount factor ( 1− Ω ( ηmφ n2 ) ) k and step-size η = O ( φ n4L2m ) , which are exactly the same orders as presented in Allen-Zhu et al . [ 2018 ] . See Theorem 1 of their ArXiv version paper for the clear comparison . We leave the proof of this conjecture as a future work . 4In shallow neural network with ReLU activation without bias terms , it is shown that µk satisfy µ0 , µ1 > 0 , µk = 0 if k = 2j + 1 with j ≥ 1 , and otherwise µk = Θ ( k−d ) . See Bietti & Mairal [ 2019 ] . However , in ReLU DNN , it is shown that these parity constraints can be removed even without bias terms and µk achieves Θ ( k−d ) decay rate for large enough k. Readers can refer Bietti & Bach [ 2021 ] for this result . 5We conjecture that this is not the tightest lower bound on λ0 . Recently , Bartlett et al . [ 2021 ] proves that λ0 & d/n in shallow neural net setting . See Lemma 5.3 in their paper . Theorem 3.8 . ( Generalization ) Let f ? ρ ∈ HNTKL . Fix a failure probability δ ∈ [ 0 , 1 ] . Set the width of the network as m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) , the step-size of gradient descent as η = O ( λ0 n2L2m ) , and the variance parameter ω ≤ O ( ( λ0δ n ) 2/3 ) . Then , if the GD iteration k ≥ Ω ( log ( n ) ηmλ0 ) or k ≤ O ( 1 ηmωL ) , with probability at least 1− δ over the randomness of initialized parametersW ( 0 ) , we have R ( f̂k , f ? ρ ) = Ω ( 1 ) . This theorem states that if the network is trained for too long or too short , the L2-prediction error of f̂W ( k ) is bounded away from 0 by some constant factor . Specifically , the former scenario indicates that the overfitting can be harmful for recovering f ? ρ ∈ HNTKL given the noisy observations . Remark 3.9 . Readers should note that the Theorem 3.8 does not consider if the GD algorithm can achieve low prediction risk R ( f̂k , f ? ρ ) over the range of iterations ( ηmωL ) −1 . k . ( ηmλ0 ) −1 log ( n ) . In the numerical experiment to be followed in Section A , we observe that for some algorithm iterations k∗ , the risk indeed decreases to the same minimum as low as the ` 2- regularized algorithm can achieve , and increases again . This observation implies that the unregularized algorithm can achieve the minimax rate of prediction risk . However , analytically deriving a data-dependent stopping time k∗ in our scenario requires further studies , since we need a sharp characterization of eigen-distribution of NTK matrix of ReLU DNN , denoted as H∞L in this paper . Readers can refer the Theorem 4.2. of Hu et al . [ 2021 ] in shallow-neural network and equation ( 6 ) in Raskutti et al . [ 2014 ] in kernel regression context on how to compute k ? with the given eigenvalues of the associated kernel matrices . Remark 3.10 . From practitioner ’ s point of view , we can simply stop GD before it hits the nearlyzero empirical risk , but monitor the performance on a held-out validation sample , and stop training early when a minimum on the validation sample has been reached . 3.4 ANALYSIS OF ` 2-REGULARIZED DNN In this subsection , we study the training dynamics of ` 2-regularized DNN and the effects of the regularization for obtaining the minimax optimal convergence rate of L2-prediction risk . In the results to be followed , we set the orders of model parameters µ , η1 , η2 in ( 6 ) , and a variance parameter of output layer , ω as follows : µ = Θ ( n d−1 2d−1 ) , η1 = Θ ( 1 m n− 3d−2 2d−1 ) , η2 = Θ ( 1 L n− 3d−2 2d−1 ) , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) . ( 12 ) Theorem 3.11 . ( Optimization ) Suppose we minimize ` 2-regularized objective function ( 5 ) via modified GD ( 6 ) . Set the network width m log3 ( m ) ≥ Ω ( L20n24 δ2 ) and model parameters as in ( 12 ) . Then , with probability at least 1− δ , the mean-squared error follows LS ( W ( k ) D ) /n ≤ ( 1− η2µL ) k · LS ( W ( 0 ) D ) /n+OP ( 1 ) , ( 13 ) for k ≥ 0 . Additionally , after k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) iterations of ( 6 ) , for some constant C > 0 , we have ∥∥∥∥∥uD ( k ) −H∞L ( Cµ · I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ OP ( 1 n ) , ( 14 ) where we denote uD ( k ) : = [ f̂W ( k ) D ( x1 ) , . . . , f̂W ( k ) D ( xn ) ] > . Several comments are in sequel . Theorem 3.11 is , to our knowledge , the first result that rigorously shows the training dynamics of ` 2-regularized ReLU DNN in overparametrized setting . Observe that the first term on the right-hand side of the inequality ( 13 ) converges linearly to 0 , and the second term is some positive constant that is bounded away from 0 . This implies that the MSE of regularized DNN is upper-bounded by some positive constant . Note that we only provide the upper bound , but the results of our numerical experiments indicate that the MSE is lower-bounded by OP ( 1 ) as well . We leave the proof of this conjecture for the future work . The inequality ( 14 ) states that the trained dynamics of the regularized neural network can approximate the optimal solution ( denoted as g ? µ ) of the following kernel ridge regression problem : min f∈HNTK { 1 2 n∑ i=1 ( yi − f ( xi ) ) 2 + Cµ 2 ‖f‖2HNTKL } , ( 15 ) where ‖ · ‖HNTKL denotes a NTK-induced RKHS norm . ( Note that the optimization problem in ( 15 ) is not normalized by sample size n. ) The inequality ( 14 ) states that after approximately ( η2µL ) −1 iterations of ( 6 ) , the error rate becomes OP ( 1 n ) . The approximation error is computed at the training data points under ` 2 norm . This should be compared with the Theorem 5.1 of Hu et al . [ 2021 ] , where they showed that the similar approximation holds “ within ” a certain range of algorithm in shallow neural network setting . In contrast , we show that the approximation holds “ after ” k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) in deep neural network . It should be noted that the difference of results comes from the regularization scheme , where we penalize the ∑L ` =1 ‖W ` −W ( 0 ) ` ‖2F , whereas Hu et al . [ 2021 ] regularized the term ‖W1‖2F . As another important comparison , Hu et al . [ 2019 ] showed the equivalence of a solution of kernel ridge regression associated with NTK and first order Taylor expansion of the regularized neural network dynamics ; note , however , that the uD ( k ) in ( 14 ) is a full neural network dynamics . LetR ( f̂ W ( k ) D , f ? ρ ) be the L2-prediction risk of the regularized estimator f̂W ( k ) D via modified GD ( 6 ) . Next theorem states the result of generalization ability of f̂ W ( k ) D . Theorem 3.12 . ( Generalization ) Let f ? ρ ∈ HNTKL . Suppose the network width mlog3 ( m ) ≥ Ω ( L20n24 δ2 ) and model parameters are set as suggested in ( 12 ) . Then , with probability tending to 1 , we have R ( f̂ W ( k ) D , f ? ρ ) = OP ( n− d 2d−1 ) . The resulting convergence rate is O ( n− d 2d−1 ) with respect to the training sample size n. Note that the rate is always faster than O ( n−1/2 ) and turns out to be the minimax optimal [ Caponnetto & De Vito , 2007 ; Blanchard & Mücke , 2018 ] for recovering f ? ρ ∈ HNTKL in the following sense : lim r→0 lim inf n→∞ inf f̂ sup ρ P [ R ( f̂ , f ? ρ ) > rn− d 2d−1 ] = 1 , ( 16 ) where ρ is a data distribution class satisfying the Assumptions ( A1 ) , ( A2 ) and f ? ρ ∈ HNTKL , and infimum is taken over all estimators D → f̂ . It is worth noting that the minimax rate in ( 16 ) is same with the minimax rate for recovering f ? ρ ∈ HNTK1 . ( i.e. , Hu et al . [ 2021 ] ) This result can be derived from the recent discovery of the equivalence between two function spaces , HNTK1 = HNTKL . See Geifman et al . [ 2020 ] and Chen & Xu [ 2020 ] . Remark 3.13 . A particular choice of µ = Θ ( n d−1 2d−1 ) in ( 12 ) is for obtaining an optimal minimax rate for prediction error in Theorem 3.12 . Specifically , the order of µ determines the L2 distance between the f ? ρ and the kernel regressor g ? µ . That is , ‖f ? ρ − g ? µ‖22 = OP ( µ n ) . With the resultH NTK 1 = HNTKL , the same proof of Lemma D.2 . in Hu et al . [ 2021 ] can be applied for proving this result . 4 CONCLUSION We analyze the convergence rate of L2-prediction error of both the unregularized and the regularized gradient descent for overparameterized DNN with ReLU activation for a regression problem . Under a positivity assumption of NTK , we show that without the adoption of early stopping , the L2-prediction error of the estimated DNN via vanilla GD is bounded away from 0 ( Theorem 3.5 ) , whereas the prediction error of the DNN via ` 2-regularized GD achieves the optimal minimax rate ( Theorem 3.12 ) . The minimax rate O ( n− d 2d−1 ) is a faster than the O ( n−1/2 ) by specifying the complexities of target function and hypothesis space . 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Stochastic gradient descent optimizes over-parameterized deep ReLU networks . arxiv e-prints , art . arXiv preprint arXiv:1811.08888 , 2018 . Difan Zou , Yuan Cao , Dongruo Zhou , and Quanquan Gu . Gradient descent optimizes overparameterized deep ReLU networks . Machine Learning , 109 ( 3 ) :467–492 , 2020 . A NUMERICAL ILLUSTRATIONS In this section , we use synthetic data to corroborate our theoretical findings . We use the He initialization [ He et al. , 2015 ] and employ ( ` 2-regularized ) GD as introduced in subsection 2.2 . For the experiments , we run 1000 epochs of GD and use a fixed step size , setting η1 = η2 = 0.001 . We uniformly generate n feature data xitrain from Sd−1 with d = 2 and generate yi from f ? ρ ( xtraini ) with εi ∼ N ( 0 , 1 ) . To create a function f ? ρ ∈ HNTKL , we use the definition in ( 10 ) with α ∈ Unif ( Sp−1 ) and with p fixed points { x̃j } pj=1 ⊂ Unif ( Sd−1 ) , where p is simply set as 1 . Note that Ker ( · , · ) in ( 10 ) can be calculated via the formulas ( 8 ) and ( 9 ) with specified network depth L. We consider a scenario where we have a network with depth L = 8 and width m = 2000 . The variance parameter of the output layer ( ω ) is set as 1 for unregularized and 0.001 for regularized cases . In Fig 1 . ( a ) , we record the training errors of regularized networks over the GD epochs k ≤ 1000 , where we have n ∈ { 100 , 300 , 500 , 1000 , 5000 } training samples . This aims to verify the inequality ( 13 ) that the MSE of regularized network is bounded away from 0 by some constant . In Fig 1 . ( b ) , the prediction risks of both unregularized and regularized networks are displayed . We approximate the risk with 1500 ∑500 j=1 ( f̂k ( x test j ) − f ? ρ ( xtestj ) ) 2 with a new test data set { xtestj , f ? ρ ( xtestj ) } 500j=1 over k ≤ 1000 for both unregularized and regularized cases . In both cases , they reach the same minimal risks , but the risk of unregularized network increase after it hits the minimal point , whereas the risk of regularized network stays stable . Theorem 3.6 tells us that for the iteration less than the order O ( 1 ηmωL ) , the prediction error is bounded away from 0 . In the experiment for unregularized case , we set η = 0.01 , m = 2000 , L = 8 , and ω = 1 . Plugging in these parameters in the bound says that the minimum can be achieved within a very few iterations . Note that the optimal risk is non-zero as long as we have finite sample sizes n , but converges to 0 at the rate O ( n− d 2d−1 ) . In Fig 1 . ( c ) , we verify that the more training sample sizes we have , the closer the risks of the regularized networks get to 0 . The risk is evaluated at the sample sizes n = { 100 , 300 , 500 , 1000 , 5000 } . We have to acknowledge that there is a discrepancy between our experiment setting and theory . Specifically , due to the limited computing power , we could not run the experiment under the regime of width m log3 ( m ) ≥ Ω ( ω7n8L18 λ80δ 2 ) . But the prediction risk behaves similarly as expected by our theorem , which can be a partial evidence that the statement in Theorem 3.6 still holds in the narrower width of the network . B PRELIMINARY NOTATIONS Before presenting the formal proofs of Lemmas and main results , we introduce several notations used frequently throughout the proofs . First , we denote x ` , i the output of the ` th hidden layer with the input data xi after applying entry-wise ReLU activation function . x ` , i = σ ( W ` σ ( W ` −1 · · ·σ ( W1xi ) · · · ) ) . Denote fW ( k ) ( x ) a value of neural network ( 2 ) evaluated at the collection of network parameters W ( k ) : = { W ( k ) ` } ` =1 , ... , L and W ( k ) ` denotes the ` th hidden layer parameter updated by kth GD iterations . Partial gradient of fW ( k ) ( x ) . We employ the following matrix product notation which was used in several other papers [ Zou et al. , 2018 ; Cao & Gu , 2019 ] : ` 2∏ r= ` 1 Ar : = { A ` 2A ` 2−1 · · ·A ` 1 if ` 1 ≤ ` 2 , I otherwise . ( 17 ) Then , the partial gradient of fW ( k ) ( x ) with respect to W ( k ) ` for 1 ≤ ` ≤ L has a following form : for i ∈ { 1 , . . . , n } , ∇W ` [ fW ( k ) ( xi ) ] = √ m · [ x ( k ) ` −1 , iv T ( L∏ r= ` +1 Σ ( k ) r , i W ( k ) r ) Σ ( k ) ` , i ] > , ` ∈ [ L ] , where Σ ( k ) ` , i : = Diag ( 1 ( 〈w ( k ) ` ,1 , x ( k ) ` −1 , i〉 ≥ 0 ) , . . . ,1 ( 〈w ( k ) ` , m , x ( k ) ` −1 , i〉 ≥ 0 ) ) ∈ Rm×m and w ( k ) ` , j denotes jth column of the matrix W ( k ) ` . Gram matrix H ( k ) . Each entries of empirical gram matrix evaluated at the kth GD update are defined as follows : Hi , j ( k ) = 1 m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr . Note that H ( 0 ) → H∞L as m → ∞ which is proved in Jacot et al . [ 2018 ] ; Yang [ 2019 ] ; Lee et al . [ 2018 ] ; Arora et al . [ 2019b ] . Perturbation region of weight matrices . Consider a collection of weight matrices W̃ = { W̃ ` } ` =1 , ... , L such that W̃ ∈ B ( W ( 0 ) , τ ) : = { W̃ ` : ‖W̃ ` −W ( 0 ) ` ‖2 ≤ τ , ∀ ` ∈ [ L ] } . ( 18 ) For all i ∈ { 1 , . . . , n } and ` = 1 , . . . , L , we denote x ` , i and x̃ ` , i as the outputs of the ` -th layer of the neural network with weight matrices W ( 0 ) and W̃ , and Σ ` , i and Σ̃ ` , i are diagonal matrices with ( Σ ` , i ) jj = 1 ( 〈w ( 0 ) ` , j , x ` −1 , i〉 ≥ 0 ) and ( Σ̃ ` , i ) jj = 1 ( 〈w̃ ` , j , x̃ ` −1 , i〉 ≥ 0 ) , respectively . C WHY IS IT HARD TO PROVE ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) ? In this subsection , we provide a heuristic argument on why it is hard to prove ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) , where W ( k ) D , ` is the model parameter of ` th layer in kth iteration of algorithm . Here , we regularize solely on the model parameter , instead on the relative to the initialization . In this case , we can write the update rule as follows : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . ( 19 ) By recursively applying above equation ( 4.3 ) , we can write W ( k ) D , ` with respect to W ( 0 ) D , ` as follows : W ( k ) D , ` = ( 1− η2µ ) kW ( 0 ) D , ` − η1 k−1∑ ` =0 ( 1− η2µ ) ` ∇W ` [ LS ( W ( k− ` −1 ) D ) ] . Then , we can control the bound as follows : ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ ( 1− ( 1− η2µ ) k ) ∥∥∥W ( 0 ) D , ` ∥∥∥ 2 + η1 η2µ max ` =0 , ... , k−1 ∥∥∥∇W ` [ LS ( W ( k− ` −1 ) D ) ] ∥∥∥ 2 . We know under the initialization setting in our paper , ‖W ( k ) D , ` ‖2 ≤ O ( 1 ) with high-probability ( see Vershynin [ 2018 ] ) , and as long as we can prove the ` 2-norm of gradient is bounded , then we can conclude ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ O ( 1 ) . However , we are not aware of works in which they control the size of ‖∇W ` [ LS ( W ( k− ` −1 ) D ) ] ‖2 where the non-convex interactions between model parameters across the hidden layers are allowed . To the best of our knowledge , we know the work Allen-Zhu et al . [ 2019 ] deals with the three layer case under this setting . But we need further investigations on whether the techniques employed in their paper can be generalized to arbitrary L-hidden layer setting . D USEFUL LEMMAS A simple fact . Suppose vj i.i.d∼ N ( 0 , ωm ) for j ∈ [ m ] . Then , with probability at least 1 − exp [ −Ω ( m ) ] , ‖v‖22 ≤ O ( ω ) . Proof . Since ∥∥v2j∥∥Ψ1 ≤ O ( ωm ) for j ∈ [ m ] , where ‖ · ‖Ψ1 denotes a sub-exponential norm , Bernstein ’ s inequality for i.i.d . centered sub-exponential random variables can be employed : For any t ≥ 0 , P ( ∣∣∣∣∣∣ m∑ j=1 ( v2j − ω m ) ∣∣∣∣∣∣ ≥ t ) ≤ 2 exp ( − cmin ( t2∑m j=1 ∥∥v2j∥∥2Ψ1 , t maxj ∥∥v2j∥∥Ψ1 ) ) , ( 20 ) where c > 0 is an absolute constant . Note that we used the fact centering does not hurt the subexponentiality of random variable . Choosing t = O ( ω ) concludes the proof . Lemma 4.1 ( Lemma 7.1 . Allen-Zhu et al . [ 2018 ] ) . With probability at least 1 − O ( nL ) · exp [ −Ω ( m/L ) ] , 3/4 ≤ ‖x ( 0 ) ` , i ‖2 ≤ 5/4 for all i ∈ { 1 , . . . , n } and ` ∈ { 1 , . . . , L } . Lemma 4.2 ( Lemma B.1 . Cao & Gu [ 2019 ] ) . If τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , then with probability at least 1 − O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , 1/2 ≤ ‖x̃ ` , i‖2 ≤ 3/2 for all W̃ ∈ B ( W ( 0 ) , τ ) , i ∈ { 1 , . . . , n } and ` ∈ { 1 , . . . , L } . Lemma 4.3 ( Allen-Zhu et al . [ 2018 ] ) . Uniformly over i ∈ { 1 , . . . , n } and 1 ≤ ` 1 ≤ ` 2 ≤ L , the following results hold : 1 . ( Lemma.7.3 , Allen-Zhu et al . [ 2018 ] ) Suppose m ≥ Ω ( nL log ( nL ) ) , then with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥∥∥ ` 2∏ r= ` 1 Σ ( 0 ) r , iW ( 0 ) r ∥∥∥∥∥ 2 ≤ O ( √ L ) . 2 . ( Lemma.7.4 , Allen-Zhu et al . [ 2018 ] ) Suppose m ≥ Ω ( nL log ( nL ) ) , then with probability at least , 1−O ( nL ) · exp [ −Ω ( m/L ) ] , ∥∥∥∥∥v > ( L∏ r= ` 1 Σ ( 0 ) r , iW ( 0 ) r ) ∥∥∥∥∥ 2 ≤ O ( √ w ) . 3 . ( Lemma.8.2 , Allen-Zhu et al . [ 2018 ] ) Suppose τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) . For all W̃ ∈ B ( W ( 0 ) , τ ) , with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥x̃ ` 1 , i − x ( 0 ) ` 1 , i∥∥∥2 ≤ O ( τL5/2√log ( m ) ) . 4 . ( Corollary.8.4 , Allen-Zhu et al . [ 2018 ] ) Suppose τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , then with probability at least , 1−O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , ∥∥∥Σ̃ ` 1 , i −Σ ( 0 ) ` 1 , i∥∥∥0 ≤ O ( mτ2/3L ) . 5 . ( Lemma.8.7 , Allen-Zhu et al . [ 2018 ] ) For all ` ∈ [ L ] , let Σ′′ ` , i ∈ [ −3 , 3 ] m×m be the diagonal matrices with at most s = O ( mτ2/3L ) non-zero entries . For all W̃ ∈ B ( W ( 0 ) , τ ) , where τ = O ( 1 L1.5 ) , with probability at least 1−O ( nL ) · exp [ −Ω ( s log ( m ) ) ] , ∥∥∥∥∥vT ( L∏ r= ` 1+1 ( Σ′′r , i + Σ ( 0 ) r , i ) W̃r , i ) ( Σ′′ ` 1 , i + Σ ( 0 ) r , i ) − vT ( L∏ r= ` 1+1 Σ ( 0 ) r , iW ( 0 ) r , i ) Σ ( 0 ) ` 1 , i ∥∥∥∥∥ 2 ≤ O ( τ1/3L2 √ ω log ( m ) ) . Lemma 4.4 ( Lemma B.3 . Cao & Gu [ 2019 ] ) . There exists an absolute constant κ such that , with probability at least 1 − O ( nL2 ) · exp [ −Ω ( mτ2/3L ) ] , i ∈ 1 , . . . , n and ` ∈ 1 , . . . , L and for all W̃ ∈ B ( W ( 0 ) , τ ) , with τ ≤ κL−6 [ log ( m ) ] −3 , it holds uniformly that∥∥∇W ` [ fW̃ ( xi ) ] ∥∥2 ≤ O ( √ωm ) . Lemma 4.5 . Suppose W̃ ∈ B ( W ( 0 ) , τ ) and τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) . For all u ∈ Rm with a cardinality ‖u‖0 ≤ s , for any 1 ≤ ` ≤ L and i ∈ { 1 , . . . , n } , with probability at least 1−O ( nL ) · exp ( − Ω ( s log ( m ) ) ) −O ( nL ) · exp ( − Ω ( mτ2/3L ) ) , ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≤ √ ωs log ( m ) m · O ( ‖u‖2 ) . Proof . Recall Lemma 4.2 . For any fixed vector u ∈ Rm , with probability at least 1 − O ( nL ) · exp [ −Ω ( mτ2/3L ) ] for τ ≤ O ( L−9/2 [ log ( m ) ] −3 ) , for any 1 ≤ ` ≤ L and i ∈ { 1 , . . . , n } , we have the event T , ∥∥∥∥∥ ( L∏ r= ` Σ̃r , iW̃r , i ) u ∥∥∥∥∥ 2 ≤ 3 ‖u‖2 . ( 21 ) Conditioned on this event happens , it is easy to see the random variable v > ( ∏L r=a Σ̃r , iW̃r , i ) u ∼ SG ( 9ω m ‖u‖ 2 2 ) . Based on this observation , we have the probability , P ( ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≥ √ ωs log ( m ) m · O ( ‖u‖2 ) ) ≤ P ( ∣∣∣∣∣v > ( L∏ r= ` Σ̃r , iW̃r , i ) u ∣∣∣∣∣ ≥ √ ωs log ( m ) m · O ( ‖u‖2 ) ∣∣∣ T ) + P ( T c ) ≤ O ( nL ) · exp ( − Ω ( s log ( m ) ) ) +O ( nL ) · exp ( − Ω ( mτ2/3L ) ) , where in the last inequality , union bounds over the indices ` and i , and over the vector u ∈ Rm with ‖u‖0 ≤ s are taken . Lemma 4.6 . Suppose τ ≤ 1 CL9/2 [ log ( m ) ] 3 for some constant C > 0 . Then , for all i ∈ [ n ] and ` ∈ [ L ] , with probability at least 1−O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , we have ∥∥∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥2 ≤ O ( τ1/3L2√ωm log ( m ) ) . Proof . By using the results from Lemma 4.3 , we can control the term : ‖∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ‖2 = √ m · ∥∥∥∥∥x ( k ) ` −1vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` − x ( 0 ) ` −1v T ( L∏ r= ` +1 Σ ( 0 ) r W ( 0 ) r ) Σ ( 0 ) ` ∥∥∥∥∥ 2 ≤ √ m · ‖x ( k ) ` −1 − x ( 0 ) ` −1 ‖2︸ ︷︷ ︸ ≤O ( τL5/2 √ log ( m ) ) · ∥∥∥∥∥vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` ∥∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ ω ) + √ m · ∥∥∥x ( 0 ) ` −1∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) · ∥∥∥∥∥vT ( L∏ r= ` +1 Σ ( k ) r W ( k ) r ) Σ ( k ) ` − v T ( L∏ r= ` +1 Σ ( 0 ) r W ( 0 ) r ) Σ ( 0 ) ` ∥∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ω log ( m ) ) ≤ O ( τ1/3L2 √ ωm log ( m ) ) , where , in the last inequality , we used the condition on τ ≤ 1 CL9/2 [ log ( m ) ] 3 < 1 . Remark 4.7 . Note that the results in Lemmas 6.3 ( second and fifth items ) , 6.4 , 6.5 , 6.6 are in the setting of vj ∼ N ( 0 , ωm ) for j ∈ [ m ] . For the notational convenience , in following Lemmas we denote fW ( k ) ( xi ) as ui ( k ) and let u ( k ) : = [ u1 ( k ) , . . . , un ( k ) ] > for k ≥ 0 . Lemma 4.8 . For some δ ∈ [ 0 , 1 ] , if m ≥ Ω ( L log ( nL/δ ) ) , then with probability at least 1 − δ , ‖u ( k ) ‖2 ≤ O ( √nω δ ) for any k ≥ 0 . Proof . Recall the Lemma 4.2 stating that ∥∥∥x ( k ) L , i∥∥∥ 2 = O ( 1 ) for any input data xi for i ∈ [ n ] . Also recall that vj ∼ N ( 0 , ωm ) for j ∈ [ m ] , xL , i ∈ R m and ui ( k ) = √ mv > xL , i ∼ N ( 0 , O ( ω ) ) . Then , we have a following via simple Markov inequality : for any t ≥ 0 , P ( ‖u ( k ) ‖2 ≥ t ) ≤ E [ ‖u ( k ) ‖2 ] t ≤ √ E [ ‖u ( k ) ‖22 ] t ≤ O ( √ nω ) t . Lemma 4.9 . For some δ ∈ [ 0 , 1 ] , if m ≥ Ω ( L log ( nL/δ ) ) , then with probability at least 1− δ , we have ‖u ( 0 ) − y‖2 ≤ O ( √ n δ ) . Proof . By Markov ’ s inequality , for any t ≥ 0 , P ( ‖u ( 0 ) − y‖2 ≥ t ) ≤ Eε , W ( 0 ) , v [ ‖u ( 0 ) − y‖22 ] t2 . ( 22 ) Note that the expectation in the nominator of ( 22 ) is taken over the random noise ε and initialized parameter W ( 0 ) , v. We can expand the nominator as follows : Eε , W ( 0 ) , v [ ‖u ( 0 ) − y‖22 ] = EW ( 0 ) , v‖u ( 0 ) ‖22 + Eε‖y‖22 − 2Eε , W ( 0 ) , v [ y > u ( 0 ) ] . ( 23 ) For the convenience of notation , let y∗ : = [ f ? ρ ( x1 ) , . . . , f ? ρ ( xn ) ] > and ε : = [ ε1 , . . . , εn ] > . Recall that we have y = y∗ + ε , and ‖y∗‖22 = O ( n ) . Also note that by Lemma 4.1 , with probability at least 1−O ( nL ) · exp [ −Ω ( m/L ) ] , for any i = 1 , . . . , n , ‖x ( 0 ) L , i‖22 = O ( 1 ) . Then , we have a random variable ui ( 0 ) = √ mv > xL , i ∼ N ( 0 , O ( ω ) ) . Now , we are ready to derive the orders of three terms on the RHS of ( 23 ) . EW ( 0 ) , v‖u ( 0 ) ‖22 = O ( n ) , Eε‖y‖22 = Eε [ ‖y∗‖22 + ‖ε‖22 − 2y > ε ] = O ( n ) , Eε , W ( 0 ) , v [ y > u ( 0 ) ] = Eε , W ( 0 ) , v [ ( y∗ + ε ) > u ( 0 ) ] = 0 . Combining the above three equalities , we conclude the proof . Lemma 4.10 . Suppose τ = O ( n√ω√ mδλ0 ) . For some δ ∈ [ 0 , 1 ] such that δ ≥ O ( nL ) · exp [ −Ω ( mτ2/3L ) ] , then with probability at least 1− δ , we have ‖H ( k ) −H ( 0 ) ‖2 ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) . Proof . By the definition of gram matrix Hi , j ( k ) for any k ≥ 0 , we have |Hi , j ( k ) −Hi , j ( 0 ) | = ∣∣∣∣∣ 1m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr − 〈 ∇W ` [ fW ( 0 ) ( xi ) ] , ∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ∣∣∣∣∣ ≤ 1 m L∑ ` =1 { ∣∣∣∣〈∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] −∇W ` [ fW ( 0 ) ( xj ) ] 〉 Tr ∣∣∣∣ + ∣∣∣∣〈∇W ` [ fW ( 0 ) ( xj ) ] , ∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] 〉 Tr ∣∣∣∣ } ≤ 1 m L∑ ` =1 { ∥∥∇W ` [ fW ( k ) ( xi ) ] ∥∥2︸ ︷︷ ︸ ≤O ( √ ωm ) · ∥∥∇W ` [ fW ( k ) ( xj ) ] −∇W ` [ fW ( 0 ) ( xj ) ] ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) + ∥∥∇W ` [ fW ( 0 ) ( xj ) ] ∥∥2︸ ︷︷ ︸ ≤O ( √ ωm ) · ∥∥∇W ` [ fW ( k ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) } ≤ O ( ω7/6n1/3L3 6 √ log3 ( m ) mδλ20 ) . In the second inequality , Lemmas 4.4 and 4.6 are used , and in the last inequality , τ = O ( n√ω√ mδλ0 ) is plugged in . With this , using the fact that Frobenius norm of a matrix is bigger than the operator norm , we bound the term ‖H ( k ) −H ( 0 ) ‖2 as follows : ‖H ( k ) −H ( 0 ) ‖2 ≤ ‖H ( k ) −H ( 0 ) ‖F ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) . Lemma 4.11 . For some δ ∈ [ 0 , 1 ] , with probability at least 1− δ , ‖H∞L −H ( 0 ) ‖2 ≤ O ( ωnL5/2 4 √ log ( nL/δ ) m ) Proof . For some δ ′ ∈ [ 0 , 1 ] , set ε = L3/2 4 √ log ( L/δ ′ ) m from Theorem 3.1. of Arora et al . [ 2019b ] . For any fixed i , j ∈ [ n ] , we have P [ ∣∣H∞i , j −Hi , j ( 0 ) ∣∣ ≤ O ( ωL5/2 4√ log ( L/δ′ ) m ) ] ≥ 1− δ′ . After applying the union bound over all i , j ∈ [ n ] , setting δ = δ ′ n2 , and using the fact that Frobenius norm of a matrix is bigger than the operator norm , we conclude the proof . For two positive semi-definite matrices A and B , if we write A B , then it means A − B is positive semi-definite matrix . Similarly , if we write A B , then it means A−B is positive definite matrix . With these notations , we introduce a following Lemma . Lemma 4.12 ( Lemma D.6 . Hu et al . [ 2021 ] ) . For two positive semi-definite matrices A and B , 1 . Suppose A is non-singular , then A B ⇐⇒ λmax ( BA−1 ) ≤ 1 and A B ⇐⇒ λmax ( BA −1 ) < 1 , where λmax ( · ) denotes the maximum eigenvalue of the input matrix . 2 . Suppose A , B and Q are positive definite matrices , A and B are exchangeable , then A B =⇒ AQA BQB . E PROOF OF THEOREM 3.5 For the convenience of notation , denote ui ( k ) = fW ( k ) ( xi ) and let u ( k ) = [ u1 ( k ) , u2 ( k ) , . . . , un ( k ) ] > . In order to achieve linear convergence rate of the training error , ‖u ( k ) − y‖22 , we decompose the term as follows : ‖u ( k + 1 ) − y‖22 = ∥∥u ( k ) − y + ( u ( k + 1 ) − u ( k ) ) ∥∥2 2 = ‖u ( k ) − y‖22 − 2 ( u ( k ) − y ) > ( u ( k + 1 ) − u ( k ) ) + ‖u ( k + 1 ) − u ( k ) ‖22 . Equipped with this decomposition , the proof consists of the following steps : 1 . Similarly with Du et al . [ 2019 ] , a term ( u ( k + 1 ) − u ( k ) ) is decomposed into two terms , where we denote them as I ( k ) 1 and I ( k ) 2 , respectively . We note that the first term I ( k ) 1 is related with a gram matrix H ( k ) and a second term I ( k ) 2 can be controlled small enough in ` 2 sense with proper choices of the step size and the width of network . 2 . A term ‖u ( k + 1 ) − u ( k ) ‖22 needs to be controlled small enough to ensure 2 ( u ( k ) − y ) > ( u ( k + 1 ) − u ( k ) ) > ‖u ( k + 1 ) − u ( k ) ‖22 so that the loss decreases . 3 . It is shown that the distance between the gram matrix H ( k ) and the NTK matrix H∞L is close enough in terms of operator norm . 4 . Lastly , we inductively show that the weights generated from gradient descent stay within a perturbation region B ( W ( 0 ) , τ ) , irrespective with the number of iterations of algorithm , We start the proof by analyzing the term u ( k + 1 ) − u ( k ) . Step 1 . Control on u ( k + 1 ) − u ( k ) . Recall ( Σ ( k ) ` , i ) jj = 1 ( 〈w ( k ) ` , j , x ( k ) ` −1 , i〉 ≥ 0 ) and we introduce a diagonal matrix Σ̃ ( k ) ` , i , whose jth entry is defined as follows : ( Σ̃ ( k ) ` , i ) jj = ( Σ ( k+1 ) ` , i −Σ ( k ) ` , i ) jj · 〈w ( k+1 ) ` , j , x ( k+1 ) ` −1 , i 〉 〈w ( k+1 ) ` , j , x ( k+1 ) ` −1 , i 〉 − 〈w ( k ) ` , j , x ( k ) ` −1 , i〉 . With this notation , the difference x ( k+1 ) L , i − x ( k ) L , i can be rewritten via the recursive applications of Σ̃ ( k ) ` , i : x ( k+1 ) L , i − x ( k ) L , i = ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) ( W ( k+1 ) L x ( k+1 ) L−1 , i −W ( k ) L x ( k ) L−1 , i ) = ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) W ( k+1 ) L ( x ( k+1 ) L−1 , i − x ( k ) L−1 , i ) + ( Σ ( k ) L , i + Σ̃ ( k ) L , i ) ( W ( k+1 ) L −W ( k ) L ) x ( k ) L−1 , i = L∑ ` =1 ( L∏ r= ` +1 ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) W ( k+1 ) r ) ( Σ ( k ) ` , i + Σ̃ ( k ) ` , i ) ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i ( 24 ) Then , we introduce following notations : D ( k ) ` , i = ( L∏ r= ` +1 Σ ( k ) r , i W ( k ) r ) Σ ( k ) ` , i , D̃ ( k ) ` , i = ( L∏ r= ` +1 ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) W ( k+1 ) r ) ( Σ ( k ) ` , i + Σ̃ ( k ) ` , i ) . Now , we can write ui ( k + 1 ) − ui ( k ) by noting that ui ( k ) = √ m · vTx ( k ) L , i : ui ( k + 1 ) − ui ( k ) = √ m · vT ( x ( k+1 ) L , i − x ( k ) L , i ) = √ m · vT L∑ ` =1 D̃ ( k ) ` , i ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i ( 25 ) = −η √ m · vT L∑ ` =1 D ( k ) ` , i ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i︸ ︷︷ ︸ I ( k ) 1 , i −η √ m · vT L∑ ` =1 ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i︸ ︷︷ ︸ I ( k ) 2 , i Here , I ( k ) 1 , i can be rewritten as follows : I ( k ) 1 , i = −η √ m · vT L∑ ` =1 D ( k ) ` , i n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] x ( k ) ` −1 , i = −η · n∑ j=1 ( uj ( k ) − yj ) · ( √ m L∑ ` =1 vTD ( k ) ` , i ∇W ` [ fW ( k ) ( xj ) ] x ( k ) ` −1 , i ) = −mη · n∑ j=1 ( uj ( k ) − yj ) · 1 m L∑ ` =1 〈 ∇W ` [ fW ( k ) ( xi ) ] , ∇W ` [ fW ( k ) ( xj ) ] 〉 Tr = −mη · n∑ j=1 ( uj ( k ) − yj ) ·Hi , j ( k ) . For I ( k ) 2 , i , we need a more careful control . First , we pay our attention on bounding the term ‖v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ‖2 as follows : By triangle inequality , we have∥∥∥∥v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∥∥∥∥ 2 ≤ ∥∥∥∥v > ( D ( k ) ` , i −D ( 0 ) ` , i ) ∥∥∥∥ 2 + ∥∥∥∥v > ( D̃ ( k ) ` , i −D ( 0 ) ` , i ) ∥∥∥∥ 2 . ( 26 ) We control the first term of the right-hand side ( R.H.S ) in ( 26 ) . By the fourth item of the Lemma 4.3 , we know ‖Σ ( k ) r , i −Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) and | ( Σ ( k ) r , i −Σ ( 0 ) r , i ) j , j | ≤ 1 for j ∈ [ m ] . Then , we can plug Σ′′r , i = Σ ( k ) r , i −Σ ( 0 ) r , i in the inequality of the fifth item of Lemma 4.3 . So , the first term of the R.H.S in ( 26 ) can be bounded by O ( τ1/3L2 √ ω log ( m ) ) . The second term of the R.H.S in ( 26 ) can be similarly controlled as the first term . Observe that | ( Σ ( k ) r , i +Σ̃ ( k ) r , i ) jj | ≤ 1 , then we have | ( Σ ( k ) r , i +Σ̃ ( k ) r , i −Σ ( 0 ) r , i ) j , j | ≤ 2 for all j ∈ [ m ] . Note that by the definition of Σ̃ ( k ) r , i , we have ‖Σ̃ ( k ) r , i ‖0 = ‖Σ ( k+1 ) r , i −Σ ( k ) r , i ‖0 ≤ ‖Σ ( k+1 ) r , i −Σ ( 0 ) r , i ‖0+‖Σ ( k ) r , i −Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) . Thus , by the triangle inequality , we have ‖Σ ( k ) r , i + Σ̃ ( k ) r , i − Σ ( 0 ) r , i ‖0 ≤ O ( mτ2/3L ) . These observations enable us to plug Σ′′r , i = Σ ( k ) r , i + Σ̃ ( k ) r , i −Σ ( 0 ) r , i in the inequality of the fifth item of Lemma 4.3 , and give the bound on the second term as O ( τ1/3L2 √ ω log ( m ) ) . We have ‖v > ( D̃ ( k ) ` , i − D ( k ) ` , i ) ‖2 ≤ O ( τ1/3L2 √ ω log ( m ) ) . Now , we control the ` 2-norm of the I ( k ) 2 as follows : ∥∥∥I ( k ) 2 ∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 2 , i ∣∣∣ ≤ η √ m · n∑ i=1 [ L∑ ` =1 ∥∥∥∥v > ( D̃ ( k ) ` , i −D ( k ) ` , i ) ∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( L2τ1/3 √ ω log ( m ) ) · ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 · ∥∥∥x ( k ) ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) : Lemma 4.2 ] ≤ O ( ηnL2τ1/3 √ ωm log ( m ) ) L∑ ` =1 ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 ≤ O ( ηnL5/2τ1/3 √ ωm log ( m ) ) √√√√ L∑ ` =1 ∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥2F ≤ O ( ηnL5/2τ1/3 √ ωm log ( m ) ) √√√√ n∑ j=1 ( uj ( k ) − yj ) 2 L∑ ` =1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥2F ≤ O ( ηnL3τ1/3ωm √ log ( m ) ) ‖u ( k ) − y‖2 . ( 27 ) Step 2 . Control on ‖u ( k + 1 ) −u ( k ) ‖22 . Recall that by ( 25 ) , x ( k+1 ) L , i −x ( k ) L , i can be written as follows : x ( k+1 ) L , i − x ( k ) L , i = L∑ ` =1 D̃ ( k ) ` , i ( W ( k+1 ) ` −W ( k ) ` ) x ( k ) ` −1 , i = −η · L∑ ` =1 D̃ ( k ) ` , i ∇W ` [ LS ( W ( k ) ) ] x ( k ) ` −1 , i . It is worth noting that , ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥2 2 = ∥∥∥∥∥∥ n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] ∥∥∥∥∥∥ 2 2 ≤ n∑ j=1 ( uj ( k ) − yj ) 2 n∑ j=1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥22 ≤ O ( nmω ) ‖u ( k ) − y‖22 . ( 28 ) Also , observe that | ( Σ ( k ) r , i + Σ̃ ( k ) r , i ) jj | ≤ 1 for all j ∈ [ m ] , so by Lemma A.3 of Zou et al . [ 2020 ] , we know ‖D̃ ( k ) ` , i ‖2 ≤ O ( √ L ) . Combining all the facts , we can conclude : ‖u ( k + 1 ) − u ( k ) ‖22 = m · n∑ i=1 ( vTx ( k+1 ) L , i − v Tx ( k ) L , i ) 2 ≤ m · ‖v‖22 n∑ i=1 ∥∥∥x ( k+1 ) L , i − x ( k ) L , i∥∥∥2 2 ≤ η2m · ‖v‖22 n∑ i=1 [ L∑ ` =1 ∥∥∥D̃ ( k ) ` , i ∥∥∥2 2 · ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥2 2 · ∥∥∥x ( k ) ` −1 , i∥∥∥2 2 ] ≤ O ( η2n2L2m2ω2 ) ‖u ( k ) − y‖22 ≤ O ( η2n2L2m2 ) ‖u ( k ) − y‖22 , ( 29 ) where in the third inequality , we additionally used the fact ‖v‖22 = O ( ω ) with probability at least 1− exp ( −Ω ( m ) ) , and the inequality ( 28 ) . In the last inequality , we used the assumption ω ≤ 1 . Step 3. λmin ( H ( k ) ) ≥ λ02 with sufficiently large m. Denote ρ ( A ) as a sprectral radius of a matrix A . Then , we have ‖H ( k ) −H∞L ‖2 ≥ ρ ( H ( k ) −H∞L ) ≥ −λmin ( H ( k ) −H∞L ) ≥ λmin ( H∞L ) − λmin ( H ( k ) ) ≥ λ0 − λmin ( H ( k ) ) , ( 30 ) where , in the second inequality , we used a triangle inequality , λmin ( H ( k ) −H∞L ) + λmin ( H∞L ) ≤ λmin ( H ( k ) ) . By Lemmas 4.10 and 4.11 , setting m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ ) and Õ ( λ 4/3 0 δ 1/3 n4/3L4 ) ≤ ω ≤ 1 , we have ‖H ( k ) −H∞L ‖2 ≤ ‖H ( k ) −H ( 0 ) ‖2 + ‖H ( 0 ) −H ∞ L ‖2 ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) +O ( ωn2L5/2 4 √ log ( nL/δ ) m ) ≤ O ( ω7/6n4/3L3 6 √ log3 ( m ) mδλ20 ) ≤ λ0 2 . ( 31 ) Thus , combining ( 30 ) and ( 31 ) yields that λmin ( H ( k ) ) ≥ λ02 . Step 4 . Concluding the proof . Recall that I ( k ) 1 = −mη · H ( k ) ( u ( k ) − y ) . Then observe that ( u ( k ) − y ) > I ( k ) 1 = −ηm · ( u ( k ) − y ) > H ( k ) ( u ( k ) − y ) ≤ −ηm · λmin ( H ( k ) ) ‖u ( k ) − y‖22 ≤ −ηm · λ0 2 ‖u ( k ) − y‖22 . ( 32 ) We set the step size η , radius of perturbation region τ and network width m as follows , η = Ω ( λ0 n2L2m ) , τ = O ( n √ ω√ mδλ0 ) , m ≥ Ω ( ω7n8L18 log3 ( m ) λ80δ ) . With the above settings , we can control the ‖u ( k + 1 ) − y‖22 by combining ( 27 ) , ( 29 ) and ( 32 ) as follows , ‖u ( k + 1 ) − y‖22 = ∥∥u ( k ) − y + ( u ( k + 1 ) − u ( k ) ) ∥∥2 2 = ‖u ( k ) − y‖22 − 2ηm · ( u ( k ) − y ) > H ( k ) ( u ( k ) − y ) − ( u ( k ) − y ) > I ( k ) 2 + ‖u ( k + 1 ) − u ( k ) ‖ 2 2 ≤ ( 1− ηmλ0 +O ( ηnL3τ1/3mω √ log ( m ) ) +O ( η2n2L2m2 ) ) ‖u ( k ) − y‖22 ≤ ( 1− ηmλ0 2 ) ‖u ( k ) − y‖22 . So far , we have shown from Step 1 to Step 4 that given the radius of perturbation region τ has the order O ( n√ω√ mδλ0 ) , then we can show the training error drops linearly to 0 with the discount factor ( 1 − ηmλ02 ) along with the proper choices of η and m. It remains us to prove the iterates W ( k ) ` for all ` ∈ [ L ] generated by GD algorithm indeed stay in the perturbation region B ( W ( 0 ) , τ ) over k ≥ 0 with τ = O ( n√ω√ mδλ0 ) . Step 5 . The order of the radius of perturbation region . We employ the induction process for the proof . The induction hypothesis is : ∀s ∈ [ k + 1 ] , ∥∥∥W ( s ) ` −W ( 0 ) ` ∥∥∥ 2 ≤ η · O ( √ nmω ) s−1∑ t=0 ( 1− ηmλ0 2 ) t 2 O ( √ n δ ) ≤ O ( n √ ω√ mδλ0 ) . ( 33 ) First , it is easy to see it holds for s = 0 . Now , suppose it holds for s = 0 , . . . , k , we consider s = k + 1.∥∥∥W ( k+1 ) ` −W ( k ) ` ∥∥∥ 2 = ∥∥∥∇W ` [ LS ( W ( k ) ) ] ∥∥∥ 2 = η · ∥∥∥∥∥∥ n∑ j=1 ( uj ( k ) − yj ) ∇W ` [ fW ( k ) ( xj ) ] ∥∥∥∥∥∥ 2 ≤ η · √√√√ n∑ j=1 ∥∥∇W ` [ fW ( k ) ( xj ) ] ∥∥22 √√√√ n∑ j=1 ( uj ( k ) − yj ) 2 ≤ η · O ( √ nmω ) √ 2LS ( W ( k ) ) ≤ η · O ( √ nmω ) ( 1− ηmλ0 2 ) k 2 O ( √ n δ ) , ( 34 ) where in the second inequality , we used Lemmas 4.4 . Note that since it is assumed that W ( k ) ` ∈ B ( W ( 0 ) , τ ) , the Lemma is applicable with m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ ) . Simi- larly , since it is assumed that the induction hypothesis holds for s = 0 , . . . , k , we can see ‖u ( k ) − y‖22 ≤ ( 1 − ηmλ02 ) k‖u ( 0 ) − y‖22 . This inequality is plugged in the last inequality with Lemma 4.9 . By combining the inequalities ( 33 ) for s ∈ [ k ] and ( 34 ) , and triangle inequality , we conclude the proof : ∥∥∥W ( k+1 ) ` −W ( 0 ) ` ∥∥∥ 2 ≤ η · O ( √ nmω ) k∑ t=0 ( 1− ηmλ0 2 ) t 2 O ( √ n δ ) ≤ O ( n √ ω√ mδλ0 ) . Proposition 5.1 . For some δ ∈ [ 0 , 1 ] , if we set the width of the network as m ≥ Ω ( ω7n8L18 log 3 ( m ) λ80δ 2 ) , and we set the step-size of gradient descent as η = O ( λ0 n2L2m ) . Then , with probability at least 1 − δ over the randomness of initialized parameters W ( 0 ) , we have for k = 0 , 1 , 2 , . . . , u ( k ) − y = ( I− ηmH∞L ) k ( u ( 0 ) − y ) + ξ ( k ) , where ‖ξ ( k ) ‖2 = k ( 1− ηmλ0 2 ) k−1 O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖y − u ( 0 ) ‖2 . Proof . Define ui ( k ) : = fW ( k ) ( xi ) , then we have u ( k + 1 ) − u ( k ) = −ηm ·H ( k ) ( u ( k ) − y ) + I ( k ) 1 = −ηm ·H∞L ( u ( k ) − y ) − ηm · ( H ( k ) −H∞L ) ( u ( k ) − y ) + I ( k ) 1 = −ηm ·H∞L ( u ( k ) − y ) + e ( k ) . By recursively applying the above equality , we can easily derive a following for any k ≥ 0 , u ( k ) − y = ( I− ηmH∞L ) k ( u ( 0 ) − y ) + k−1∑ t=0 ( I− ηmH∞L ) t e ( k − 1− t ) ︸ ︷︷ ︸ =ξ ( k ) . ( 35 ) Now , we want to show ξ ( k ) can be controlled in arbitrarily small number . First , e ( k ) needs to be bounded in an ` 2 norm : ‖e ( k ) ‖2 ≤ ηm ‖H ∞ L −H ( k ) ‖2 ‖u ( k ) − y‖2 + ∥∥∥I ( k ) 2 ∥∥∥ 2 ≤ ηm · O ( ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖u ( k ) − y‖2 , where , in the second inequality , τ = O ( n √ ω√ mδλ0 ) is plugged in ( 27 ) . Equipped with the bound on ‖e ( k ) ‖2 , we can easily bound the ‖ξ ( k ) ‖2 as follows : ∥∥∥∥∥ k−1∑ t=0 ( I− ηmH∞L ) t e ( k − 1− t ) ∥∥∥∥∥ 2 ≤ k−1∑ t=0 ‖I− ηmH∞L ‖ t 2 ‖e ( k − 1− t ) ‖2 ≤ k−1∑ t=0 ( 1− ηmλ0 ) t O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖u ( k − 1− t ) − y‖2 ≤ k−1∑ t=0 ( 1− ηmλ0 ) t O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ( 1− ηmλ0 2 ) k−1−t ‖u ( 0 ) − y‖2 = k ( 1− ηmλ0 2 ) k−1 O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ 2 ) ‖u ( 0 ) − y‖2 . ( 36 ) Note that in the third inequality , we used the result from Theorem 1 . F PROOF OF THEOREM 3.8 We begin the proof by decomposing the error f̂W ( k ) ( x ) − f∗ ( x ) for any fixed x ∈ Unif ( Sd−1 ) into two terms as follows : f̂W ( k ) ( x ) − f∗ ( x ) = ( f̂W ( k ) ( x ) − g∗ ( x ) ) ︸ ︷︷ ︸ ∆1 + ( g∗ ( x ) − f∗ ( x ) ) ︸ ︷︷ ︸ ∆2 . ( 37 ) Here , we denote the solution of kernel regression with kernel H∞L as g ∗ ( x ) , which is a minimum RKHS norm interpolant of the noise-free data set { xi , f ? ρ ( xi ) } ni=1 . To avoid the confusion of the notation , we write Ker ( x , X ) = ( H∞L ( x , x1 ) , . . . , H ∞ L ( x , xn ) ) n i=1 ∈ Rn and let y∗ = [ f ? ρ ( x1 ) , . . . , f ? ρ ( xn ) ] > . Then , we have a following closed form solution g∗ ( x ) as , g∗ ( x ) : = Ker ( x , X ) ( H∞L ) −1 y∗ . With the decomposition ( 37 ) , the proof sketch of Theorem 3.8 is as follows . 1 . Note that for any ` ∈ [ L ] , we have f̂W ( k ) ( x ) = 〈vec ( ∇W ` [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ) 〉 . We can write the term vec ( W ( k ) ` ) with respect to vec ( W ( 0 ) ` ) , H∞L and the residual term via recursive applications of GD update rule and the result from proposition 2.1 . Readers can refer ( 38 ) . Using the equality ( 38 ) , we can further decompose ∆1 into three terms . That is , ∆1 = ∆11 + ∆12 + ∆13 . Then , using the boundedness of ` 2-norm of network gradient and the fact that the size of ‖ξ ( k ) ‖2 can be controlled with wide enough network , we can control the size of ‖∆12‖2 and ‖∆13‖2 aribtarily small . 2 . In the term ∆2 , the g ? is an interpolant based on noiseless data . For large enough data points , g ? converges fastly to f ? at the rate OP ( 1√n ) . 3 . Lastly , the ∆11 is the only term that is involved with random error ε , and we show that ‖∆11‖2 is bounded away from 0 for small and large GD iteration index k. Step 1 . Control on ∆1 . For n data points ( x1 , . . . , xn ) and for the kth updated parameter W ( k ) , denote : ∇W ` [ fW ( k ) ( X ) ] = [ vec ( ∇W ` [ fW ( k ) ( x1 ) ] ) , · · · , vec ( ∇W ` [ fW ( k ) ( xn ) ] ) ] . Note that when ` = 1 , ∇W ` [ fW ( k ) ( X ) ] ∈ Rmd×n and when ` = 2 , . . . , L , ∇W ` [ fW ( k ) ( X ) ] ∈ Rm2×n . With this notation , we can rewrite the Gradient Descent update rule as vec ( W ( k+1 ) ` ) = vec ( W ( k ) ` ) − η∇W ` [ fW ( k ) ( X ) ] ( u ( k ) − y ) , k ≥ 0 . Applying Proposition 3.8 , we can get : vec ( W ( k ) ` ) − vec ( W ( 0 ) ` ) = k−1∑ j=0 ( vec ( W ( j+1 ) ` ) − vec ( W ( j ) ` ) ) = −η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ( u ( j ) − y ) = η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) − η · k−1∑ j=0 ∇W ` [ fW ( k ) ( X ) ] ξ ( j ) = η · k−1∑ j=0 ∇W ` [ fW ( 0 ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) − η · k−1∑ j=0 ∇W ` [ fW ( k ) ( X ) ] ξ ( j ) + η · k−1∑ j=0 ( [ ∇W ` [ fW ( j ) ( X ) ] −∇W ` [ fW ( 0 ) ( X ) ] ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) ) = η · k−1∑ j=0 ∇W ` [ fW ( 0 ) ( X ) ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) + ξ ′ ( k ) . ( 38 ) First , we control ` 2-norm of the first term of ξ ′ ( k ) as follows : Note that ‖∇W ` [ fW ( j ) ( X ) ] ‖F ≤ O ( √ nmω ) by Lemma 4.4 for 0 ≤ j ≤ k − 1 . Then , we have∥∥∥∥∥∥η · k−1∑ j=0 ∇W ` [ fW ( j ) ( X ) ] ξ ( j ) ∥∥∥∥∥∥ 2 ≤ k−1∑ j=0 O ( η √ nmω ) O ( j ( 1− ηmλ0 2 ) j−1 ) O ( ηm · ω7/6n4/3L3 6 √ log3 ( m ) mλ20δ ) ‖y − u ( 0 ) ‖2 ≤ O ( n11/6L3ω5/3 m2/3λ 7/3 0 δ 1/6 √ log ( m ) ) ‖y − u ( 0 ) ‖2 . ( 39 ) In the second inequality , ∑∞ j=1 j ( 1− ηmλ02 ) j = O ( 1 η2m2λ20 ) is used . Then , we control ` 2-norm of the second term of ξ ′ ( k ) as follows : ∥∥∥∥∥∥η · k−1∑ j=0 [ ∇W ` [ fW ( j ) ( X ) ] −∇W ` [ fW ( 0 ) ( X ) ] ] ( I− ηmH∞L ) j ( y − u ( 0 ) ) ∥∥∥∥∥∥ 2 ≤ k−1∑ j=0 η ‖I− ηmH∞L ‖ j 2 ‖y − u ( 0 ) ‖2 √√√√ n∑ i=1 ∥∥∇W ` [ fW ( j ) ( xi ) ] −∇W ` [ fW ( 0 ) ( xi ) ] ∥∥22 ≤ k−1∑ j=0 η ( 1− ηmλ0 ) jO ( n1/3m1/3L2ω2/3 λ 1/3 0 δ 1/6 √ log ( m ) ) O ( √ n ) ‖y − u ( 0 ) ‖2 ≤ O ( n5/6L2ω2/3 m2/3λ 4/3 0 δ 1/6 √ log ( m ) ) ‖y − u ( 0 ) ‖2 , ( 40 ) where in the second inequality , we used Lemmas 4.6 with τ = O ( n√ω√ mδλ0 ) . Now , we are ready to control ∆1 term . By using the equality ( 38 ) , we can decompose the term ∆1 as follows : Let us denote Gk = ∑k−1 j=0 ηm ( I − ηmH∞L ) j . Note that for any ` ∈ [ L ] , f̂W ( k ) ( x ) = 〈vec ( ∇W ` [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ) 〉 and recall that y = y∗ + ε . Then , for any fixed ` ′ ∈ [ L ] , we have : ∆1 = [ 〈 vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( k ) ` ′ ) 〉 −Ker ( x , X ) ( H∞L ) −1 y∗ ] + Ker ( x , X ) Gky −Ker ( x , X ) Gky = [ Ker ( x , X ) [ Gk − ( H∞L ) −1 ] y∗ + Ker ( x , X ) Gkε ] ︸ ︷︷ ︸ =∆11 + [ 1 m L∑ ` =1 vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] −Ker ( x , X ) ] Gky − 1 m ∑ ` : ` 6= ` ′ vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] Gky︸ ︷︷ ︸ =∆12 + [ 〈 vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( 0 ) ` ′ ) 〉 + vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ξ ′ ( k ) − 1 m vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ] .︸ ︷︷ ︸ =∆13 ( 41 ) Our goal in this step is to control ‖∆12‖2 and ‖∆13‖2 . Then , in the third step , we will show ‖∆11‖2 is the term , which governs the behavior of the prediction risk with respect to algorithm iteration k. First , we bound the ` 2 norm of the first term in ∆12 as : ∥∥∥∥∥ [ 1 m L∑ ` =1 vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] −Ker ( x , X ) ] Gky ∥∥∥∥∥ 2 ≤ 1 mL L∑ ` =1 ∥∥vec ( ∇W ` [ fW ( k ) ( x ) ] ) − vec ( ∇W ` [ fW ( 0 ) ( x ) ] ) ∥∥2︸ ︷︷ ︸ ≤O ( τ1/3L2 √ ωm log ( m ) ) : Lemma 4.6 ∥∥∇W ` [ fW ( 0 ) ( X ) ] ∥∥F︸ ︷︷ ︸ ≤O ( √ ωnm ) : Lemma 4.4 ‖Gky‖2 + 1 L √√√√ n∑ i=1 ( 1 m L∑ ` =1 〈 ∇W ` [ fW ( 0 ) ( x ) ] , ∇W ` [ fW ( 0 ) ( xi ) ] 〉 Tr −Ker ( x , xi ) ) 2 ‖Gky‖2 ≤ { O ( n5/6L2ω7/6 m1/6δ1/6λ 1/3 0 √ log ( m ) ) +O ( ωn1/2L3/2 4 √ log ( nL/δ ) m ) } ‖Gk‖2 ‖y‖2 ≤ O ( n5/6L2ω7/6 m1/6δ1/6λ 4/3 0 √ log ( m ) · ‖y‖2 ) +O ( ωn1/2L3/2 λ0 4 √ log ( nL/δ ) m · ‖y‖2 ) , ( 42 ) where , in the second inequality , we plugged τ = O ( n√ω√ mδλ0 ) in the result of Lemma 4.6 and used Lemma 4.11 . In the last inequality , we used ‖Gk‖2 ≤ O ( 1 λ0 ) . Similarly , we can control the ` 2 norm of the second term in ∆12 as follows : ∥∥∥∥∥∥ 1m ∑ ` : ` 6= ` ′ vec ( ∇W ` [ fW ( k ) ( x ) ] ) > ∇W ` [ fW ( 0 ) ( X ) ] Gky ∥∥∥∥∥∥ 2 ≤ 1 m ∑ ` : ` 6= ` ′ ‖vec ( ∇W ` [ fW ( k ) ( x ) ] ) ‖2︸ ︷︷ ︸ ≤O ( √ ωm ) · ‖∇W ` [ fW ( 0 ) ( X ) ] ‖F︸ ︷︷ ︸ ≤O ( √ ωmn ) · ‖Gk‖2︸ ︷︷ ︸ ≤O ( 1 λ0 ) ‖y‖2 ≤ O ( ωL √ n λ0 ) · ‖y‖2 . ( 43 ) We turn our attention to controlling ‖∆13‖2 . The first term in ∆13 ; Recall that∥∥∥vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) ∥∥∥ 2 ≤ O ( √ mω ) by Lemma 4.4 . Then , the random variable vec ( ∇W ` [ fW ( k ) ( x ) ] ) > vec ( W ( 0 ) ` ) is simply a N ( 0 , O ( ω ) ) for 1 ≤ ` ≤ L. A straightforward application of Chernoff bound for normal random variable and taking union bound over the layer 1 ≤ ` ≤ L yield that : with probability at least 1− δ , ∣∣∣vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > vec ( W ( 0 ) ` ′ ) ∣∣∣ ≤ O ( √ω log ( L δ ) ) . ( 44 ) The ` 2 norm of the third term in ∆13 can be bounded as follows : ∥∥∥∥ 1mvec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ∥∥∥∥ 2 ≤ 1 m ∥∥∥vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) ∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ mω ) ∥∥∥∇W ` ′ [ fW ( 0 ) ( X ) ] ∥∥∥ F︸ ︷︷ ︸ ≤O ( √ ωmn ) ‖Gku ( 0 ) ‖2︸ ︷︷ ︸≤O ( √nωλ0δ ) ≤ O ( nω3/2 λ0δ ) . ( 45 ) In the last inequality , we used the Lemma 4.8 and ‖Gk‖2 ≤ O ( 1 λ0 ) . By combin- ing ( 39 ) , ( 40 ) , ( 44 ) , ( 45 ) with ∥∥∥∇W ` ′ [ fW ( 0 ) ( x ) ] ∥∥∥ F ≤ O ( √ mω ) , we have a following : ‖∆13‖2 ≤ ∥∥∥〈vec ( ∇W ` ′ [ fW ( k ) ( x ) ] ) , vec ( W ( 0 ) ` ′ ) 〉∥∥∥ 2 + ∥∥∥ ( vec ( ∇W ` ′ [ fW ( 0 ) ( x ) ] ) ) > ξ′ ( k ) ∥∥∥ 2 + ∥∥∥∥ 1m ( vec ( ∇W ` ′ [ fW ( 0 ) ( x ) ] ) > ∇W ` ′ [ fW ( 0 ) ( X ) ] Gku ( 0 ) ∥∥∥∥ 2 ≤ O ( √ ω log ( L δ ) ) +O ( n11/6L3ω13/6 ‖y − u ( 0 ) ‖2 m1/6λ 4/3 0 δ 1/6 √ log ( m ) ) +O ( n5/6L2ω7/6 ‖y − u ( 0 ) ‖2 m1/6λ 7/3 0 δ 1/6 √ log ( m ) ) +O ( nω3/2 λ0δ ) = O ( n11/6L3ω13/6 ‖y − u ( 0 ) ‖2 m1/6λ 4/3 0 δ 1/6 √ log ( m ) ) +O ( n5/6L2ω7/6 ‖y − u ( 0 ) ‖2 m1/6λ 7/3 0 δ 1/6 √ log ( m ) ) +O ( nω3/2 λ0δ ) . ( 46 ) Step 2 . Control on ∆2 . First , note that there is a recent finding that the reproducing kernel Hilbert spaces of NTKs with any number of layers ( i.e. , L ≥ 1 ) have the same set of functions , if kernels are defined on Sd−1 . See Chen & Xu [ 2020 ] . Along with this result , we can apply the proof used in Lemma.D.2 . in Hu et al . [ 2021 ] for proving a following : ‖∆2‖2 = OP ( 1√ n ) . ( 47 ) Step 3 . The behavior of L2 risk is characterized by the term ∆11 . Recall the decompositions ( 37 ) and ( 41 ) , then we have : f̂W ( k ) ( x ) − f∗ ( x ) = ∆11 + ( ∆12 + ∆13 + ∆2 ) : = ∆11 + Θ . ( 48 ) Our goal in this step is mainly two-folded : ( i ) Control Eε ‖Θ‖22 arbitrarily small with proper choices of step-size of GD η and width of the network m. ( ii ) Show that how Eε ‖∆11‖22 affect the behavior of prediction risk over the GD iterations k. First , note that we have Eε ‖y‖22 = Eε ‖y ∗ + ε‖22 ≤ 2 ( y∗ ) > y∗ + 2Eε ‖ε‖22 = O ( n ) . ( 49 ) Second , recall Lemma 4.9 and note that over the random initialization , with probability at least 1−δ , Eε ‖y − u ( 0 ) ‖22 ≤ O ( n δ ) . ( 50 ) Now , by combining the bounds ( 42 ) , ( 46 ) and ( 47 ) , we have Eε ‖Θ‖22 ≤ 3Eε ( ‖∆12‖22 + ‖∆13‖ 2 2 + ‖∆2‖ 2 2 ) ≤ Eε [ O ( n5/3L4ω7/3 m1/3λ 8/3 0 δ 1/3 log ( m ) · ‖y‖22 ) +O ( ω2nL3 λ20 √ log ( nL/δ ) m · ‖y‖22 ) +O ( n11/3L6ω13/3 ‖y − u ( 0 ) ‖22 m1/3λ 8/3 0 δ 1/3 log ( m ) ) +O ( n5/3L4ω7/3 ‖y − u ( 0 ) ‖22 m1/3λ 14/3 0 δ 1/3 log ( m ) ) ] +O ( n2ω3 λ20δ 2 ) +O ( 1 n ) ≤ O ( ω2n2L3 λ20 √ log ( nL/δ ) m ) +O ( n14/3L6ω13/3 m1/3λ 8/3 0 δ 4/3 log ( m ) ) +O ( n8/3L4ω7/3 m1/3λ 14/3 0 δ 4/3 log ( m ) ) +O ( n2ω3 λ20δ 2 ) +O ( 1 n ) , ( 51 ) where in the third inequality , we used ( 49 ) and ( 50 ) . Case 1 . When k is large , the L2 risk is bounded away from zero by some constant . Now we control Eε ‖∆11‖22 . Recall the definitions ‖f‖ 2 2 : = ∫ x∈Sd−1 |f ( x ) | 2dx and Gk = ∑k−1 j=0 ηm ( I− ηmH∞L ) j . Let us denote S = y∗y∗ > . Then , we have Eε ‖∆11‖22 = ∫ x∈Sd−1 Ker ( x , X ) [ ( Gk − ( H∞L ) −1 ) y∗y∗ > ( Gk − ( H∞L ) −1 ) +G2k ] Ker ( X , x ) dx = ∫ x∈Sd−1 Ker ( x , X ) ( H∞L ) −1 Mk ( H∞L ) −1Ker ( X , x ) dx where Mk = ( I− ηmH∞L ) k S ( I− ηmH∞L ) k + ( I− ( I− ηmH∞L ) k ) 2 = [ ( I− ηmH∞L ) k − ( S + I ) −1 ] ( S + I ) [ ( I− ηmH∞L ) k − ( S + I ) −1 ] + I − ( S + I ) −1 . For the algorithm iterations k ≥ ( log ( n ) ηmλ0 ) C0 with some constant C0 > 1 , we have ( I − ηmH∞L ) k ( 1− ηmλ0 ) k · I exp ( −ηmλ0k ) · I exp ( −C0 log ( n ) ) = 1 nC0 · I . Since 1 + ‖y‖22 ≤ C1n for some constant C1 , we have λmax ( 1 nC0 · ( S + I ) ) = 1 + ‖y‖22 nC0 ≤ C1 nC0−1 < 1 . ( 52 ) Using the first item of Lemma ( 4.12 ) with the inequality ( 52 ) , we have ( I − ηmH∞L ) k 1 nC0 · I ≺ ( S + I ) −1 . ( 53 ) The above inequality ( 53 ) lead to a following result : ( S + I ) −1 − ( I − ηmH∞L ) k ( S + I ) −1 − 1nC0 · I . ( 54 ) It is obvious that both ( S + I ) −1 − ( I − ηmH∞L ) k and ( S + I ) −1 − 1nC0 · I are positive definite matrices due to ( 54 ) , and it is also easy to see that they are exchangeable . By using the second item of Lemma ( 4.12 ) , we have Mk = [ ( I− ηmH∞L ) k − ( S + I ) −1 ] ( S + I ) [ ( I− ηmH∞L ) k − ( S + I ) −1 ] + I − ( S + I ) −1 [ ( S + I ) −1 − 1 nC0 · I ] ( S + I ) [ ( S + I ) −1 − 1 nC0 · I ] + I − ( S + I ) −1 = 1 n2C0 S + ( 1− 1 nC0 ) 2 · I . Then , we have Eε ‖∆11‖22 1 n2C0 A+ ( 1− 1 nC0 ) 2 B c0B , where c0 ∈ ( 0 , 1 ) is a constant and A = ∫ x∈Sd−1 [ Ker ( x , X ) ( H∞L ) −1 y∗ ] 2 dx , and B = ∫ x∈Sd−1 [ Ker ( x , X ) ( H∞L ) −1 ] 2 dx . ( 55 ) By triangle inequality with the decomposition ( 48 ) and the bound on Eε ‖Θ‖22 in ( 51 ) , we have : Eε ∥∥∥f̂W ( k ) − f∗∥∥∥2 2 = Eε ‖∆11 + Θ‖22 ≥ 1 2 Eε ‖∆11‖2 − Eε ‖Θ‖ 2 2 ≥ c0 2 B −O ( 1 n ) −O ( n2ω3 λ20δ 2 ) − Õ ( 1 m1/3 poly ( ω , n , L , 1 λ0 , 1 δ ) ) . ( 56 ) For the third term in ( 56 ) , we can choose ω ≤ C2 ( λ0δ n ) 2/3 for some constant C2 > 0 such that the term can be bounded by c08 ∥∥∥Ker ( · , X ) ( H∞L ) −1∥∥∥2 2 . Similarly , the width m can be chosen large enough such that the fourth term in ( 56 ) is upper-bounded by c08 ‖Ker ( · , X ) ( H∞L ) −1‖22 . Using the above choices of k , ω , and m , we can further bound ( 56 ) : Eε ∥∥fW ( k ) − f∗∥∥22 ≥ c04 ∥∥∥Ker ( · , X ) ( H∞L ) −1∥∥∥22 −O ( 1 n ) . ( 57 ) Note that Eε‖f̂∞ − g∗‖22 = ‖Ker ( · , X ) ( H∞L ) −1‖22 where g∗ : = 0 and f̂∞ denotes the noise interpolator . Then , by Theorem 4.2. of Hu et al . [ 2021 ] , we know that Eε‖f̂∞−g∗‖22 ≥ c1 for some constant c1 > 0 . Then , we can take n large enough such that the term O ( 1 n ) is upper-bounded by c0c1 8 , and finish the proof . Case 2 . When k is small , the L2 risk is bounded away from zero by some constant . Recall the definition of ∆11 in the decomposition ( 41 ) , ∆11 : = Ker ( x , X ) Gk [ y∗ + ε ] −Ker ( x , X ) H∞L y∗ : = ∆∗11 −Ker ( x , X ) H∞L y∗ . ( 58 ) We denote the eigen-decomposition of the matrix H∞L : = ∑n i=1 λivivi > , then we can easily see a following : Gk = ηm k−1∑ j=0 ( n∑ i=1 ( 1− ηmλi ) jvivi > ) ηm k−1∑ j=0 n∑ i=1 vivi > ηmk · I . By using the above inequality , we have Eε ‖∆∗11‖ 2 2 = ∫ x∈Sd−1 Ker ( x , X ) Gk ( S + I ) GkKer ( X , x ) dx ≤ η2m2k2 ( ∫ x∈Sd−1 [ Ker ( x , X ) y∗ ] 2 dx+ ‖Ker ( · , X ) ‖22 ) = O ( η2m2k2ω2n2L2 ) . Recall the decompositions ( 37 ) and ( 41 ) , then we have : Eε ∥∥∥f̂W ( k ) − f∗∥∥∥2 2 = Eε ‖∆∗11 + Θ−Ker ( · , X ) H∞L y∗‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 − Eε ‖∆ ∗ 11 + Θ‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 − 2Eε ‖∆ ∗ 11‖ 2 2 − 2Eε ‖Θ‖ 2 2 ≥ 1 2 ‖Ker ( · , X ) H∞L y∗‖ 2 2 −O ( η2m2k2ω2n2L2 ) −O ( 1 n ) −O ( n2ω3 λ20δ 2 ) − Õ ( 1 m1/3 poly ( ω , n , L , 1 λ0 , 1 δ ) ) . ( 59 ) For some constant C ′ 1 > 0 , let k ≤ C ′ 1 ( 1 ηmnωL ) such that the second term in the bound ( 59 ) can be bounded by 18‖Ker ( · , X ) ( H∞L ) −1 y∗‖22 . For the fourth term in ( 59 ) , we can choose ω ≤ C ′ 2 ( λ0δ n ) 2/3 for some constant C ′ 2 > 0 such that the term can be bounded by 1 8 ∥∥∥Ker ( · , X ) ( H∞L ) −1y∗∥∥∥2 2 . Sim- ilarly , the width m can be chosen large enough such that the fifth term in ( 59 ) is upper-bounded by 1 8‖Ker ( · , X ) ( H∞L ) −1 y∗‖22 . Using the above choices of k , ω , and m , we can further bound ( 59 ) : Eε ∥∥fW ( k ) − f∗∥∥22 ≥ 14 ∥∥∥Ker ( · , X ) ( H∞L ) −1y∗∥∥∥22 −O ( 1 n ) ≥ C ′ 3 ∥∥f ? ρ∥∥22 −O ( 1n ) . ( 60 ) In the second inequality , we used ( 47 ) with triangle inequality . In the third inequality , we can take n large enough such that the term O ( 1 n ) is upper-bounded by C ′ 3 2 ∥∥f ? ρ∥∥22 . Lastly , by using the assumption that f ? ρ is a square-integrable function , we finish the proof . G PROOF OF THEOREM 3.11-TRAINING ERROR For the convenience of notation , we denote uD , i ( k ) = fW ( k ) D ( xi ) and let uD ( k ) = [ u1 , D ( k ) , . . . , un , D ( k ) ] > . In order to analyze the training error of ` 2-regularized estimator , ‖uD ( k ) − y‖22 , we decompose the term as follows : ‖uD ( k + 1 ) − y‖22 = ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖ 2 2 + ‖ ( 1− η2µL ) uD ( k ) − y‖ 2 2 − 2 ( y − ( 1− η2µL ) uD ( k ) ) > ( uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ) ( 61 ) Equipped with this decomposition , the proof consists of the following steps : 1 . We decompose the decayed prediction difference uD ( k+ 1 ) − ( 1− η2µL ) uD ( k ) into two terms . We note that the first term is related with a gram matrix HD ( k ) and denote a second term as I ( k ) D . 2 . The term I ( k ) D can be further decomposed into three terms , where we denote them as I ( k ) 2 , D , I ( k ) 3 , D and I ( k ) 5 , D . The crux for controlling the ` 2-norm of the above three terms is to utilize the results from the Appendix A.4 . The applications of Lemmas in the Appendix A.4 is possible , since we can inductively guarantee that ‖W ( k ) D , ` − W ( 0 ) D , ` ‖2 is sufficiently small enough for large enough m. 3 . Given the decomposition ( 61 ) , we further decompose it into four terms as follows : ( 61 ) = ‖ ( 1− η2µL ) uD ( k ) − y‖22︸ ︷︷ ︸ : =T1 + ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖22︸ ︷︷ ︸ : =T2 + 2mη1 ( y − ( 1− η2µL ) uD ( k ) ) > HD ( k ) ( uD ( k ) − y ) ︸ ︷︷ ︸ : =T3 −2 ( y − ( 1− η2µL ) uD ( k ) ) > I ( k ) D︸ ︷︷ ︸ : =T4 . ( 62 ) In this step , we obtain the upper-bound of ‖Ti‖2 for i = 1 , 2 , 3 , 4 obtained in Step 4 . 4 . We combine the upper-bounds of ‖Ti‖2 for i = 1 , 2 , 3 , 4 in step 3 and obtain the bound on ‖uD ( k + 1 ) − y‖22 with respect to ‖uD ( k ) − y‖22 and ‖y‖2 . 5 . Lastly , we inductively show that the weights generated from regularized gradient descent stay within a perturbation region B ( W ( 0 ) , τ ) , irrespective with the number of iterations of algorithm . We start the proof by analyzing the term u ( k + 1 ) − ( 1− η2µL ) u ( k ) . Step 1 . Dynamics of uD ( k + 1 ) − ( 1−η2µL ) ·uD ( k ) . Recall ( Σ ( k ) D , ` , i ) jj = 1 ( 〈w ( k ) D , ` , j , x ( k ) D , ` −1 , i〉 ≥ 0 ) and we introduce a diagonal matrix Σ̃ ( k ) D , ` , i , whose jth entry is defined as follows : ( Σ̃ ( k ) D , ` , i ) jj = ( Σ ( k+1 ) D , ` , i −Σ ( k ) D , ` , i ) jj · 〈w ( k+1 ) D , ` , j , x ( k+1 ) D , ` −1 , i〉 〈w ( k+1 ) D , ` , j , x ( k+1 ) D , ` −1 , i〉 − 〈w ( k ) D , ` , j , x ( k ) D , ` −1 , i〉 . With this notation , the difference x ( k+1 ) D , L , i − x ( k ) D , L , i can be rewritten via the recursive applications of Σ̃ ( k ) D , ` , i : Then , we introduce following notations : D ( k ) D , ` , i = ( L∏ r= ` +1 Σ ( k ) D , r , iW ( k ) D , r ) Σ ( k ) D , ` , i , D̃ ( k ) D , ` , i = ( L∏ r= ` +1 ( Σ ( k ) D , r , i + Σ̃ ( k ) D , r , i ) W ( k+1 ) D , r ) ( Σ ( k ) D , ` , i + Σ̃ ( k ) D , ` , i ) . Now , we can write uD , i ( k + 1 ) − uD , i ( k ) by noting that uD , i ( k ) = √ m · vTx ( k ) D , L , i : uD , i ( k + 1 ) − uD , i ( k ) = √ m · vT ( x ( k+1 ) D , L , i − x ( k ) D , L , i ) = √ m · vT L∑ ` =1 D̃ ( k ) D , ` , i ( W ( k+1 ) D , ` −W ( k ) D , ` ) x ( k ) D , ` −1 , i = √ m · vT L∑ ` =1 D̃ ( k ) D , ` , i ( − η1∇W ` [ LS ( W ( k ) D ) ] − η2µW ( k ) D , ` + η2µW ( 0 ) D , ` ) x ( k ) D , ` −1 , i = −η1 √ m · vT L∑ ` =1 D ( k ) D , ` , i∇W ` [ LS ( W ( k ) D ) ] x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 1 , D , i −η1 √ m · vT L∑ ` =1 ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ∇W ` [ LS ( W ( k ) D ) ] x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 2 , D , i −η2µ √ m · vT L∑ ` =1 ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ( W ( k ) D , ` −W ( 0 ) D , ` ) x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 3 , D , i −η2µ √ m · vT L∑ ` =1 D ( k ) D , ` , iW ( k ) D , ` x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 4 , D , i +η2µ √ m · vT L∑ ` =1 D ( k ) D , ` , iW ( 0 ) D , ` x ( k ) D , ` −1 , i︸ ︷︷ ︸ I ( k ) 5 , D , i ( 63 ) where in the second equality , we used the recursive relation ( 24 ) , and in the third equality , modified GD update rule ( 6 ) is applied . Furthermore , I ( k ) 1 , D , i can be rewritten as follows : I ( k ) 1 , D , i = −η1 √ m · vT L∑ ` =1 D ( k ) D , ` , i n∑ j=1 ( uD , j ( k ) − yj ) ∇W ` [ f W ( k ) D ( xj ) ] x ( k ) D , ` −1 , i = −η1 · n∑ j=1 ( uD , j ( k ) − yj ) · ( √ m L∑ ` =1 vTD ( k ) D , ` , i∇W ` [ f W ( k ) D ( xj ) ] x ( k ) D , ` −1 , i ) = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) · 1 m L∑ ` =1 〈 ∇W ` [ f W ( k ) D ( xi ) ] , ∇W ` [ f W ( k ) D ( xj ) ] 〉 Tr = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) ·HD , i , j ( k ) . ( 64 ) With I ( k ) 4 , i = ( −η2µL ) · uD , i ( k ) and ( 64 ) , we can rewrite ( 63 ) as follows : uD , i ( k + 1 ) − ( 1− η2µL ) uD , i ( k ) = −mη1 · n∑ j=1 ( uD , j ( k ) − yj ) ·HD , i , j ( k ) + I ( k ) 2 , D , i + I ( k ) 3 , D , i + I ( k ) 5 , D , i . ( 65 ) Step 2 . Control of the size ∥∥∥I ( k ) D ∥∥∥ 2 . Let I ( k ) D = [ I ( k ) 2 , D,1 + I ( k ) 3 , D,1 + I ( k ) 5 , D,1 , . . . , I ( k ) 2 , D , n + I ( k ) 3 , D , n + I ( k ) 5 , D,1 ] > . Now , we control the bound on the ∥∥∥I ( k ) D ∥∥∥2 2 . Recall that in Eq . ( 27 ) , we have ∥∥∥I ( k ) 2 , D∥∥∥ 2 ≤ O ( η1nL 3τ1/3ωm √ log ( m ) ) ‖uD ( k ) − y‖2 . ( 66 ) Similarly , ∥∥∥I ( k ) 3 , D∥∥∥ 2 can be bounded : ∥∥∥I ( k ) 3 , D∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 3 , D , i∣∣∣ ≤ η2µ√m · n∑ i=1 [ L∑ ` =1 ∥∥∥∥v > ( D̃ ( k ) D , ` , i −D ( k ) D , ` , i ) ∥∥∥∥ 2︸ ︷︷ ︸ ≤O ( L2τ1/3 √ ω log ( m ) ) · ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2︸ ︷︷ ︸ ≤τ · ∥∥∥x ( k ) D , ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) ] ≤ O ( η2µnL 3τ4/3 √ ωm log ( m ) ) . ( 67 ) Lastly ∥∥∥I ( k ) 5 , D∥∥∥ 2 can be bounded : ∥∥∥I ( k ) 5 , D∥∥∥ 2 ≤ n∑ i=1 ∣∣∣I ( k ) 5 , D , i∣∣∣ ≤ n∑ i=1 ∣∣∣∣∣η2µ√m · vT L∑ ` =1 D ( k ) D , ` , iW ( k ) D , ` x ( k ) D , ` −1 , i ∣∣∣∣∣+ n∑ i=1 ∣∣∣∣∣η2µ√m · vT L∑ ` =1 D ( k ) D , ` , i ( W ( k ) D , ` −W ( 0 ) D , ` ) x ( k ) D , ` −1 , i ∣∣∣∣∣ ≤ η2µL · n∑ i=1 |ui , D ( k ) |+ η2µ √ m · n∑ i=1 [ L∑ ` =1 ‖v‖2︸ ︷︷ ︸ ≤O ( √ ω ) · ∥∥∥D ( k ) D , ` , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( √ L ) · ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2︸ ︷︷ ︸ ≤τ · ∥∥∥x ( k ) D , ` −1 , i∥∥∥ 2︸ ︷︷ ︸ ≤O ( 1 ) ] ≤ O ( η2µnL √ ω log ( L/δ ) ) +O ( η2µnL 3/2τ √ mω ) , ( 68 ) where in the last inequality , we employed the same logic used in ( 44 ) with the Lemma 4.2 to obtain the upper-bound on the |ui , D ( k ) | . We set the orders of the parameters µ , η1 , η2 , τ , and ω as follows : µ = Θ ( n d−1 2d−1 ) , η1 = Θ ( 1 m n− 3d−2 2d−1 ) , η2 = Θ ( 1 L n− 3d−2 2d−1 ) , τ = O ( L √ ω√ mδ n d 2d−1 ) , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) . ( 69 ) Plugging the choices of parameters ( 69 ) with sufficiently large m in ( 66 ) , ( 67 ) and ( 68 ) yields ∥∥∥I ( k ) D ∥∥∥ 2 ≤ O ( L37/12n− 9d−8 12d−6 √ log ( m ) m1/6δ1/3 ) · ‖uD ( k ) − y‖2 +OP ( 1 n2 ) . ( 70 ) Step 3 . Upper-bound of ‖Ti‖2 on i = 1 , 2 , 3 , 4 . First , we work on getting the upper-bound on λmax ( HD ( k ) ) . By the Gershgorin ’ s circle theorem [ Varga , 2004 ] , we know the maximum eigenvalue of symmetric positive semi-definite matrix is upper-bounded by the maximum absolute column sum of the matrix . Using this fact , we can bound the λmax ( HD ( k ) ) as : λmax ( HD ( k ) ) ≤ max i=1 , ... , n n∑ j=1 |HD , i , j ( k ) | ≤ max i=1 , ... , n n∑ j=1 ∣∣∣∣∣ 1m L∑ ` =1 〈 ∇W ` [ f W ( k ) D ( xi ) ] , ∇W ` [ f W ( k ) D ( xj ) ] 〉 Tr ∣∣∣∣∣ ≤ max i=1 , ... , n n∑ j=1 1 m L∑ ` =1 ∥∥∥∇W ` [ fW ( k ) D ( xi ) ] ∥∥∥F︸ ︷︷ ︸ ≤O ( √ mω ) ∥∥∥∇W ` [ fW ( k ) D ( xj ) ] ∥∥∥F︸ ︷︷ ︸ ≤O ( √ mω ) ≤ O ( nLω ) . ( 71 ) Recall the decomposition ( 62 ) . Our goal is to obtain the upper-bound on Ti for i = 1 , 2 , 3 , 4 . Control on T1 . By using the inequality 2η2µL ( 1−η2µL ) y > ( y−uD ( k ) ) ≤ η2µL ‖y‖22+η2µL ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 , we have ‖y − ( 1− η2µL ) uD ( k ) ‖22 = ∥∥ ( 1− η2µL ) ( y − uD ( k ) ) + η2µLy∥∥22 = ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 + η 2 2µ 2L2 ‖y‖22 + 2η2µL ( 1− η2µL ) y > ( y − uD ( k ) ) ≤ ( η2µL+ η22µ2L2 ) ‖y‖ 2 2 + ( 1 + η2µL ) ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 . ( 72 ) Control on T2 . Recall the equality ( 65 ) . Then , through applications of the Young ’ s inequality ‖a+ b‖22 ≤ 2 ‖a‖ 2 2 + 2 ‖b‖ 2 2 for a , b ∈ Rn , we have ‖uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) ‖22 = ∥∥∥−mη1 ·HD ( k ) ( uD ( k ) − y ) + I ( k ) D ∥∥∥2 2 ≤ 2m2η21λmax ( HD ( k ) ) 2 ‖y − uD ( k ) ‖22 + 2 ∥∥∥I ( k ) D ∥∥∥2 2 . ( 73 ) Similarly with T1 and T2 , we can control T3 and T4 as follows : Control on T3 . Recall HD ( k ) is a Gram matrix by definition . Then , by using the fact λmin ( HD ( k ) ) ≥ 0 and Cauchy-Schwarz inequality , we have 2mη1 ( y − ( 1− η2µL ) uD ( k ) ) > HD ( k ) ( uD ( k ) − y ) = −2mη1 ( 1− η2µL ) ( y − uD ( k ) ) > HD ( k ) ( y − uD ( k ) ) + ( 2mη1η2µL ) · y > HD ( k ) ( uD ( k ) − y ) ≤ ( 2mη1η2µL ) · λmax ( HD ( k ) ) ‖y − uD ( k ) ‖22 + ( 2mη1η2µL ) · ( λmax ( HD ( k ) ) ‖y‖2 ‖y − uD ( k ) ‖2 ) − 2mη1λmin ( HD ( k ) ) ‖y − uD ( k ) ‖22 = ( 4mη1η2µL ) · λmax ( HD ( k ) ) ‖y − uD ( k ) ‖22 + ( 4mη1η2µL ) · λmax ( HD ( k ) ) ‖y‖22 . ( 74 ) Control on T4 . By a simple Cauchy-Schwarz and Young ’ s inequality , we have − 2 ( y − ( 1− η2µL ) uD ( k ) ) > I ( k ) D = −2 ( 1− η2µL ) ( y − uD ( k ) ) > I ( k ) D + 2η2µL · y > ID ( k ) ≤ 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ‖y‖22 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ( 75 ) Step 4 . Upper-bound of the decomposition on training error ( 62 ) . Before getting the upper bound of the decomposition ( 62 ) , we first work on obtaining the bound of ( 76 ) . Set κ = O ( 1 n2 ) and notice η2µL = O ( 1 n ) by ( 69 ) , then we have 2 ∥∥∥I ( k ) D ∥∥∥2 2 + 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ( 76 ) = ( 2 + η2µL ) ∥∥∥I ( k ) D ∥∥∥2 2 + 2κ ( 1− η2µL ) ‖y − uD ( k ) ‖2 · 1 κ ∥∥∥I ( k ) D ∥∥∥ 2 ≤ ( 2 + η2µL+ 1 κ2 ) ∥∥∥I ( k ) D ∥∥∥2 2 + κ2 ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 = 1 κ2 · ∥∥∥I ( k ) D ∥∥∥2 2 + κ2 ( 1− η2µL ) 2 ‖y − uD ( k ) ‖22 ≤ { 1 κ2 · O ( L37/6n− 9d−8 6d−3 log ( m ) m1/3δ2/3 ) + κ2 ( 1− η2µL ) 2 } · ‖y − uD ( k ) ‖22 + 1κ2 · OP ( 1 n4 ) ≤ ( η2µL ) 4 ( 1− η2µL ) 2 · ‖y − uD ( k ) ‖22 + η2µL · ‖y‖22 , ( 77 ) where in the second inequality , the Eq . ( 70 ) is used with ( a + b ) 2 ≤ 2a2 + 2b2 for a , b ∈ R , and in the last inequality , we used ‖y‖22 = O ( n ) and the sufficiently large m to control the order of the coefficient terms of ‖y − uD ( k ) ‖22 . Specifically , we choose m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Now , by combining the inequalities ( 72 ) , ( 73 ) , ( 74 ) , ( 75 ) , ( 71 ) and ( 77 ) , we obtain the upper-bound on the decomposition ( 62 ) ; ‖uD ( k + 1 ) − y‖22 ≤ ( 2η2µL+ η 2 2µ 2L2 + 4mη1η2µL · λmax ( HD ( k ) ) ) · ‖y‖22 + ( ( 1 + η2µL ) ( 1− η2µL ) 2 + 2m2η21λmax ( HD ( k ) ) 2 + 4mη1η2µL · λmax ( HD ( k ) ) ) · ‖y − uD ( k ) ‖22 + ( 2 ∥∥∥I ( k ) D ∥∥∥2 2 + 2 ( 1− η2µL ) ‖y − uD ( k ) ‖2 ∥∥∥I ( k ) D ∥∥∥ 2 + η2µL ∥∥∥I ( k ) D ∥∥∥2 2 ) ≤ { 3η2µL+ η 2 2µ 2L2 +O ( ωmnη1η2µL 2 ) } · ‖y‖22 + { ( 1 + η2µL+ η 4 2µ 4L4 ) ( 1− η2µL ) 2 +O ( ω2m2n2η21L 2 ) +O ( ωmnη1η2µL 2 ) } · ‖y − uD ( k ) ‖22 : = A · ‖y‖22 + ( 1− B ) · ‖y − uD ( k ) ‖ 2 2 . ( 78 ) With the order choices of µ , η1 and η2 as in ( 69 ) , it is easy to see the leading terms of both A and B are same as η2µL = o ( 1n ) . Then , by recursively applying the inequality ( 78 ) , we can get the upper-bound on the training error . ‖y − uD ( k + 1 ) ‖22 ≤ A · ‖y‖ 2 2 + ( 1− B ) · ‖y − uD ( k ) ‖ 2 2 ≤ A‖y‖22 · ( k∑ j=0 ( 1− B ) j ) + ( 1− B ) k+1 · ‖y − uD ( 0 ) ‖22 ≤ A B · ‖y‖22 + ( 1− B ) k+1 · ‖y − uD ( 0 ) ‖22 ≤ O ( n ) + ( 1− η2µL ) k+1 · ‖y − uD ( 0 ) ‖22 . ( 79 ) In the last inequality , we used AB = o ( 1 ) , B ≥ η2µL and ‖y‖ 2 2 = O ( n ) . Step 5 . The order of the radius of perturbation region . It remains us to prove the radius of perturbation region τ has the order OP ( L √ ω√ m n d 2d−1 ) . First , recall that the ` 2-regularized GD update rule is as : W ( k ) D , ` = ( 1− η2µ ) W ( k−1 ) D , ` − η1∇W ` [ LS ( W ( k−1 ) D ) ] + η2µW ( 0 ) D , ` , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . ( 80 ) Similarly with the proof in the Theorem 3.5 , we employ the induction process for the proof . The induction hypothesis is∥∥∥W ( s ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 ≤ O ( η1n √ mω√ δη2µ ) , ∀s ∈ [ k + 1 ] . ( 81 ) It is easy to see it holds for s = 0 , and suppose it holds for s = 0 , 1 , . . . , k , we consider k+1 . Using the update rule ( 80 ) , we have∥∥∥W ( k+1 ) D , ` −W ( k ) D , ` ∥∥∥ 2 ≤ η2µ ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 + η1 ∥∥∥∇W ` [ LS ( W ( k ) D ) ] ∥∥∥ 2 = η2µ ∥∥∥W ( k ) D , ` −W ( 0 ) D , ` ∥∥∥ 2 + η1 ∥∥∥∥∥ n∑ i=1 ( yi − uD , i ( k ) ) ∇W ` [ fWD ( k ) ( xi ) ] ∥∥∥∥∥ 2 ≤ O ( η1n √ mω√ δ ) +O ( η1 √ nmω ) · ‖y − uD ( k ) ‖2 ≤ O ( η1n √ mω√ δ ) +O ( η1 √ nmω ) · { O ( √ n ) + ( 1− η2µL ) k 2O ( √ n δ ) } ≤ O ( η1n √ mω√ δη2µ ) . In the first inequality , we use the induction hypothesis for s = k , and Lemma 4.4 . In the second inequality , since the induction hypothesis holds for s = 0 , 1 , . . . , k , we employ ‖y − uD ( k ) ‖2 ≤ O ( √ n ) + ( 1− η2µL ) k 2 ‖y − uD ( 0 ) ‖2 with the Lemma 4.9 . In the last inequality , we use η2µ < 1 . By triangle inequality , the induction holds for s = k + 1 . Plugging the proper choices of η1 , η2 and µ as suggested in ( 69 ) to O ( η1n √ mω√ δη2µ ) yields ‖W ( k ) D , ` −W ( 0 ) D , ` ‖2 ≤ OP ( L √ ω√ m n d 2d−1 ) . H PROOF OF THEOREM 3.11-KERNEL RIDGE REGRESSOR APPROXIMATION We present a following proof sketch on the approximation of regularized DNN estimator to kernel ridge regressor . 1 . The key idea for proving the second result in Theorem 3.8 is to write the distance between ui , D ( k ) ( where D is to denote the prediction is obtained from regularized GD rule ) and kernel regressor B : = H∞L ( Cµ · I + H∞L ) −1 y in terms of NTK matrix H∞L , which is as follows : uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) . Above equality describes how the regularized estimator evolves to fit the kernel regressor as iteration of algorithm goes by . 2 . We can bound the ` 2-norm of residual term eD ( k ) as O ( 1/n ) , and show that the ` 2 norm of the first term on the RHS of equation ( 4.3 ) decays at the rateO ( √ n ( 1− η2µL ) k ) . Here the √ n comes from the bound ‖B‖2 ≤ O ( √ n ) , since we know ‖u ( 0 ) ‖2 hasO ( √ nω ) with small ω ≤ 1 . This yields the claim . Recall the equality ( 65 ) . Then , we have uD ( k + 1 ) − ( 1− η2µL ) uD ( k ) = −mη1 ·HD ( k ) ( uD ( k ) − y ) + I ( k ) 2 , D + I ( k ) 3 , D + I ( k ) 5 , D = −mη1 ·H∞L ( uD ( k ) − y ) −mη1 · ( HD ( k ) −H∞L ) ( uD ( k ) − y ) + I ( k ) 2 , D + I ( k ) 3 , D + I ( k ) 5 , D = −mη1 ·H∞L ( uD ( k ) − y ) + ξD ( k ) . ( 82 ) With τ = O ( L √ ω√ mδ n d 2d−1 ) , similarly with Lemma 4.10 and a direct employment of the result from Lemma 4.11 , we can control the distance from HD ( k ) to H∞L under operator norm as follows : ‖HD ( k ) −H∞L ‖2 ≤ ‖HD ( k ) −H ( 0 ) ‖2 + ‖H ( 0 ) −H ∞ L ‖2 ≤ O ( ω7/6L10/3n 7d−3 6d−3 6 √ log3 ( m ) mδ2 ) +O ( ωL5/2n 4 √ log ( nL/δ ) m ) ≤ O ( L19/12n− 21d−8 12d−6 6 √ log3 ( m ) mδ2 ) +O ( Ln− 18d−6 12d−6 4 √ log ( nL/δ ) m ) ≤ O ( L19/12n− 21d−8 12d−6 6 √ log3 ( m ) mδ2 ) , ( 83 ) where in the third inequality , ω = O ( 1 L3/2 n− 5d−2 2d−1 ) is plugged-in . The last inequality holds with d ≥ 2 with large enough n and the condition on width m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Then , the ` 2 norm of ξD ( k ) can be bounded as : ‖ξD ( k ) ‖2 ≤ mη1 · ‖H ∞ L −HD ( k ) ‖2 ‖uD ( k ) − y‖2 + ∥∥∥I ( k ) D ∥∥∥ 2 ≤ O ( L19/12n− 12d−5 6d−3 √ log ( m ) m1/6δ1/3 ) · ‖uD ( k ) − y‖2︸ ︷︷ ︸ ≤O ( √ n/δ ) +OP ( 1 n2 ) ≤ O ( L19/12n− 18d−7 12d−6 √ log ( m ) m1/6δ5/6 ) +OP ( 1 n2 ) = OP ( 1 n2 ) , ( 84 ) where in the second inequality , we used ( 83 ) with η1 = O ( 1 mn − 3d−22d−1 ) to control the first term and employed Eq . ( 70 ) to control the second term . In the last equality , we usedm ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Now , by setting B : = ( η2µL η1m I + H∞L ) −1 H∞L y , we can easily convert the equality ( 82 ) as follows : for k ≥ 1 , uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) ( uD ( k − 1 ) −B ) + ξD ( k − 1 ) . ( 85 ) The recursive applications of the equality ( 85 ) yields uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + k∑ j=0 ( ( 1− η2µL ) · I −mη1H∞L ) j ξD ( k − j − 1 ) = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) . ( 86 ) Now , we bound the ` 2 norm of eD ( k ) in ( 86 ) : ‖eD ( k ) ‖2 = ∥∥∥∥∥∥ k∑ j=0 ( ( 1− η2µL ) · I −mη1H∞L ) j ξD ( k − j − 1 ) ∥∥∥∥∥∥ 2 ≤ k∑ j=0 ∥∥ ( 1− η2µL ) · I −mη1H∞L ∥∥j2 ‖ξD ( k − j − 1 ) ‖2 ≤ k∑ j=0 ( 1− η2µL ) j ‖ξD ( k − j − 1 ) ‖2 = O ( 1n ) , ( 87 ) in the last inequality , we used η2µL = O ( 1 n ) and Eq . ( 84 ) . Now , we control the ` 2-norm of the first term in ( 86 ) as : ∥∥∥∥∥ ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) ∥∥∥∥∥ 2 ≤ ( 1− η2µL ) k ‖uD ( 0 ) −B‖2 ≤ O ( √ n ( 1− η2µL ) k ) , ( 88 ) where in the second inequality , we used ‖uD ( 0 ) ‖2 ≤ O ( √ nω/δ ) and the fact that ‖B‖2 ≤ ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 H∞L ∥∥∥∥∥ 2 · ‖y‖2 ≤ O ( √ n ) . By combining ( 87 ) and ( 88 ) and using a fact ( 1− η2µL ) k ≤ exp ( −η2µLk ) , we conclude that after k ≥ Ω ( ( η2µL ) −1 log ( n3/2 ) ) , the error ‖uD ( k ) −B‖2 decays at the rate O ( 1 n ) . I PROOF OF THEOREM 3.12 We begin the proof by decomposing the error f̂ W ( k ) D ( x ) − f∗ ( x ) for any fixed x ∈ Unif ( Sd−1 ) into two terms as follows : f̂ W ( k ) D ( x ) − f∗ ( x ) = ( f̂ W ( k ) D ( x ) − g∗µ ( x ) ) ︸ ︷︷ ︸ ∆D,1 + ( g∗µ ( x ) − f∗ ( x ) ) ︸ ︷︷ ︸ ∆D,2 . ( 89 ) Here , we devise a solution of kernel ridge regression g∗µ ( x ) in the decomposition ( 89 ) : g∗µ ( x ) : = Ker ( x , X ) ( Cµ · I + H∞L ) −1 y , for some constant C > 0 . Specifically , in the proof to follow , we choose η1 and η2 such that C = η2Lη1m for the theoretical convenience . Our goal is to show that all the terms ‖∆D,1‖ 2 2 , and ‖∆D,2‖22 have the order either equal to or smaller than O ( n− d 2d−1 ) with the proper choices on m , µ , η1 and η2 . Since the high-level proof idea is similar with that of Theorem 3.8 , we omit the step-by-step proof sketch of Theorem 3.12 . The most notable difference between the proof strategies of the two theorems is that the regularized DNN approximate the kernel ridge regressor of noisy data , whereas in Theorem 3.8 , unregularized DNN approximate the interpolant based on noiseless data . Step 1 . Control on ∆D,2 . First , note that there is a recent finding that the reproducing kernel Hilbert spaces induced from NTKs with any number of layers ( i.e. , L ≥ 1 ) have the same set of functions , if kernels are defined on Sd−1 . See Chen & Xu [ 2020 ] . Along with this result , under the choice of model parameters as suggested in ( 69 ) , we can apply exactly the same proof used in Theorem.3.2 in Hu et al . [ 2021 ] for proving a following : ‖∆D,2‖22 : = ∥∥g∗µ − f∗∥∥22 = OP ( n− d2d−1 ) , ∥∥g∗µ∥∥2H = OP ( 1 ) . ( 90 ) Step 2 . Control on ∆D,1 . For n data points ( x1 , . . . , xn ) and for the kth updated parameter W ( k ) D , denote : ∇W ` [ f W ( k ) D ( X ) ] = [ vec ( ∇W ` [ f W ( k ) D ( x1 ) ] ) , · · · , vec ( ∇W ` [ f W ( k ) D ( xn ) ] ) ] . Note that when ` = 1 , ∇W ` [ f W ( k ) D ( X ) ] ∈ Rmd×n and when ` = 2 , . . . , L , ∇W ` [ f W ( k ) D ( X ) ] ∈ Rm2×n . With this notation , we can write the vectorized version of the update rule ( 80 ) as : vec ( W ( k ) D , ` ) = vec ( W ( 0 ) D , ` ) − η1 k−1∑ j=0 ( 1− η2µ ) j∇W ` [ fWD ( k−j−1 ) ( X ) ] ( uD ( k − j − 1 ) − y ) , ∀1 ≤ ` ≤ L and ∀k ≥ 1 . Using the equality , we can get the decomposition : vec ( W ( k ) D , ` ) = vec ( W ( 0 ) D , ` ) ︸ ︷︷ ︸ : =E1 −η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( uD ( k − j − 1 ) − y ) ︸ ︷︷ ︸ : =E2 −η1 k−1∑ j=0 ( 1− η2µ ) j [ ∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ] ( uD ( k − j − 1 ) − y ) .︸ ︷︷ ︸ : =E3 ( 91 ) Let zD , k ( x ) : = vec ( ∇W ` [ f W ( k ) D ( x ) ] ) , and note that f W ( k ) D ( x ) = 〈zD , k ( x ) , vec ( W ( k ) D , ` ) 〉 . Then , by the definition of ∆D,1 and the decomposition ( 91 ) , we have ∆D,1 = 1 L L∑ ` =1 〈zD , k ( x ) , E1 + E2 + E3〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y = 1 L L∑ ` =1 〈zD , k ( x ) , E1〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E3〉 + 1 L L∑ ` =1 〈zD , k ( x ) , E2〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y︸ ︷︷ ︸ : =C ( 92 ) First , we focus on controlling the ` 2 bound on the first two terms in ( 92 ) . Observe that the first term can be bounded as : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E1〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E1〉|2 . ( 93 ) Recall that ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) by Lemma 4.4 . Then , the random variable zD , k ( x ) > vec ( W ( 0 ) D , ` ) | zD , k ( x ) is simply a N ( 0 , O ( ω ) ) for 1 ≤ ` ≤ L. A straightforward application of Chernoff bound for normal random variable and taking union bound over the layer 1 ≤ ` ≤ L yield that : with probability at least 1− δ , 1 L L∑ ` =1 ∣∣∣zD , k ( x ) > vec ( W ( 0 ) D , ` ) ∣∣∣2 ≤ O ( ω log ( L δ ) ) . ( 94 ) The ` 2 norm of the second term in ( 92 ) can be similarly bounded as ( 93 ) in addition with the CauchySchwarz inequality : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E3〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E3〉|2 ≤ 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 . ( 95 ) The ‖E3‖2 is bounded as : ‖E3‖2 = ∥∥∥∥∥∥η1 k−1∑ j=0 ( 1− η2µ ) j [ ∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ] ( uD ( k − j − 1 ) − y ) ∥∥∥∥∥∥ 2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · ∥∥∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ∥∥2 ‖uD ( k − j − 1 ) − y‖2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · ∥∥∇W ` [ fWD ( k−j−1 ) ( X ) ] −∇W ` [ fWD ( 0 ) ( X ) ] ∥∥F ‖uD ( k − j − 1 ) − y‖2 = η1 k−1∑ j=0 ( 1− η2µ ) j · √√√√ n∑ i=1 ∥∥∇W ` [ fWD ( k−j−1 ) ( xi ) ] −∇W ` [ fWD ( 0 ) ( xi ) ] ∥∥2F ‖uD ( k − j − 1 ) − y‖2 ≤ η1 k−1∑ j=0 ( 1− η2µ ) j · √√√√2 n∑ i=1 ∥∥∇W ` [ fWD ( k−j−1 ) ( xi ) ] −∇W ` [ fWD ( 0 ) ( xi ) ] ∥∥22 ‖uD ( k − j − 1 ) − y‖2 ≤ η1 η2µ · O ( τ1/3L2 √ ωmn log ( m ) ) · O ( √ n ) ≤ O ( L10/3ω1/6 m2/3δ1/3 n 4d 6d−3 √ log ( m ) ) . ( 96 ) In the first , second and third inequalities , we used a simple fact that for the matrix A ∈ Rd1×d2 with rank r , then ‖A‖2 ≤ ‖A‖F ≤ √ r‖A‖2 . Recall that the rank of the matrix∇W ` [ fWD ( k−j−1 ) ( x ) ] − ∇W ` [ fWD ( 0 ) ( x ) ] is at most 2 . In the second to the last inequality , we use the result of Lemma 4.6 and the ‖uD ( i ) − y‖2 ≤ O ( √ n ) for any i ≥ 1 . In the last inequality , we plug the correct orders as set in ( 69 ) to τ , η1 , η2 and µ . Back to the inequality ( 95 ) , using the ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) and ( 96 ) , we can get 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 ≤ OP ( L20/3ω4/3 m1/3 n 8d 6d−3 log ( m ) ) . ( 97 ) Before controlling the ` 2 norm of C in ( 92 ) , recall that we set B : = ( η2µL η1m I + H∞L ) −1 H∞L y and the dynamics of uD ( k ) −B can be expressed in terms of H∞L as follows : For any k ≥ 1 , uD ( k ) −B = ( ( 1− η2µL ) · I −mη1H∞L ) k ( uD ( 0 ) −B ) + eD ( k ) , ( 98 ) with ‖eD ( k ) ‖2 ≤ O ( 1 n ) . Using ( 98 ) , we can further decompose the term E2 in ( 91 ) as : E2 : = −η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( uD ( k − j − 1 ) − y ) = η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 B − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 uD ( 0 ) − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) jeD ( k − j − 1 ) − η1∇W ` [ fWD ( 0 ) ( X ) ] k−1∑ j=0 ( 1− η2µ ) j ( B− y ) = E2,1 + E2,2 + E2,3 + E2,4 . ( 99 ) Then , we can re-write the error term C in ( 92 ) as : C = 1 L L∑ ` =1 〈zD , k ( x ) , E2,1〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E2,2〉+ 1 L L∑ ` =1 〈zD , k ( x ) , E2,3〉 + { 1 L L∑ ` =1 〈zD , k ( x ) , E2,4〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y } ︸ ︷︷ ︸ : =D . ( 100 ) Our goal is to control the ` 2 norm of each summand in the equality ( 100 ) . For the first three terms in ( 100 ) , a simple Cauchy-Schwarz inequality can be applied : for i = 1 , 2 , 3 : ∣∣∣∣∣ 1L L∑ ` =1 〈zD , k ( x ) , E2 , i〉 ∣∣∣∣∣ 2 ≤ 1 L L∑ ` =1 |〈zD , k ( x ) , E2 , i〉|2 ≤ 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 · ‖E2 , i‖ 2 2 . We work on obtaining the bound of ∑L ` =1 ‖E2,1‖ 2 2 . Let Tk be defined as Tk : = k−1∑ j=0 ( 1− η2µ ) j ( ( 1− η2µL ) · I −mη1H∞L ) k−j−1 . Then , we have L∑ ` =1 ‖E2,1‖22 = η 2 1 L∑ ` =1 ( B > T > k ∇W ` [ fWD ( 0 ) ( X ) ] > ∇W ` [ fWD ( 0 ) ( X ) ] TkB ) = mη21B > T > k H ( 0 ) TkB = mη21B > T > k ( H ( 0 ) −H∞L ) TkB +mη21B > T > k H∞L TkB ≤ mη21 ‖H ( 0 ) −H∞L ‖2 ·B > T 2k B +mη21B > T > k H∞L TkB . ( 101 ) To obtain the upper-bound on ( 101 ) , we need to control the terms T > k H∞L Tk and B > T 2k B . Let us denote H∞L = ∑n i=1 λiviv > i be the eigen-decomposition of H ∞ L . Using 1− η2µL ≤ 1− η2µ , note that Tk = k−1∑ j=0 ( 1− η2µ ) j ( 1− η2µL ) k−j−1 ( I − mη1 1− η2µL H∞L ) k−j−1 ( 1− η2µ ) k−1 k−1∑ i=0 ( I − mη1 1− η2µ H∞L ) i = ( 1− η2µ ) k−1 n∑ j=0 ( 1− ( 1− mη11−η2µλj ) k mη1 1−η2µλj ) vjv > j ( 1− η2µ ) k mη1λ0 · I . ( 102 ) A similar logic can be applied to bound T > k H∞L Tk : T > k H∞L Tk ( 1− η2µ ) k−1 n∑ j=0 ( 1− ( 1− mη11−η2µλj ) k mη1 1−η2µλj ) 2 λjvjv > j ( 1− η2µ ) 2k m2η21 · ( H∞L ) −1 . ( 103 ) Recall the definition of the notation B : = H∞L ( η2µL η1m I + H∞L ) −1 y . Then , we can bound the term B > T > k H∞L TkB : B > T > k H∞L TkB ≤ ( 1− η2µ ) 2k m2η21 ·B > ( H∞L ) −1 B = ( 1− η2µ ) 2k m2η21 · y > ( η2µL η1m I + H∞L ) −1 H∞L ( η2µL η1m I + H∞L ) −1 y = O ( ( 1− η2µ ) 2k m2η21 ) , ( 104 ) where in the last equality , we used ∥∥g∗µ∥∥2H = OP ( 1 ) in ( 90 ) . Now we turn our attention to bound the term B > T 2k B , B > T 2k B ≤ ( 1− η2µ ) 2k m2η21λ 2 0 y > ( η2µL η1m I + H∞L ) −1 ( H∞L ) 2 ( η2µL η1m I + H∞L ) −1 y = O ( ( 1− η2µ ) 2k n m2η21λ 2 0 ) , ( 105 ) where we used ‖y‖22 = O ( n ) in the last inequality . Combining the bounds ( 104 ) , ( 105 ) and the result from Lemma 4.11 , we can further bound ( 101 ) and have : L∑ ` =1 ‖E2,1‖22 ≤ O ( ω ( 1− η2µ ) 2k mλ20 n2L5/2 4 √ log ( nL/δ ) m + ( 1− η2µ ) 2k m ) ≤ O ( ( 1− η2µ ) 2k m ) , ( 106 ) where in the second inequality , we used m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Similarly , we can bound∑L ` =1 ‖E2,2‖ 2 2 : L∑ ` =1 ‖E2,2‖22 = η 2 1 L∑ ` =1 ( uD ( 0 ) > T > k ∇W ` [ fWD ( 0 ) ( X ) ] > ∇W ` [ fWD ( 0 ) ( X ) ] TkuD ( 0 ) ) = mη21uD ( 0 ) > T > k H ( 0 ) TkuD ( 0 ) = mη21uD ( 0 ) > T > k ( H ( 0 ) −H∞L ) TkuD ( 0 ) +mη21uD ( 0 ) > T > k H∞L TkuD ( 0 ) ≤ mη21 ‖H ( 0 ) −H∞L ‖2 · uD ( 0 ) > T 2k uD ( 0 ) +mη21uD ( 0 ) > T > k H∞L TkuD ( 0 ) ≤ mη21 ( 1− η2µ ) 2k m2η21λ 2 0 O ( ωnL5/2 4 √ log ( nL/δ ) m ) ‖uD ( 0 ) ‖22 +mη21 ( 1− η2µ ) 2k m2η21 uD ( 0 ) > ( H∞L ) −1uD ( 0 ) ≤ O ( ( 1− η2µ ) 2k n2ω2L5/2 mλ20δ 2 4 √ log ( nL/δ ) m + nω ( 1− η2µ ) 2k mλ0δ2 ) = OP ( nω ( 1− η2µ ) 2k mλ0 ) . ( 107 ) Here , in the second inequality , we used the inequalities ( 102 ) and ( 103 ) and Lemma 4.11 . In the third inequality , we used the Lemma 4.8 , ‖u ( 0 ) ‖2 = O ( √nω δ ) with probability at least 1− δ . In the last equality , we used m ≥ Ω ( L19n20 log 3 ( m ) δ2 ) . Next , we bound ∑L ` =1 ‖E2,3‖ 2 2 as : L∑ ` =1 ‖E2,3‖22 = mη 2 1 · ( k−1∑ j=0 ( 1− η2µ ) j ( ek−j−1 ) ) > HD ( 0 ) ( k−1∑ j=0 ( 1− η2µ ) j ( ek−j−1 ) ) ≤ mη 2 1 η22µ 2 · λmax ( HD ( k ) ) · ‖ek−j−1‖22 ≤ mη21 η22µ 2 · O ( ωnL ) · O ( 1 n2 ) = O ( L3 m ω · n− 4d−3 2d−1 ) . ( 108 ) Now , we focus on obtaining the ` 2 norm bound on D in ( 100 ) . Recall the definition of the notation B : = H∞L ( η2µL η1m I + H∞L ) −1 y . A simple calculation yields that B− y = H∞L ( η2µL η1m I + H∞L ) −1 y − y = −η2µL mη1 ( η2µL η1m I + H∞L ) −1 y . Then , we can re-write the expression of the D as : D : = ( η2µL mη1 ) · η1 1 L L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 k−1∑ j=0 ( 1− η2µ ) j ( η2µL η1m I + H∞L ) −1 y −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y = ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y − ( 1− η2µ ) k 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y = ( 1 m L∑ ` =1 〈 z0 , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y − ( 1− η2µ ) k 1 m L∑ ` =1 〈 z0 , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y + ( 1− ( 1− η2µ ) k ) ( 1 m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ) ( η2µL η1m I + H∞L ) −1 y , ( 109 ) where in the second equality , ∑k−1 j=0 ( 1− η2µ ) j = 1− ( 1−η2µ ) k η2µ is used . The ` 2 norm of first term in the ( 109 ) can be bounded as : ∥∥∥∥∥ ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ∥∥∥∥∥ ( 1 m L∑ ` =1 〈 zD , k ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 = √√√√ n∑ i=1 ( 1 m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( xi ) ] 〉 −Ker ( x , xi ) ) 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ O ( ω √ nL5/2 4 √ log ( nL/δ ) m ) · O ( η1m η2µL √ n ) = O ( ωη1mn η2µL L5/2 4 √ log ( nL/δ ) m ) = O ( ωL5/2n d 2d−1 4 √ log ( nL/δ ) m ) , ( 110 ) where , in the second inequality , we used Lemma 4.11 , and also we used∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ √ y > ( η2µL η1m I + H∞L ) −2 y ≤ √ η21m 2 η22µ 2L2 · ‖y‖22 = O ( η1m η2µL √ n ) . ( 111 ) The ` 2 norm of the second term in ( 109 ) can be easily bounded as : ∥∥∥∥∥ ( 1− η2µ ) k 1m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ∥∥∥∥∥ ( 1− η2µ ) k ( 1 m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 + ∥∥∥∥∥ ( 1− η2µ ) kKer ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− η2µ ) k ∥∥∥∥∥ 1m L∑ ` =1 〈 z0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 −Ker ( x , X ) ∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 + ( 1− η2µ ) k ∥∥∥∥∥Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− η2µ ) k · O ( ω√nL3/2 4√ log ( nL/δ ) m ) · O ( η1mη2µL√n ) +O ( ( 1− η2µ ) k ) ≤ ( 1− η2µ ) k · O ( ωL3/2n d2d−1 4√ log ( nL/δ ) m ) +O ( ( 1− η2µ ) k ) . ( 112 ) Lastly , the ` 2 norm of the third term in ( 109 ) is bounded as : ∥∥∥∥∥ ( 1− ( 1− η2µ ) k ) ( 1 m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉 ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ∥∥∥∥∥ 1m L∑ ` =1 〈 zD , k ( x ) − zD,0 ( x ) , ∇W ` [ fWD ( 0 ) ( X ) ] 〉∥∥∥∥∥ 2 · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ( 1 m L∑ ` =1 ‖zD , k ( x ) − zD,0 ( x ) ‖F ∥∥∇W ` [ fWD ( 0 ) ( X ) ] ∥∥F ) · ∥∥∥∥∥ ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 ≤ ( 1− ( 1− η2µ ) k ) · ( L m O ( τ1/3L2 √ ωm log ( m ) ) · O ( √ ωmn ) ) · O ( η1m η2µL √ n ) ≤ ( 1− ( 1− η2µ ) k ) · O ( ω7/6L10/3n 4d6d−3 √log ( m ) m1/6δ1/3 ) ≤ O ( ω7/6L10/3n 4d 6d−3 √ log ( m ) m1/6δ1/3 ) , ( 113 ) where in the fourth inequality , τ = OP ( L √ ω√ m n d 2d−1 ) is plugged in . Combining the inequali- ties ( 110 ) , ( 112 ) and ( 113 ) , we get the bound on ‖D‖2 in ( 109 ) : ‖D‖22 ≤ O ( ω2L5n 2d 2d−1 √ log ( nL/δ ) m ) + ( 1− η2µ ) 2kO ( ω2L3n 2d2d−1√ log ( nL/δ ) m ) +O ( ( 1− η2µ ) 2k ) +O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) ≤ O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) +O ( ( 1− η2µ ) 2k ) . ( 114 ) Step 3 . Combining all pieces . Recall ‖zD , k ( x ) ‖2 ≤ O ( √ mω ) . With this fact , combining the bounds ( 94 ) , ( 97 ) , ( 106 ) , ( 107 ) , ( 108 ) and ( 114 ) , we can bound the ‖∆D,1‖22 via the decomposition ( 92 ) as follows : ‖∆D,1‖22 ≤ 1 L L∑ ` =1 ∣∣∣zD , k ( x ) > vec ( W ( 0 ) D , ` ) ∣∣∣2 + 1L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E3‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2,1‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2,2‖ 2 2 + 1 L L∑ ` =1 ‖zD , k ( x ) ‖22 ‖E2.3‖ 2 2 + ∥∥∥∥∥ 1L L∑ ` =1 〈zD , k ( x ) , E2,4〉 −Ker ( x , X ) ( η2µL η1m I + H∞L ) −1 y ∥∥∥∥∥ 2 2 ≤ O ( ω log ( L δ ) ) +OP ( L20/3ω4/3 m1/3 n 8d 6d−3 log ( m ) ) +OP ( ω ( 1− η2µ ) 2k L ) +OP ( nω2 ( 1− η2µ ) 2k Lλ0 ) +OP ( L2 m ω · n− 4d−3 2d−1 ) +O ( ω7/3L20/3n 8d 6d−3 log ( m ) m1/3δ2/3 ) +OP ( ( 1− η2µ ) 2k ) ≤ OP ( n− d 2d−1 ) . | This paper considers fully connected DNNs with finite equal width $m$ in each layer. Gradient descent is used to minimize the unregularized or L2-norm regularized training error. For unregularized situations, this paper proves that the training error decreases to zero at a linear rate and the generalization error is bounded away from zero by some constant factor if the training process is too long or too short when there exists noise. For the L2-norm regularized situation, this paper provides an upper bound on the training error that consists of two terms, one converges linearly to 0, the other is a constant. Additionally, this paper also proves that the training dynamics of the regularized neural network can approximate the corresponding kernel ridge regression. For the generalization performance of the regularized situation, this paper proves that with enough large width, the prediction risk decreases with the number of training samples $n$ at least at the speed of $n^{-d/(2d-1)}$, which is a little bit faster than the existing results $n^{-1/2}$. Numerical results are provided to support those theoretical results. | SP:232f05c3b3a9b5ca1cea27b2aea1293a7da190c8 |
Robust Imitation via Mirror Descent Inverse Reinforcement Learning | 1 INTRODUCTION . Inverse reinforcement learning ( IRL ) is an algorithm of learning ground-truth rewards from demonstrations of an expert that acts optimally with respect to an unknown reward function . Traditional IRL approaches ( Ng & Russell , 2000 ; Abbeel & Ng , 2004 ; Ziebart et al. , 2008 ) solve the imitation problem based on iterative algorithms , alternating between the reward estimation process and a reinforcement learning ( RL ) algorithm . Recent adversarial imitation learning ( AIL ) studies ( Ho & Ermon , 2016 ; Fu et al. , 2017 ) focus on formulating a statistical divergence minimization with a fine-tuned representation of the target expert probability distribution ( Ghasemipour et al. , 2020 ) . One method to approximate the distance between probability distributions is using Bregman divergences , a family of metric-like functions induced by a strongly convex function . Jeon et al . ( 2020 ) generalized the AIL framework to an optimization problem with respect to a Bregman divergence between policy distributions where the method shares on the essence of regularized Markov decision processes ( Geist et al. , 2019 ) . Through the lens of differential geometries , this implies that a policy distribution and a reward function can be associated with geometric constraints specified by a convex regularizer ( Shima , 2007 ) . Apparently , the limitation of regularized IRL studies naturally comes from a geometric property in which minimizing a Bregman divergence does not guarantee unbiased progression due to the constraints of the underlying space ( Butnariu & Resmerita , 2006 ) . The success of machine learning has been the key to learning such divergences with high precision . When the demonstration size is sufficiently large , the discriminator approximation of AIL can model the entire expert policy distribution internally using nonlinear approximators such as neural networks ( Finn et al. , 2016 ; Fu et al. , 2017 ) . However , there are challenging tasks that require long action sequences to be solved . When states can only be rarely visited by imitation learning agents , the uncertainty of estimated expert policy often substantially affects the discrimination quality . Therefore , the development of a robust IRL method that is tolerant to unreliable discriminative signals induced by imperfect demonstrations is necessary . The reasonable question is how to learn with cost estimations that are not precisely determined throughout the learning process . There have been considerable achievements in dealing with temporal costs in the online learning domain ( Fiat & Woeginger , 1998 ; Hazan , 2019 ) . An online learning algorithm predicts a sequence of parameters { wt } Tt=1 ⊂ W that is incurred by a cost functionFt ( · ) at each iteration . The most ordinary approach in online learning is stochastic gradient descent ( SGD ) : wt+1 = wt − ηt∇Ft ( wt ) . SGD is a desirable algorithm when the setW is the Euclidean geometry since it can provide unbiased progression for minimizing Ft ( Raskutti & Mukherjee , 2015 ) . However , policies appear in manifolds of probability distributions ; a gradient may not be the direction of the steepest descent in this case due to geometric constraints ( Amari , 1998 ; 2016 ) . In this paper , we propose an iterative imitation learning framework that interprets the RL-IRL scheme similar to proximal optimization methods ( Boyd et al. , 2004 ; Amari , 2016 ) . We identify two issues in the AIL studies that have characteristics of unconstrained updates : ( 1 ) a divergence does not guarantee informative global directions to match the expert policy due to the geometry , and ( 2 ) a representation of divergence often can not be accurately obtained due to insufficient data . Our method is motivated by a template of optimization algorithms specified by a convex function , called mirror descent ( MD ; Nemirovsky & Yudin 1983 ) . For parameters sequences , cost functions , and step sizes , an MD update for a strongly convex function Ω is derived as ∇Ω ( wt+1 ) = ∇Ω ( wt ) − ηt∇Ft ( wt ) , ( 1 ) where ∇Ω is a bijective transformation that links the primal space and the dual space of gradients . Different from these standard MD formulations , our methodology draws a sequence of functions on a space formulated by a regularized reward operator which is an alternative to the dual space . Consequently , the reward functions are projected optimization targets for the space of policies . Our Contributions . We propose a novel IRL algorithm which facilitates the agent in robustly imitating the expert . Our work is complementary to previous regularized IRL studies ; we introduce NEW R1 , R2 a geometric perspective for optimizing rewards and derive solutions with theoretical guarantees . • Instead of a monolithic estimation process of a global solution , we derive a sequence of reward functions that provides local optimization targets for the space of policies ( Section 3 ) . • We prove that rewards derived by an MD algorithm guarantee convergent divergence minimization performance along with a rigorous regret bound ( Section 4 ) . • We propose mirror descent adversarial inverse reinforcement learning ( MD-AIRL ) , a novel IRL algorithm that can be easily implemented on top of the standard AIL framework ( Section 5 ) . • We validate the outperforming performance of MD-AIRL on benchmarks with large discrete ac- tion spaces and continuous action spaces ( Section 6 ) . 2 PRELIMINARIES . Notation . For finite setsX and Y , we define Y X as a set of functions fromX to Y . ∆X ( ∆YX ) is a set of ( conditional ) probabilities over X ( conditioned on Y ) . For a function f ∈ RS×A and a policy π ∈ ∆SA , fs and πs denote shorthand notation of f ( s , · ) and π ( ·|s ) , respectively . A Markov decision process ( MDP ) is defined as a tuple ( S , A , P , r , γ ) with the state space S , the action space A , the transition kernel P ∈ ∆S×AS , the reward function r ∈ RS×A and the discount factor γ ∈ [ 0 , 1 ) . A Bregman divergence with respect to a convex function Ω for a state s is defined as DΩ ( πs∥π̂s ) = Ω ( πs ) − Ω ( π̂s ) − ⟨∇Ω ( π̂s ) , πs − π̂s⟩A . We write Ω ( π ) = [ Ω ( πs ) ] s∈S and similarly for∇Ω . Regularized RL & IRL . We consider the RL-IRL framework in regularized MDPs ( Geist et al. , 2019 ) , where the policy is optimized along with a causal convex regularizer . The objective is to find π which maximizes the expected discounted sum of rewards with a strongly convex regularizer Ω : maximize π∈∆SA JΩ ( π , r ) : = Eπ [ ∑∞ i=0 γi { r ( si , ai ) − Ω ( π ( ·|si ) ) } ] , ( 2 ) where the subscript π on the expectation denotes samples generated from the MDP and policy π . Consider the convex conjugate of qsE = ∇Ω ( πsE ) : Ω∗ ( qsE ) = maxπs∈∆A⟨πs , qsE⟩A − Ω ( πs ) . Differentiating both sides with respect to qsE , the gradient of conjugate∇Ω∗ maps qE to the unique optimal policy πE . Let Π be a bounded , open set for ∆SA ⊂ Π and R : Π → RS×A be a reward operator which maps a policy to a reward function . The output r̂ = R ( πE ) is a representation of the expert ’ s behavior ; but finding such operator is an ill-posed problem because every function r̂ that makes the state-action value function qE is a valid solution of regularized IRL ( Geist et al. , 2019 ) . Recently , Jeon et al . ( 2020 ) proposed a tractable solution , which we refer to the function as the regularized reward function . We rewrite the previous notation by defining the regularized reward operator ΨΩ . Definition 1 ( Regularized reward operators ) . Define the regularized reward operator ΨΩ : π 7→ ψπ ψπ ( s , a ) : = Ω ′ ( s , a ; π ) − 〈 πs , ∇Ω ( πs ) 〉 A +Ω ( π s ) ∀s ∈ S , a ∈ A , ( 3 ) for Ω′ ( s , · ; π ) : = ∇Ω ( πs ) = [ ∇pΩ ( p ) ] p=π ( ·|s ) . By using the operator in discriminative networks , Jeon et al . ( 2020 ) proposed regularized adversarial IRL ( RAIRL ) as a generalization of AIL in terms of minimizing a specific Bregman divergence . Mirror Descent . Let w be a parameter on a set W and Ft : W → R be a convex cost function from a class of functions F . Replacing the L2 proximity term of proximal gradient descent with the Bregman divergence ( Gutman & Peña , 2018 ) , the proximal form of the MD update is given as wMDt+1 = argmin w∈W 〈 ∇Ft ( wMDt ) , w − wMDt 〉 W + αtDΩ ( w ∥∥wMDt ) , ( 4 ) where αt : = 1/ηt denotes an inverse of the step size ηt . MD was developed as a generalization of SGD , where the local geometry is specified by a Bregman divergence ( Gunasekar et al. , 2020 ) . 3 ITERATIVE RL-IRL AS A PROXIMAL OPTIMIZATION METHOD . We interpret the RL-IRL framework as a variant of proximal optimization methods and consider two sequences { πt } ∞t=1 and { ψt } ∞t=1 that denote the learning policies and reward functions , respectively . Associated reward function . An updated point in MD can be uniquely projected to the desired space ( ∆SA in our case ) using a Bregman projection operator PΩ that locates iterative points to the feasible region , i.e . πt+1 = PΩ ( π̃t+1 ) : = argminπ∈∆SA [ DΩ ( π∥π̃t+1 ) ] s∈S for π̃t+1 ∈ Π . To avoid these computations , we preemptively constrain a reward operator to satisfy the projection invariance with respect to regularized RL process of the associated state-action value function q̂ : PΩ ( ∇Ω∗ ( q̂ ) ) = ∇Ω∗ ( q̂ ) . According to Lemma 1 of Jeon et al . ( 2020 ) , a regularized reward function ψπ can replace q̂ , since the induced Bregman divergence allows the learning in a greedy manner . FIX R3 The projection invariance of the operator ΨΩ can be shown by a bijective relation with ∇Ω∗ in the space of ∆SA by the following lemma . Lemma 1 ( Natural isomorphism ) . Let ψ ∈ ΨΩ ( ∆SA ) for ΨΩ ( Π ) = { ψ | ψ ( s , a ) = ψπ ( s , a ) , ∀s∈ S , a∈A , π∈Π } . Then , ∇Ω∗ ( ψ ) is unique with respect to ψ , and for every π = ∇Ω∗ ( ψ ) , π ∈ ∆SA . Policy Space ∆SA Π πt πt+1 Reward Space ψt ψt+1 ΨΩ ∇Ω∗ FIX R3 As a result , we consider an updated reward function ψt+1 ∈ ΨΩ ( ∆SA ) as a projected target of MD , FIX R3 which can be associated by parameterization of ∆SA . For instance , a bijective mapping of ψt+1 can NEW R3 represent a softmax policy for a discrete space , or a Gaussian policy for a continuous space , using its parameters . The subsequent RL process at t-th step becomes finding the next iteration πt+1 by NEW R3 FIX R3maximize π∈∆SA JΩ ( π , ψt+1 ) = Eπ [ ∑∞ i=0 γiDΩ ( πsi ∥∥πsit+1 ) ] , πt+1 = ∇Ω∗ ( ψt+1 ) . ( 5 ) The equation shows that an arbitrary regularized RL algorithm with the regularizer Ω forms a cumulative discounted sum of Bregman divergences , thus the mapping to πt+1 can be achieved . Online imitation learning . The necessity of online learning setup comes from observing that our RL-IRL processes do not retain ψE = ΨΩ ( πE ) , a representation of πE , during training . Instead , consider a random process { π̄E , t } ∞t=1 where the estimation π̄E , t resides in a closed , convex neighborhood of πE , estimated by a separate estimation algorithm such as maximum likelihood methods . Substituting ψE to ψπ̄E , t in JΩ ( π , ψE ) turns the RL objective into an online learning problem : FIX R3minimize π∈∆SA Ft ( π ) = −JΩ ( π , ψπ̄E , t ) = Eπ [ ∑∞ i=0 γiDΩ ( πsi ∥∥π̄siE , t ) ] . ( 6 ) For a deeper understanding of our setup , we consider an example in Figure 2 . Assume that policies of the learning agent and the expert follow multivariate Gaussian distributions initially at π1 = N ( [ 0 , 0 ] T , I ) and πE = N ( [ 5 , 3 ] T , ΣE ) for |ΣE| < 1 . The policy πt is trained by a cost function DΩ ( ·∥π̄E , t ) with the proximity constraint 1ηDΩ ( ·∥πt ) where the Gaussian policy π̄E , t is also fitted using stochastic samples from πE at each iteration , starting from π̄E,1 = π1 . We first observe that the step size constant η affects the training speed in the early phase . Notably , the performance of certain cases exceeds the baselines of DΩ ( π̄E , t∥πE ) by choosing the step size effectively low as η < 1 . This suggests that there is a clear advantage of the online imitation learning setup for unreliable π̄E , t . MD update rules . Plugging each divergence of the cumulative cost Ft to the template of Equation ( 4 ) , the optimization process for the subsequent ψt+1 of the MD formulation is derived as1 ψt+1 : = ΨΩ ( πt+1 ) , π s t+1 = argmin πs∈∆A 〈 ∇DΩ ( πst ∥π̄sE , t ) ︸ ︷︷ ︸ ∇Ω ( πst ) −∇Ω ( π̄sE , t ) , πs − πst 〉 A + αtDΩ ( π s∥πst ) = argmin πs∈∆A DΩ ( π s∥π̄sE , t ) −DΩ ( πs∥πst ) + αtDΩ ( πs∥πst ) = argmin πs∈∆A ηtDΩ ( π s∥π̄sE , t ) ︸ ︷︷ ︸ estimated expert + ( 1− ηt ) DΩ ( πs∥πst ) ︸ ︷︷ ︸ learning agent , ηt = 1/αt , ∀s ∈ S , ( 7 ) where the gradient of DΩ is taken with respect to its first argument πst . The objective of reward NEW R3 learning is analogous to finding an interpolation at each iteration where the point is controlled by the step size ηt . Figure 3 shows that the region of πt ( defined by a norm ) gradually decreases 1Bregman divergences are generally intractable to be computed , unless the policy is a specific parametric model ( e.g . exponential families ) . See the works of Nielsen & Nock ( 2011 ) ; Jeon et al . ( 2020 ) and Appendix B. when ηt > ηt+1 . Note that solving the optimization of Ft requires interaction between πt and the dynamics of the given environment ; thus , the RL process in Equation ( 5 ) plays the essential role of sequential learning by the value measures of the reward function . | This paper presents a novel mirror-descent adversarial inverse reinforcement learning (MD-AIRL) algorithm. MD-AIRL considers the reward function as an iterative sequence in a proximal method. MD-AIRL has been introduced with dense theoretical analysis and validated with diverse experiments covering both discrete and continuous action spaces. | SP:a353062940ca72746921b1a50646d9584b4b9e39 |
Robust Imitation via Mirror Descent Inverse Reinforcement Learning | 1 INTRODUCTION . Inverse reinforcement learning ( IRL ) is an algorithm of learning ground-truth rewards from demonstrations of an expert that acts optimally with respect to an unknown reward function . Traditional IRL approaches ( Ng & Russell , 2000 ; Abbeel & Ng , 2004 ; Ziebart et al. , 2008 ) solve the imitation problem based on iterative algorithms , alternating between the reward estimation process and a reinforcement learning ( RL ) algorithm . Recent adversarial imitation learning ( AIL ) studies ( Ho & Ermon , 2016 ; Fu et al. , 2017 ) focus on formulating a statistical divergence minimization with a fine-tuned representation of the target expert probability distribution ( Ghasemipour et al. , 2020 ) . One method to approximate the distance between probability distributions is using Bregman divergences , a family of metric-like functions induced by a strongly convex function . Jeon et al . ( 2020 ) generalized the AIL framework to an optimization problem with respect to a Bregman divergence between policy distributions where the method shares on the essence of regularized Markov decision processes ( Geist et al. , 2019 ) . Through the lens of differential geometries , this implies that a policy distribution and a reward function can be associated with geometric constraints specified by a convex regularizer ( Shima , 2007 ) . Apparently , the limitation of regularized IRL studies naturally comes from a geometric property in which minimizing a Bregman divergence does not guarantee unbiased progression due to the constraints of the underlying space ( Butnariu & Resmerita , 2006 ) . The success of machine learning has been the key to learning such divergences with high precision . When the demonstration size is sufficiently large , the discriminator approximation of AIL can model the entire expert policy distribution internally using nonlinear approximators such as neural networks ( Finn et al. , 2016 ; Fu et al. , 2017 ) . However , there are challenging tasks that require long action sequences to be solved . When states can only be rarely visited by imitation learning agents , the uncertainty of estimated expert policy often substantially affects the discrimination quality . Therefore , the development of a robust IRL method that is tolerant to unreliable discriminative signals induced by imperfect demonstrations is necessary . The reasonable question is how to learn with cost estimations that are not precisely determined throughout the learning process . There have been considerable achievements in dealing with temporal costs in the online learning domain ( Fiat & Woeginger , 1998 ; Hazan , 2019 ) . An online learning algorithm predicts a sequence of parameters { wt } Tt=1 ⊂ W that is incurred by a cost functionFt ( · ) at each iteration . The most ordinary approach in online learning is stochastic gradient descent ( SGD ) : wt+1 = wt − ηt∇Ft ( wt ) . SGD is a desirable algorithm when the setW is the Euclidean geometry since it can provide unbiased progression for minimizing Ft ( Raskutti & Mukherjee , 2015 ) . However , policies appear in manifolds of probability distributions ; a gradient may not be the direction of the steepest descent in this case due to geometric constraints ( Amari , 1998 ; 2016 ) . In this paper , we propose an iterative imitation learning framework that interprets the RL-IRL scheme similar to proximal optimization methods ( Boyd et al. , 2004 ; Amari , 2016 ) . We identify two issues in the AIL studies that have characteristics of unconstrained updates : ( 1 ) a divergence does not guarantee informative global directions to match the expert policy due to the geometry , and ( 2 ) a representation of divergence often can not be accurately obtained due to insufficient data . Our method is motivated by a template of optimization algorithms specified by a convex function , called mirror descent ( MD ; Nemirovsky & Yudin 1983 ) . For parameters sequences , cost functions , and step sizes , an MD update for a strongly convex function Ω is derived as ∇Ω ( wt+1 ) = ∇Ω ( wt ) − ηt∇Ft ( wt ) , ( 1 ) where ∇Ω is a bijective transformation that links the primal space and the dual space of gradients . Different from these standard MD formulations , our methodology draws a sequence of functions on a space formulated by a regularized reward operator which is an alternative to the dual space . Consequently , the reward functions are projected optimization targets for the space of policies . Our Contributions . We propose a novel IRL algorithm which facilitates the agent in robustly imitating the expert . Our work is complementary to previous regularized IRL studies ; we introduce NEW R1 , R2 a geometric perspective for optimizing rewards and derive solutions with theoretical guarantees . • Instead of a monolithic estimation process of a global solution , we derive a sequence of reward functions that provides local optimization targets for the space of policies ( Section 3 ) . • We prove that rewards derived by an MD algorithm guarantee convergent divergence minimization performance along with a rigorous regret bound ( Section 4 ) . • We propose mirror descent adversarial inverse reinforcement learning ( MD-AIRL ) , a novel IRL algorithm that can be easily implemented on top of the standard AIL framework ( Section 5 ) . • We validate the outperforming performance of MD-AIRL on benchmarks with large discrete ac- tion spaces and continuous action spaces ( Section 6 ) . 2 PRELIMINARIES . Notation . For finite setsX and Y , we define Y X as a set of functions fromX to Y . ∆X ( ∆YX ) is a set of ( conditional ) probabilities over X ( conditioned on Y ) . For a function f ∈ RS×A and a policy π ∈ ∆SA , fs and πs denote shorthand notation of f ( s , · ) and π ( ·|s ) , respectively . A Markov decision process ( MDP ) is defined as a tuple ( S , A , P , r , γ ) with the state space S , the action space A , the transition kernel P ∈ ∆S×AS , the reward function r ∈ RS×A and the discount factor γ ∈ [ 0 , 1 ) . A Bregman divergence with respect to a convex function Ω for a state s is defined as DΩ ( πs∥π̂s ) = Ω ( πs ) − Ω ( π̂s ) − ⟨∇Ω ( π̂s ) , πs − π̂s⟩A . We write Ω ( π ) = [ Ω ( πs ) ] s∈S and similarly for∇Ω . Regularized RL & IRL . We consider the RL-IRL framework in regularized MDPs ( Geist et al. , 2019 ) , where the policy is optimized along with a causal convex regularizer . The objective is to find π which maximizes the expected discounted sum of rewards with a strongly convex regularizer Ω : maximize π∈∆SA JΩ ( π , r ) : = Eπ [ ∑∞ i=0 γi { r ( si , ai ) − Ω ( π ( ·|si ) ) } ] , ( 2 ) where the subscript π on the expectation denotes samples generated from the MDP and policy π . Consider the convex conjugate of qsE = ∇Ω ( πsE ) : Ω∗ ( qsE ) = maxπs∈∆A⟨πs , qsE⟩A − Ω ( πs ) . Differentiating both sides with respect to qsE , the gradient of conjugate∇Ω∗ maps qE to the unique optimal policy πE . Let Π be a bounded , open set for ∆SA ⊂ Π and R : Π → RS×A be a reward operator which maps a policy to a reward function . The output r̂ = R ( πE ) is a representation of the expert ’ s behavior ; but finding such operator is an ill-posed problem because every function r̂ that makes the state-action value function qE is a valid solution of regularized IRL ( Geist et al. , 2019 ) . Recently , Jeon et al . ( 2020 ) proposed a tractable solution , which we refer to the function as the regularized reward function . We rewrite the previous notation by defining the regularized reward operator ΨΩ . Definition 1 ( Regularized reward operators ) . Define the regularized reward operator ΨΩ : π 7→ ψπ ψπ ( s , a ) : = Ω ′ ( s , a ; π ) − 〈 πs , ∇Ω ( πs ) 〉 A +Ω ( π s ) ∀s ∈ S , a ∈ A , ( 3 ) for Ω′ ( s , · ; π ) : = ∇Ω ( πs ) = [ ∇pΩ ( p ) ] p=π ( ·|s ) . By using the operator in discriminative networks , Jeon et al . ( 2020 ) proposed regularized adversarial IRL ( RAIRL ) as a generalization of AIL in terms of minimizing a specific Bregman divergence . Mirror Descent . Let w be a parameter on a set W and Ft : W → R be a convex cost function from a class of functions F . Replacing the L2 proximity term of proximal gradient descent with the Bregman divergence ( Gutman & Peña , 2018 ) , the proximal form of the MD update is given as wMDt+1 = argmin w∈W 〈 ∇Ft ( wMDt ) , w − wMDt 〉 W + αtDΩ ( w ∥∥wMDt ) , ( 4 ) where αt : = 1/ηt denotes an inverse of the step size ηt . MD was developed as a generalization of SGD , where the local geometry is specified by a Bregman divergence ( Gunasekar et al. , 2020 ) . 3 ITERATIVE RL-IRL AS A PROXIMAL OPTIMIZATION METHOD . We interpret the RL-IRL framework as a variant of proximal optimization methods and consider two sequences { πt } ∞t=1 and { ψt } ∞t=1 that denote the learning policies and reward functions , respectively . Associated reward function . An updated point in MD can be uniquely projected to the desired space ( ∆SA in our case ) using a Bregman projection operator PΩ that locates iterative points to the feasible region , i.e . πt+1 = PΩ ( π̃t+1 ) : = argminπ∈∆SA [ DΩ ( π∥π̃t+1 ) ] s∈S for π̃t+1 ∈ Π . To avoid these computations , we preemptively constrain a reward operator to satisfy the projection invariance with respect to regularized RL process of the associated state-action value function q̂ : PΩ ( ∇Ω∗ ( q̂ ) ) = ∇Ω∗ ( q̂ ) . According to Lemma 1 of Jeon et al . ( 2020 ) , a regularized reward function ψπ can replace q̂ , since the induced Bregman divergence allows the learning in a greedy manner . FIX R3 The projection invariance of the operator ΨΩ can be shown by a bijective relation with ∇Ω∗ in the space of ∆SA by the following lemma . Lemma 1 ( Natural isomorphism ) . Let ψ ∈ ΨΩ ( ∆SA ) for ΨΩ ( Π ) = { ψ | ψ ( s , a ) = ψπ ( s , a ) , ∀s∈ S , a∈A , π∈Π } . Then , ∇Ω∗ ( ψ ) is unique with respect to ψ , and for every π = ∇Ω∗ ( ψ ) , π ∈ ∆SA . Policy Space ∆SA Π πt πt+1 Reward Space ψt ψt+1 ΨΩ ∇Ω∗ FIX R3 As a result , we consider an updated reward function ψt+1 ∈ ΨΩ ( ∆SA ) as a projected target of MD , FIX R3 which can be associated by parameterization of ∆SA . For instance , a bijective mapping of ψt+1 can NEW R3 represent a softmax policy for a discrete space , or a Gaussian policy for a continuous space , using its parameters . The subsequent RL process at t-th step becomes finding the next iteration πt+1 by NEW R3 FIX R3maximize π∈∆SA JΩ ( π , ψt+1 ) = Eπ [ ∑∞ i=0 γiDΩ ( πsi ∥∥πsit+1 ) ] , πt+1 = ∇Ω∗ ( ψt+1 ) . ( 5 ) The equation shows that an arbitrary regularized RL algorithm with the regularizer Ω forms a cumulative discounted sum of Bregman divergences , thus the mapping to πt+1 can be achieved . Online imitation learning . The necessity of online learning setup comes from observing that our RL-IRL processes do not retain ψE = ΨΩ ( πE ) , a representation of πE , during training . Instead , consider a random process { π̄E , t } ∞t=1 where the estimation π̄E , t resides in a closed , convex neighborhood of πE , estimated by a separate estimation algorithm such as maximum likelihood methods . Substituting ψE to ψπ̄E , t in JΩ ( π , ψE ) turns the RL objective into an online learning problem : FIX R3minimize π∈∆SA Ft ( π ) = −JΩ ( π , ψπ̄E , t ) = Eπ [ ∑∞ i=0 γiDΩ ( πsi ∥∥π̄siE , t ) ] . ( 6 ) For a deeper understanding of our setup , we consider an example in Figure 2 . Assume that policies of the learning agent and the expert follow multivariate Gaussian distributions initially at π1 = N ( [ 0 , 0 ] T , I ) and πE = N ( [ 5 , 3 ] T , ΣE ) for |ΣE| < 1 . The policy πt is trained by a cost function DΩ ( ·∥π̄E , t ) with the proximity constraint 1ηDΩ ( ·∥πt ) where the Gaussian policy π̄E , t is also fitted using stochastic samples from πE at each iteration , starting from π̄E,1 = π1 . We first observe that the step size constant η affects the training speed in the early phase . Notably , the performance of certain cases exceeds the baselines of DΩ ( π̄E , t∥πE ) by choosing the step size effectively low as η < 1 . This suggests that there is a clear advantage of the online imitation learning setup for unreliable π̄E , t . MD update rules . Plugging each divergence of the cumulative cost Ft to the template of Equation ( 4 ) , the optimization process for the subsequent ψt+1 of the MD formulation is derived as1 ψt+1 : = ΨΩ ( πt+1 ) , π s t+1 = argmin πs∈∆A 〈 ∇DΩ ( πst ∥π̄sE , t ) ︸ ︷︷ ︸ ∇Ω ( πst ) −∇Ω ( π̄sE , t ) , πs − πst 〉 A + αtDΩ ( π s∥πst ) = argmin πs∈∆A DΩ ( π s∥π̄sE , t ) −DΩ ( πs∥πst ) + αtDΩ ( πs∥πst ) = argmin πs∈∆A ηtDΩ ( π s∥π̄sE , t ) ︸ ︷︷ ︸ estimated expert + ( 1− ηt ) DΩ ( πs∥πst ) ︸ ︷︷ ︸ learning agent , ηt = 1/αt , ∀s ∈ S , ( 7 ) where the gradient of DΩ is taken with respect to its first argument πst . The objective of reward NEW R3 learning is analogous to finding an interpolation at each iteration where the point is controlled by the step size ηt . Figure 3 shows that the region of πt ( defined by a norm ) gradually decreases 1Bregman divergences are generally intractable to be computed , unless the policy is a specific parametric model ( e.g . exponential families ) . See the works of Nielsen & Nock ( 2011 ) ; Jeon et al . ( 2020 ) and Appendix B. when ηt > ηt+1 . Note that solving the optimization of Ft requires interaction between πt and the dynamics of the given environment ; thus , the RL process in Equation ( 5 ) plays the essential role of sequential learning by the value measures of the reward function . | This paper proposes an alternative optimization technique (Mirror Descent) for adversarial inverse reinforcement learning, claiming it will resolve some of the issues with previous methods. -The authors prove the convergence of the proposed optimization method and provide a regret bound. -They experimentally show the benefits of the proposed methods in discrete and continuous action spaces. | SP:a353062940ca72746921b1a50646d9584b4b9e39 |
Robust Imitation via Mirror Descent Inverse Reinforcement Learning | 1 INTRODUCTION . Inverse reinforcement learning ( IRL ) is an algorithm of learning ground-truth rewards from demonstrations of an expert that acts optimally with respect to an unknown reward function . Traditional IRL approaches ( Ng & Russell , 2000 ; Abbeel & Ng , 2004 ; Ziebart et al. , 2008 ) solve the imitation problem based on iterative algorithms , alternating between the reward estimation process and a reinforcement learning ( RL ) algorithm . Recent adversarial imitation learning ( AIL ) studies ( Ho & Ermon , 2016 ; Fu et al. , 2017 ) focus on formulating a statistical divergence minimization with a fine-tuned representation of the target expert probability distribution ( Ghasemipour et al. , 2020 ) . One method to approximate the distance between probability distributions is using Bregman divergences , a family of metric-like functions induced by a strongly convex function . Jeon et al . ( 2020 ) generalized the AIL framework to an optimization problem with respect to a Bregman divergence between policy distributions where the method shares on the essence of regularized Markov decision processes ( Geist et al. , 2019 ) . Through the lens of differential geometries , this implies that a policy distribution and a reward function can be associated with geometric constraints specified by a convex regularizer ( Shima , 2007 ) . Apparently , the limitation of regularized IRL studies naturally comes from a geometric property in which minimizing a Bregman divergence does not guarantee unbiased progression due to the constraints of the underlying space ( Butnariu & Resmerita , 2006 ) . The success of machine learning has been the key to learning such divergences with high precision . When the demonstration size is sufficiently large , the discriminator approximation of AIL can model the entire expert policy distribution internally using nonlinear approximators such as neural networks ( Finn et al. , 2016 ; Fu et al. , 2017 ) . However , there are challenging tasks that require long action sequences to be solved . When states can only be rarely visited by imitation learning agents , the uncertainty of estimated expert policy often substantially affects the discrimination quality . Therefore , the development of a robust IRL method that is tolerant to unreliable discriminative signals induced by imperfect demonstrations is necessary . The reasonable question is how to learn with cost estimations that are not precisely determined throughout the learning process . There have been considerable achievements in dealing with temporal costs in the online learning domain ( Fiat & Woeginger , 1998 ; Hazan , 2019 ) . An online learning algorithm predicts a sequence of parameters { wt } Tt=1 ⊂ W that is incurred by a cost functionFt ( · ) at each iteration . The most ordinary approach in online learning is stochastic gradient descent ( SGD ) : wt+1 = wt − ηt∇Ft ( wt ) . SGD is a desirable algorithm when the setW is the Euclidean geometry since it can provide unbiased progression for minimizing Ft ( Raskutti & Mukherjee , 2015 ) . However , policies appear in manifolds of probability distributions ; a gradient may not be the direction of the steepest descent in this case due to geometric constraints ( Amari , 1998 ; 2016 ) . In this paper , we propose an iterative imitation learning framework that interprets the RL-IRL scheme similar to proximal optimization methods ( Boyd et al. , 2004 ; Amari , 2016 ) . We identify two issues in the AIL studies that have characteristics of unconstrained updates : ( 1 ) a divergence does not guarantee informative global directions to match the expert policy due to the geometry , and ( 2 ) a representation of divergence often can not be accurately obtained due to insufficient data . Our method is motivated by a template of optimization algorithms specified by a convex function , called mirror descent ( MD ; Nemirovsky & Yudin 1983 ) . For parameters sequences , cost functions , and step sizes , an MD update for a strongly convex function Ω is derived as ∇Ω ( wt+1 ) = ∇Ω ( wt ) − ηt∇Ft ( wt ) , ( 1 ) where ∇Ω is a bijective transformation that links the primal space and the dual space of gradients . Different from these standard MD formulations , our methodology draws a sequence of functions on a space formulated by a regularized reward operator which is an alternative to the dual space . Consequently , the reward functions are projected optimization targets for the space of policies . Our Contributions . We propose a novel IRL algorithm which facilitates the agent in robustly imitating the expert . Our work is complementary to previous regularized IRL studies ; we introduce NEW R1 , R2 a geometric perspective for optimizing rewards and derive solutions with theoretical guarantees . • Instead of a monolithic estimation process of a global solution , we derive a sequence of reward functions that provides local optimization targets for the space of policies ( Section 3 ) . • We prove that rewards derived by an MD algorithm guarantee convergent divergence minimization performance along with a rigorous regret bound ( Section 4 ) . • We propose mirror descent adversarial inverse reinforcement learning ( MD-AIRL ) , a novel IRL algorithm that can be easily implemented on top of the standard AIL framework ( Section 5 ) . • We validate the outperforming performance of MD-AIRL on benchmarks with large discrete ac- tion spaces and continuous action spaces ( Section 6 ) . 2 PRELIMINARIES . Notation . For finite setsX and Y , we define Y X as a set of functions fromX to Y . ∆X ( ∆YX ) is a set of ( conditional ) probabilities over X ( conditioned on Y ) . For a function f ∈ RS×A and a policy π ∈ ∆SA , fs and πs denote shorthand notation of f ( s , · ) and π ( ·|s ) , respectively . A Markov decision process ( MDP ) is defined as a tuple ( S , A , P , r , γ ) with the state space S , the action space A , the transition kernel P ∈ ∆S×AS , the reward function r ∈ RS×A and the discount factor γ ∈ [ 0 , 1 ) . A Bregman divergence with respect to a convex function Ω for a state s is defined as DΩ ( πs∥π̂s ) = Ω ( πs ) − Ω ( π̂s ) − ⟨∇Ω ( π̂s ) , πs − π̂s⟩A . We write Ω ( π ) = [ Ω ( πs ) ] s∈S and similarly for∇Ω . Regularized RL & IRL . We consider the RL-IRL framework in regularized MDPs ( Geist et al. , 2019 ) , where the policy is optimized along with a causal convex regularizer . The objective is to find π which maximizes the expected discounted sum of rewards with a strongly convex regularizer Ω : maximize π∈∆SA JΩ ( π , r ) : = Eπ [ ∑∞ i=0 γi { r ( si , ai ) − Ω ( π ( ·|si ) ) } ] , ( 2 ) where the subscript π on the expectation denotes samples generated from the MDP and policy π . Consider the convex conjugate of qsE = ∇Ω ( πsE ) : Ω∗ ( qsE ) = maxπs∈∆A⟨πs , qsE⟩A − Ω ( πs ) . Differentiating both sides with respect to qsE , the gradient of conjugate∇Ω∗ maps qE to the unique optimal policy πE . Let Π be a bounded , open set for ∆SA ⊂ Π and R : Π → RS×A be a reward operator which maps a policy to a reward function . The output r̂ = R ( πE ) is a representation of the expert ’ s behavior ; but finding such operator is an ill-posed problem because every function r̂ that makes the state-action value function qE is a valid solution of regularized IRL ( Geist et al. , 2019 ) . Recently , Jeon et al . ( 2020 ) proposed a tractable solution , which we refer to the function as the regularized reward function . We rewrite the previous notation by defining the regularized reward operator ΨΩ . Definition 1 ( Regularized reward operators ) . Define the regularized reward operator ΨΩ : π 7→ ψπ ψπ ( s , a ) : = Ω ′ ( s , a ; π ) − 〈 πs , ∇Ω ( πs ) 〉 A +Ω ( π s ) ∀s ∈ S , a ∈ A , ( 3 ) for Ω′ ( s , · ; π ) : = ∇Ω ( πs ) = [ ∇pΩ ( p ) ] p=π ( ·|s ) . By using the operator in discriminative networks , Jeon et al . ( 2020 ) proposed regularized adversarial IRL ( RAIRL ) as a generalization of AIL in terms of minimizing a specific Bregman divergence . Mirror Descent . Let w be a parameter on a set W and Ft : W → R be a convex cost function from a class of functions F . Replacing the L2 proximity term of proximal gradient descent with the Bregman divergence ( Gutman & Peña , 2018 ) , the proximal form of the MD update is given as wMDt+1 = argmin w∈W 〈 ∇Ft ( wMDt ) , w − wMDt 〉 W + αtDΩ ( w ∥∥wMDt ) , ( 4 ) where αt : = 1/ηt denotes an inverse of the step size ηt . MD was developed as a generalization of SGD , where the local geometry is specified by a Bregman divergence ( Gunasekar et al. , 2020 ) . 3 ITERATIVE RL-IRL AS A PROXIMAL OPTIMIZATION METHOD . We interpret the RL-IRL framework as a variant of proximal optimization methods and consider two sequences { πt } ∞t=1 and { ψt } ∞t=1 that denote the learning policies and reward functions , respectively . Associated reward function . An updated point in MD can be uniquely projected to the desired space ( ∆SA in our case ) using a Bregman projection operator PΩ that locates iterative points to the feasible region , i.e . πt+1 = PΩ ( π̃t+1 ) : = argminπ∈∆SA [ DΩ ( π∥π̃t+1 ) ] s∈S for π̃t+1 ∈ Π . To avoid these computations , we preemptively constrain a reward operator to satisfy the projection invariance with respect to regularized RL process of the associated state-action value function q̂ : PΩ ( ∇Ω∗ ( q̂ ) ) = ∇Ω∗ ( q̂ ) . According to Lemma 1 of Jeon et al . ( 2020 ) , a regularized reward function ψπ can replace q̂ , since the induced Bregman divergence allows the learning in a greedy manner . FIX R3 The projection invariance of the operator ΨΩ can be shown by a bijective relation with ∇Ω∗ in the space of ∆SA by the following lemma . Lemma 1 ( Natural isomorphism ) . Let ψ ∈ ΨΩ ( ∆SA ) for ΨΩ ( Π ) = { ψ | ψ ( s , a ) = ψπ ( s , a ) , ∀s∈ S , a∈A , π∈Π } . Then , ∇Ω∗ ( ψ ) is unique with respect to ψ , and for every π = ∇Ω∗ ( ψ ) , π ∈ ∆SA . Policy Space ∆SA Π πt πt+1 Reward Space ψt ψt+1 ΨΩ ∇Ω∗ FIX R3 As a result , we consider an updated reward function ψt+1 ∈ ΨΩ ( ∆SA ) as a projected target of MD , FIX R3 which can be associated by parameterization of ∆SA . For instance , a bijective mapping of ψt+1 can NEW R3 represent a softmax policy for a discrete space , or a Gaussian policy for a continuous space , using its parameters . The subsequent RL process at t-th step becomes finding the next iteration πt+1 by NEW R3 FIX R3maximize π∈∆SA JΩ ( π , ψt+1 ) = Eπ [ ∑∞ i=0 γiDΩ ( πsi ∥∥πsit+1 ) ] , πt+1 = ∇Ω∗ ( ψt+1 ) . ( 5 ) The equation shows that an arbitrary regularized RL algorithm with the regularizer Ω forms a cumulative discounted sum of Bregman divergences , thus the mapping to πt+1 can be achieved . Online imitation learning . The necessity of online learning setup comes from observing that our RL-IRL processes do not retain ψE = ΨΩ ( πE ) , a representation of πE , during training . Instead , consider a random process { π̄E , t } ∞t=1 where the estimation π̄E , t resides in a closed , convex neighborhood of πE , estimated by a separate estimation algorithm such as maximum likelihood methods . Substituting ψE to ψπ̄E , t in JΩ ( π , ψE ) turns the RL objective into an online learning problem : FIX R3minimize π∈∆SA Ft ( π ) = −JΩ ( π , ψπ̄E , t ) = Eπ [ ∑∞ i=0 γiDΩ ( πsi ∥∥π̄siE , t ) ] . ( 6 ) For a deeper understanding of our setup , we consider an example in Figure 2 . Assume that policies of the learning agent and the expert follow multivariate Gaussian distributions initially at π1 = N ( [ 0 , 0 ] T , I ) and πE = N ( [ 5 , 3 ] T , ΣE ) for |ΣE| < 1 . The policy πt is trained by a cost function DΩ ( ·∥π̄E , t ) with the proximity constraint 1ηDΩ ( ·∥πt ) where the Gaussian policy π̄E , t is also fitted using stochastic samples from πE at each iteration , starting from π̄E,1 = π1 . We first observe that the step size constant η affects the training speed in the early phase . Notably , the performance of certain cases exceeds the baselines of DΩ ( π̄E , t∥πE ) by choosing the step size effectively low as η < 1 . This suggests that there is a clear advantage of the online imitation learning setup for unreliable π̄E , t . MD update rules . Plugging each divergence of the cumulative cost Ft to the template of Equation ( 4 ) , the optimization process for the subsequent ψt+1 of the MD formulation is derived as1 ψt+1 : = ΨΩ ( πt+1 ) , π s t+1 = argmin πs∈∆A 〈 ∇DΩ ( πst ∥π̄sE , t ) ︸ ︷︷ ︸ ∇Ω ( πst ) −∇Ω ( π̄sE , t ) , πs − πst 〉 A + αtDΩ ( π s∥πst ) = argmin πs∈∆A DΩ ( π s∥π̄sE , t ) −DΩ ( πs∥πst ) + αtDΩ ( πs∥πst ) = argmin πs∈∆A ηtDΩ ( π s∥π̄sE , t ) ︸ ︷︷ ︸ estimated expert + ( 1− ηt ) DΩ ( πs∥πst ) ︸ ︷︷ ︸ learning agent , ηt = 1/αt , ∀s ∈ S , ( 7 ) where the gradient of DΩ is taken with respect to its first argument πst . The objective of reward NEW R3 learning is analogous to finding an interpolation at each iteration where the point is controlled by the step size ηt . Figure 3 shows that the region of πt ( defined by a norm ) gradually decreases 1Bregman divergences are generally intractable to be computed , unless the policy is a specific parametric model ( e.g . exponential families ) . See the works of Nielsen & Nock ( 2011 ) ; Jeon et al . ( 2020 ) and Appendix B. when ηt > ηt+1 . Note that solving the optimization of Ft requires interaction between πt and the dynamics of the given environment ; thus , the RL process in Equation ( 5 ) plays the essential role of sequential learning by the value measures of the reward function . | The paper proposes an algorithm MD-AIRL, where each iteration involves policy and reward updates, and an expert policy estimation. The algorithm is motivated by MD, especially the reward update is modeled as a projected update of an MD update. The sufficiency and necessities of the choices of stepsizes to guarantee convergence are theoretically justified. A practical algorithm with numerical experiments are also provided. | SP:a353062940ca72746921b1a50646d9584b4b9e39 |
Unsupervised Semantic Segmentation by Distilling Feature Correspondences | 1 INTRODUCTION . Semantic segmentation is the process of classifying each individual pixel of an image into a known ontology . Though semantic segmentation models can detect and delineate objects at a much finer granularity than classification or object detection systems , these systems are hindered by the difficulties of creating labelled training data . In particular , segmenting an image can take over 100× more effort for a human annotator than classifying or drawing bounding boxes ( Zlateski et al. , 2018 ) . Furthermore , in complex domains such as medicine , biology , or astrophysics , ground-truth segmentation labels may be unknown , ill-defined , or require considerable domain-expertise to provide ( Yu et al. , 2018 ) . Recently , several works introduced semantic segmentation systems that could learn from weaker forms of labels such as classes , tags , bounding boxes , scribbles , or point annotations ( Ren et al. , 2020 ; Pan et al. , 2021 ; Liu et al. , 2020 ; Bilen et al. ) . However , comparatively few works take up the challenge of semantic segmentation without any form of human supervision or motion cues . Attempts such as Independent Information Clustering ( IIC ) ( Ji et al. , 2019 ) and PiCIE ( Cho et al. , 2021 ) aim to learn semantically meaningful features through transformation equivariance , while imposing a clustering step to improve the compactness of the learnt features . In contrast to these previous methods , we utilize pre-trained features from unsupervised feature learning frameworks and focus on distilling them into a compact and discrete structure while preserving their relationships across the image corpora . This is motivated by the observation that correlations between unsupervised features , such as ones learnt by DINO ( Caron et al. , 2021 ) , are already semantically consistent , both within the same image and across image collections . As a result , we introduce STEGO ( Self-supervised Transformer with Energy-based Graph Optimization ) , which is capable of jointly discovering and segmenting objects without human supervision . STEGO distills pretrained unsupervised visual features into semantic clusters using a novel contrastive loss . STEGO dramatically improves over prior art and is a considerable step towards closing the gap with supervised segmentation systems . We include a short video detailing the work at https : //tinyurl.com/wrkx4rz9 . Specifically , we make the following contributions : • Show that unsupervised deep network features have correlation patterns that are largely consistent with true semantic labels . • Introduce STEGO , a novel transformer-based architecture for unsupervised semantic segmentation . • Demonstrate that STEGO achieves state of the art performance on both the CocoStuff ( +14 mIoU ) and Cityscapes ( +9 mIoU ) segmentation challenges . • Justify STEGO ’ s design with an ablation study on the CocoStuff dataset . 2 RELATED WORK . Self-supervised Visual Feature Learning Learning meaningful visual features without human annotations is a longstanding goal of computer vision . Approaches to this problem often optimize a surrogate task , such as denoising ( Vincent et al. , 2008 ) , inpainting ( Pathak et al. , 2016 ) , jigsaw puzzles , colorization ( Zhang et al. , 2017 ) , rotation prediction ( Gidaris et al. , 2018 ) , and most recently , contrastive learning over multiple augmentations ( Hjelm et al. , 2018 ; Chen et al. , 2020a ; a ; c ; Oord et al. , 2018 ) . Contrastive learning approaches , whose performance surpass all other surrogate tasks , assume visual features are invariant under a certain set of image augmentation operations . These approaches maximize feature similarities between an image and its augmentations , while minimizing similarity between negative samples , which are usually randomly sampled images . Some notable examples of positive pairs include temporally adjacent images in videos ( Oord et al. , 2018 ) , image augmentations ( Chen et al. , 2020a ; c ) , and local crops of a single image ( Hjelm et al. , 2018 ) . Many works highlight the importance of large numbers of negative samples during training . To this end Wu et al . ( 2018 ) propose keeping a memory bank of negative samples and Chen et al . ( 2020c ) propose momentum updates that can efficiently simulate large negative batch sizes . Recently some works have aimed to produce spatially dense feature maps as opposed to a single global vector per image . In this vein , VADeR ( Pinheiro et al. , 2020 ) contrasts local per-pixel features based on random compositions of image transformations that induce known correspondences among pixels which act as positive pairs for contrastive training . Instead of trying to learn visual features and clustering from scratch , STEGO treats pretrained self-supervised features as input and is agnostic to the underlying feature extractor . This makes it easy to integrate future advances in self-supervised feature learning into STEGO . Unsupervised Semantic Segmentation Many unsupervised semantic segmentation approaches use techniques from self-supervised feature learning . IIC ( Ji et al. , 2019 ) maximizes mutual information of patch-level cluster assignments between an image and its augmentations . Contrastive Clustering ( Li et al. , 2020 ) , and SCAN ( Van Gansbeke et al. , 2020 ) improve on IIC ’ s image clustering results with supervision from negative samples and nearest neighbors but do not attempt semantic segmentation . PiCIE ( Cho et al. , 2021 ) improves on IIC ’ s semantic segmentation results by using invariance to photometric effects and equivariance to geometric transformations as an inductive bias . In PiCIE , a network minimizes the distance between features under different transformations , where the distance is defined by an in-the-loop k-means clustering process . SegSort ( Hwang et al. , 2019 ) adopts a different approach . First , SegSort learns good features using superpixels as proxy segmentation maps , then uses Expectation-Maximization to iteratively refine segments over a spherical embedding space . In a similar vein , MaskContrast ( Van Gansbeke et al. , 2021 ) achieves promising results on PascalVOC by first using an off-the-shelf saliency model to generate a binary mask for each image . MaskContrast then contrasts learned features within and across the saliency masks . In contrast , our method focuses refining existing pretrained self-supervised visual features to distill their correspondence information and encourage cluster formation . This is similar to the work of Collins et al . ( 2018 ) who show that low rank factorization of deep network features can be useful for unsupervised co-segmentation . We are not aware of any previous work that achieves the goal of high-quality , pixel-level unsupervised semantic segmentation on large scale datasets with diverse images . Visual Transformers Convolutional neural networks ( CNNs ) have long been state of the art for many computer vision tasks , but the nature of the convolution operator makes it hard to model longrange interactions . To circumvent such shortcomings , Wang et al . ( 2018 ) ; Zhang et al . ( 2019 ) use self-attention operations within a CNN to model long range interactions . Transformers ( Vaswani et al. , 2017 ) , or purely self-attentive networks , have made significant progress in NLP and have recently been used for many computer vision tasks ( Dosovitskiy et al. , 2020 ; Touvron et al. , 2021 ; Ranftl et al. , 2021 ; Caron et al. , 2021 ) . Visual Transformers ( ViT ) ( Vaswani et al. , 2017 ) apply self-attention mechanisms to image patches and positional embeddings in order to generate features and predictions . Several modifications of ViT have been proposed to improve supervised learning , unsupervised learning , multi-scale processing , and dense predictions . In particular , DINO ( Caron et al. , 2021 ) uses a ViT within a self-supervised learning framework that performs self-distillation with exponential moving average updates . Caron et al . ( 2021 ) show that DINO ’ s class-attention can produce localized and semantically meaningful salient object segmentations . Our work shows that DINO ’ s features not only detect salient objects but can be used to extract dense and semantically meaningful correspondences between images . In STEGO , we refine the features of this pre-trained backbone to yield semantic segmentation predictions when clustered . We focus on DINO ’ s embeddings because of their quality but note that STEGO can work with any deep network features . 3 METHODS . 3.1 FEATURE CORRESPONDENCES PREDICT CLASS CO-OCCURRENCE . Recent progress in self-supervised visual feature learning has yielded methods with powerful and semantically relevant features that improve a variety of downstream tasks . Though most works aim to generate a single vector for an image , many works show that intermediate dense features are semantically relevant ( Hamilton et al. , 2021 ; Collins et al. , 2018 ; Zhou et al. , 2016 ) . To use this information , we focus on the “ correlation volume ” ( Teed & Deng , 2020 ) between the dense feature maps . For convolutional or transformer architectures , these dense feature maps can be the activation map of a specific layer . Additionally , the Q , K or V matrices in transformers can also serve as candidate features , though we find these attention tensors do not perform as well in practice . More formally , let f ∈ RCHW , g ∈ RCIJ be the feature tensors for two different images where C represents the channel dimension and ( H , W ) , ( I , J ) represent spatial dimensions . We form the feature correspondence tensor : Fhwij : = ∑ c fchw |fhw| gcij |gij | , ( 1 ) whose entries represent the cosine similarity between the feature at spatial position ( h , w ) of feature tensor f and position ( i , j ) of feature tensor g. In the special case where f = g these correspondences measure the similarity between two regions of the same image . We note that this quantity appears often as the “ cost-volume ” within the optical flow literature , and Hamilton et al . ( 2021 ) show this acts a higher-order generalization of Class Activation Maps ( Zhou et al. , 2016 ) for contrastive architectures and visual search engines . By examining slices of the correspondence tensor , F , at a given ( h , w ) we are able to visualize how two images relate according the featurizer . For example , Figure 2 shows how three different points from the source image ( shown in blue , red , and green ) are in correspondence with relevant semantic areas within the image and its K-nearest neighbors with respect to the DINO ( Caron et al. , 2021 ) as the feature extractor . This feature correspondence tensor not only allows us to visualize image correspondences but is strongly correlated with the true label co-occurrence tensor . In particular , we can form the ground truth label co-occurrence tensor given a pair of ground-truth semantic segmentation labels k ∈ CHW , l ∈ CIJ where C represents the set of possible classes : Lhwij : = { 1 , if lhw = kij 0 , if lhw 6= kij By examining how well the feature correspondences , F , predict the ground-truth label cooccurrences , L , we can measure how compatible the features are with the semantic segmentation labels . More specifically we treat the feature correspondences as a probability logit and compute the average precision when used as a classifier for L. This approach not only acts as a quick diagnostic tool to determine the efficacy of features , but also allows us to compare with other forms of supervision such as the fully connected Conditional Random Field ( CRF ) ( Krähenbühl & Koltun , 2011 ) , which uses correspondences between pixels to refine low-resolution label predictions . In Figure 3 we plot precision-recall curves for the DINO backbone , the MoCoV2 backbone , the CRF Kernel , and our trained STEGO architecture . Interestingly , we find that DINO is already a spectacular predictor of label co-occurrence within the Coco stuff dataset despite never seeing the labels . In particular , DINO recalls 50 % of true label co-occurrences with a precision of 90 % and significantly outperforms both MoCoV2 feature correspondences and the CRF kernel . One curious note is that our final trained model is a better label predictor than the supervisory signal it learns from . We attribute this to the distillation process discussed in Section 3.2 which amplifies this supervisory signal and drives consistency across the entire dataset . Finally , we stress that our comparison to ground truth labels within this section is solely to provide intuition about the quality of feature correspondences as a supervisory signal . We do not use the ground truth labels to tune any parameters of STEGO . | The paper introduces STEGO (Self-supervised Transformer with Energy-based Graph Optimization), a novel feature correlation refinement method that builds on top of modern self-supervised visual backbones (visual-transformers) that generate dense semantically-correlated features in an effort to improve scene semantic segmentation without any type of labels (unsupervised). Different from previous works, the authors decouple the feature learning from cluster compactification and introduce a novel contrastive loss function (a combination of three correlation factors: KNN, self and random images) in order to further constrain the features to form compact semantic clusters without damaging their consistency throughout the dataset. The authors demonstrate state-of-the-art results on two popular benchmarks CocoStuff and Cityscapes. The results are quite striking in both a quantitative (method improves segmentation by a large margin (+14mIoU and +9mIoU) compared to recently published work) and also a qualitative manner (clearly distinguish the structure of the objects in the scene). | SP:e89dfe9b1ca26fb502b32ddf1d4564f6227b63ac |
Unsupervised Semantic Segmentation by Distilling Feature Correspondences | 1 INTRODUCTION . Semantic segmentation is the process of classifying each individual pixel of an image into a known ontology . Though semantic segmentation models can detect and delineate objects at a much finer granularity than classification or object detection systems , these systems are hindered by the difficulties of creating labelled training data . In particular , segmenting an image can take over 100× more effort for a human annotator than classifying or drawing bounding boxes ( Zlateski et al. , 2018 ) . Furthermore , in complex domains such as medicine , biology , or astrophysics , ground-truth segmentation labels may be unknown , ill-defined , or require considerable domain-expertise to provide ( Yu et al. , 2018 ) . Recently , several works introduced semantic segmentation systems that could learn from weaker forms of labels such as classes , tags , bounding boxes , scribbles , or point annotations ( Ren et al. , 2020 ; Pan et al. , 2021 ; Liu et al. , 2020 ; Bilen et al. ) . However , comparatively few works take up the challenge of semantic segmentation without any form of human supervision or motion cues . Attempts such as Independent Information Clustering ( IIC ) ( Ji et al. , 2019 ) and PiCIE ( Cho et al. , 2021 ) aim to learn semantically meaningful features through transformation equivariance , while imposing a clustering step to improve the compactness of the learnt features . In contrast to these previous methods , we utilize pre-trained features from unsupervised feature learning frameworks and focus on distilling them into a compact and discrete structure while preserving their relationships across the image corpora . This is motivated by the observation that correlations between unsupervised features , such as ones learnt by DINO ( Caron et al. , 2021 ) , are already semantically consistent , both within the same image and across image collections . As a result , we introduce STEGO ( Self-supervised Transformer with Energy-based Graph Optimization ) , which is capable of jointly discovering and segmenting objects without human supervision . STEGO distills pretrained unsupervised visual features into semantic clusters using a novel contrastive loss . STEGO dramatically improves over prior art and is a considerable step towards closing the gap with supervised segmentation systems . We include a short video detailing the work at https : //tinyurl.com/wrkx4rz9 . Specifically , we make the following contributions : • Show that unsupervised deep network features have correlation patterns that are largely consistent with true semantic labels . • Introduce STEGO , a novel transformer-based architecture for unsupervised semantic segmentation . • Demonstrate that STEGO achieves state of the art performance on both the CocoStuff ( +14 mIoU ) and Cityscapes ( +9 mIoU ) segmentation challenges . • Justify STEGO ’ s design with an ablation study on the CocoStuff dataset . 2 RELATED WORK . Self-supervised Visual Feature Learning Learning meaningful visual features without human annotations is a longstanding goal of computer vision . Approaches to this problem often optimize a surrogate task , such as denoising ( Vincent et al. , 2008 ) , inpainting ( Pathak et al. , 2016 ) , jigsaw puzzles , colorization ( Zhang et al. , 2017 ) , rotation prediction ( Gidaris et al. , 2018 ) , and most recently , contrastive learning over multiple augmentations ( Hjelm et al. , 2018 ; Chen et al. , 2020a ; a ; c ; Oord et al. , 2018 ) . Contrastive learning approaches , whose performance surpass all other surrogate tasks , assume visual features are invariant under a certain set of image augmentation operations . These approaches maximize feature similarities between an image and its augmentations , while minimizing similarity between negative samples , which are usually randomly sampled images . Some notable examples of positive pairs include temporally adjacent images in videos ( Oord et al. , 2018 ) , image augmentations ( Chen et al. , 2020a ; c ) , and local crops of a single image ( Hjelm et al. , 2018 ) . Many works highlight the importance of large numbers of negative samples during training . To this end Wu et al . ( 2018 ) propose keeping a memory bank of negative samples and Chen et al . ( 2020c ) propose momentum updates that can efficiently simulate large negative batch sizes . Recently some works have aimed to produce spatially dense feature maps as opposed to a single global vector per image . In this vein , VADeR ( Pinheiro et al. , 2020 ) contrasts local per-pixel features based on random compositions of image transformations that induce known correspondences among pixels which act as positive pairs for contrastive training . Instead of trying to learn visual features and clustering from scratch , STEGO treats pretrained self-supervised features as input and is agnostic to the underlying feature extractor . This makes it easy to integrate future advances in self-supervised feature learning into STEGO . Unsupervised Semantic Segmentation Many unsupervised semantic segmentation approaches use techniques from self-supervised feature learning . IIC ( Ji et al. , 2019 ) maximizes mutual information of patch-level cluster assignments between an image and its augmentations . Contrastive Clustering ( Li et al. , 2020 ) , and SCAN ( Van Gansbeke et al. , 2020 ) improve on IIC ’ s image clustering results with supervision from negative samples and nearest neighbors but do not attempt semantic segmentation . PiCIE ( Cho et al. , 2021 ) improves on IIC ’ s semantic segmentation results by using invariance to photometric effects and equivariance to geometric transformations as an inductive bias . In PiCIE , a network minimizes the distance between features under different transformations , where the distance is defined by an in-the-loop k-means clustering process . SegSort ( Hwang et al. , 2019 ) adopts a different approach . First , SegSort learns good features using superpixels as proxy segmentation maps , then uses Expectation-Maximization to iteratively refine segments over a spherical embedding space . In a similar vein , MaskContrast ( Van Gansbeke et al. , 2021 ) achieves promising results on PascalVOC by first using an off-the-shelf saliency model to generate a binary mask for each image . MaskContrast then contrasts learned features within and across the saliency masks . In contrast , our method focuses refining existing pretrained self-supervised visual features to distill their correspondence information and encourage cluster formation . This is similar to the work of Collins et al . ( 2018 ) who show that low rank factorization of deep network features can be useful for unsupervised co-segmentation . We are not aware of any previous work that achieves the goal of high-quality , pixel-level unsupervised semantic segmentation on large scale datasets with diverse images . Visual Transformers Convolutional neural networks ( CNNs ) have long been state of the art for many computer vision tasks , but the nature of the convolution operator makes it hard to model longrange interactions . To circumvent such shortcomings , Wang et al . ( 2018 ) ; Zhang et al . ( 2019 ) use self-attention operations within a CNN to model long range interactions . Transformers ( Vaswani et al. , 2017 ) , or purely self-attentive networks , have made significant progress in NLP and have recently been used for many computer vision tasks ( Dosovitskiy et al. , 2020 ; Touvron et al. , 2021 ; Ranftl et al. , 2021 ; Caron et al. , 2021 ) . Visual Transformers ( ViT ) ( Vaswani et al. , 2017 ) apply self-attention mechanisms to image patches and positional embeddings in order to generate features and predictions . Several modifications of ViT have been proposed to improve supervised learning , unsupervised learning , multi-scale processing , and dense predictions . In particular , DINO ( Caron et al. , 2021 ) uses a ViT within a self-supervised learning framework that performs self-distillation with exponential moving average updates . Caron et al . ( 2021 ) show that DINO ’ s class-attention can produce localized and semantically meaningful salient object segmentations . Our work shows that DINO ’ s features not only detect salient objects but can be used to extract dense and semantically meaningful correspondences between images . In STEGO , we refine the features of this pre-trained backbone to yield semantic segmentation predictions when clustered . We focus on DINO ’ s embeddings because of their quality but note that STEGO can work with any deep network features . 3 METHODS . 3.1 FEATURE CORRESPONDENCES PREDICT CLASS CO-OCCURRENCE . Recent progress in self-supervised visual feature learning has yielded methods with powerful and semantically relevant features that improve a variety of downstream tasks . Though most works aim to generate a single vector for an image , many works show that intermediate dense features are semantically relevant ( Hamilton et al. , 2021 ; Collins et al. , 2018 ; Zhou et al. , 2016 ) . To use this information , we focus on the “ correlation volume ” ( Teed & Deng , 2020 ) between the dense feature maps . For convolutional or transformer architectures , these dense feature maps can be the activation map of a specific layer . Additionally , the Q , K or V matrices in transformers can also serve as candidate features , though we find these attention tensors do not perform as well in practice . More formally , let f ∈ RCHW , g ∈ RCIJ be the feature tensors for two different images where C represents the channel dimension and ( H , W ) , ( I , J ) represent spatial dimensions . We form the feature correspondence tensor : Fhwij : = ∑ c fchw |fhw| gcij |gij | , ( 1 ) whose entries represent the cosine similarity between the feature at spatial position ( h , w ) of feature tensor f and position ( i , j ) of feature tensor g. In the special case where f = g these correspondences measure the similarity between two regions of the same image . We note that this quantity appears often as the “ cost-volume ” within the optical flow literature , and Hamilton et al . ( 2021 ) show this acts a higher-order generalization of Class Activation Maps ( Zhou et al. , 2016 ) for contrastive architectures and visual search engines . By examining slices of the correspondence tensor , F , at a given ( h , w ) we are able to visualize how two images relate according the featurizer . For example , Figure 2 shows how three different points from the source image ( shown in blue , red , and green ) are in correspondence with relevant semantic areas within the image and its K-nearest neighbors with respect to the DINO ( Caron et al. , 2021 ) as the feature extractor . This feature correspondence tensor not only allows us to visualize image correspondences but is strongly correlated with the true label co-occurrence tensor . In particular , we can form the ground truth label co-occurrence tensor given a pair of ground-truth semantic segmentation labels k ∈ CHW , l ∈ CIJ where C represents the set of possible classes : Lhwij : = { 1 , if lhw = kij 0 , if lhw 6= kij By examining how well the feature correspondences , F , predict the ground-truth label cooccurrences , L , we can measure how compatible the features are with the semantic segmentation labels . More specifically we treat the feature correspondences as a probability logit and compute the average precision when used as a classifier for L. This approach not only acts as a quick diagnostic tool to determine the efficacy of features , but also allows us to compare with other forms of supervision such as the fully connected Conditional Random Field ( CRF ) ( Krähenbühl & Koltun , 2011 ) , which uses correspondences between pixels to refine low-resolution label predictions . In Figure 3 we plot precision-recall curves for the DINO backbone , the MoCoV2 backbone , the CRF Kernel , and our trained STEGO architecture . Interestingly , we find that DINO is already a spectacular predictor of label co-occurrence within the Coco stuff dataset despite never seeing the labels . In particular , DINO recalls 50 % of true label co-occurrences with a precision of 90 % and significantly outperforms both MoCoV2 feature correspondences and the CRF kernel . One curious note is that our final trained model is a better label predictor than the supervisory signal it learns from . We attribute this to the distillation process discussed in Section 3.2 which amplifies this supervisory signal and drives consistency across the entire dataset . Finally , we stress that our comparison to ground truth labels within this section is solely to provide intuition about the quality of feature correspondences as a supervisory signal . We do not use the ground truth labels to tune any parameters of STEGO . | The paper proposes a novel transformer-based unsupervised feature distillation method for semantic segmetation- STEGO (Self-supervised Transformer with Energy-based Graph Optimization). The pipeline has two separate stages, feature learning and cluster compactification (feature distillation). The feature network is DINO (Caron et al., 2021) - pretrained, frozen. The main contribution is the cluster compactification(distillation) network. It features a novel 3-term loss function: Knn, self correltation and random image corelation. The authors also propose a number of strategies for performance improvement: clamping the segmentation feature correspondence tensor at 0, five-cropping the dataset (corners+center), spatial centering on the feature correspondences and conditional random fields for better segmentation at the object edges. Training the segmentation head is fast (a couple of hours on a recent GPU with the ViT-small transformer backbone) and the results are state-of-the art on semantic segmentation on CocoStuff and Cityscapes (+14mIou@CocoStuff, +7mIoU@Citiscapes). | SP:e89dfe9b1ca26fb502b32ddf1d4564f6227b63ac |
Unsupervised Semantic Segmentation by Distilling Feature Correspondences | 1 INTRODUCTION . Semantic segmentation is the process of classifying each individual pixel of an image into a known ontology . Though semantic segmentation models can detect and delineate objects at a much finer granularity than classification or object detection systems , these systems are hindered by the difficulties of creating labelled training data . In particular , segmenting an image can take over 100× more effort for a human annotator than classifying or drawing bounding boxes ( Zlateski et al. , 2018 ) . Furthermore , in complex domains such as medicine , biology , or astrophysics , ground-truth segmentation labels may be unknown , ill-defined , or require considerable domain-expertise to provide ( Yu et al. , 2018 ) . Recently , several works introduced semantic segmentation systems that could learn from weaker forms of labels such as classes , tags , bounding boxes , scribbles , or point annotations ( Ren et al. , 2020 ; Pan et al. , 2021 ; Liu et al. , 2020 ; Bilen et al. ) . However , comparatively few works take up the challenge of semantic segmentation without any form of human supervision or motion cues . Attempts such as Independent Information Clustering ( IIC ) ( Ji et al. , 2019 ) and PiCIE ( Cho et al. , 2021 ) aim to learn semantically meaningful features through transformation equivariance , while imposing a clustering step to improve the compactness of the learnt features . In contrast to these previous methods , we utilize pre-trained features from unsupervised feature learning frameworks and focus on distilling them into a compact and discrete structure while preserving their relationships across the image corpora . This is motivated by the observation that correlations between unsupervised features , such as ones learnt by DINO ( Caron et al. , 2021 ) , are already semantically consistent , both within the same image and across image collections . As a result , we introduce STEGO ( Self-supervised Transformer with Energy-based Graph Optimization ) , which is capable of jointly discovering and segmenting objects without human supervision . STEGO distills pretrained unsupervised visual features into semantic clusters using a novel contrastive loss . STEGO dramatically improves over prior art and is a considerable step towards closing the gap with supervised segmentation systems . We include a short video detailing the work at https : //tinyurl.com/wrkx4rz9 . Specifically , we make the following contributions : • Show that unsupervised deep network features have correlation patterns that are largely consistent with true semantic labels . • Introduce STEGO , a novel transformer-based architecture for unsupervised semantic segmentation . • Demonstrate that STEGO achieves state of the art performance on both the CocoStuff ( +14 mIoU ) and Cityscapes ( +9 mIoU ) segmentation challenges . • Justify STEGO ’ s design with an ablation study on the CocoStuff dataset . 2 RELATED WORK . Self-supervised Visual Feature Learning Learning meaningful visual features without human annotations is a longstanding goal of computer vision . Approaches to this problem often optimize a surrogate task , such as denoising ( Vincent et al. , 2008 ) , inpainting ( Pathak et al. , 2016 ) , jigsaw puzzles , colorization ( Zhang et al. , 2017 ) , rotation prediction ( Gidaris et al. , 2018 ) , and most recently , contrastive learning over multiple augmentations ( Hjelm et al. , 2018 ; Chen et al. , 2020a ; a ; c ; Oord et al. , 2018 ) . Contrastive learning approaches , whose performance surpass all other surrogate tasks , assume visual features are invariant under a certain set of image augmentation operations . These approaches maximize feature similarities between an image and its augmentations , while minimizing similarity between negative samples , which are usually randomly sampled images . Some notable examples of positive pairs include temporally adjacent images in videos ( Oord et al. , 2018 ) , image augmentations ( Chen et al. , 2020a ; c ) , and local crops of a single image ( Hjelm et al. , 2018 ) . Many works highlight the importance of large numbers of negative samples during training . To this end Wu et al . ( 2018 ) propose keeping a memory bank of negative samples and Chen et al . ( 2020c ) propose momentum updates that can efficiently simulate large negative batch sizes . Recently some works have aimed to produce spatially dense feature maps as opposed to a single global vector per image . In this vein , VADeR ( Pinheiro et al. , 2020 ) contrasts local per-pixel features based on random compositions of image transformations that induce known correspondences among pixels which act as positive pairs for contrastive training . Instead of trying to learn visual features and clustering from scratch , STEGO treats pretrained self-supervised features as input and is agnostic to the underlying feature extractor . This makes it easy to integrate future advances in self-supervised feature learning into STEGO . Unsupervised Semantic Segmentation Many unsupervised semantic segmentation approaches use techniques from self-supervised feature learning . IIC ( Ji et al. , 2019 ) maximizes mutual information of patch-level cluster assignments between an image and its augmentations . Contrastive Clustering ( Li et al. , 2020 ) , and SCAN ( Van Gansbeke et al. , 2020 ) improve on IIC ’ s image clustering results with supervision from negative samples and nearest neighbors but do not attempt semantic segmentation . PiCIE ( Cho et al. , 2021 ) improves on IIC ’ s semantic segmentation results by using invariance to photometric effects and equivariance to geometric transformations as an inductive bias . In PiCIE , a network minimizes the distance between features under different transformations , where the distance is defined by an in-the-loop k-means clustering process . SegSort ( Hwang et al. , 2019 ) adopts a different approach . First , SegSort learns good features using superpixels as proxy segmentation maps , then uses Expectation-Maximization to iteratively refine segments over a spherical embedding space . In a similar vein , MaskContrast ( Van Gansbeke et al. , 2021 ) achieves promising results on PascalVOC by first using an off-the-shelf saliency model to generate a binary mask for each image . MaskContrast then contrasts learned features within and across the saliency masks . In contrast , our method focuses refining existing pretrained self-supervised visual features to distill their correspondence information and encourage cluster formation . This is similar to the work of Collins et al . ( 2018 ) who show that low rank factorization of deep network features can be useful for unsupervised co-segmentation . We are not aware of any previous work that achieves the goal of high-quality , pixel-level unsupervised semantic segmentation on large scale datasets with diverse images . Visual Transformers Convolutional neural networks ( CNNs ) have long been state of the art for many computer vision tasks , but the nature of the convolution operator makes it hard to model longrange interactions . To circumvent such shortcomings , Wang et al . ( 2018 ) ; Zhang et al . ( 2019 ) use self-attention operations within a CNN to model long range interactions . Transformers ( Vaswani et al. , 2017 ) , or purely self-attentive networks , have made significant progress in NLP and have recently been used for many computer vision tasks ( Dosovitskiy et al. , 2020 ; Touvron et al. , 2021 ; Ranftl et al. , 2021 ; Caron et al. , 2021 ) . Visual Transformers ( ViT ) ( Vaswani et al. , 2017 ) apply self-attention mechanisms to image patches and positional embeddings in order to generate features and predictions . Several modifications of ViT have been proposed to improve supervised learning , unsupervised learning , multi-scale processing , and dense predictions . In particular , DINO ( Caron et al. , 2021 ) uses a ViT within a self-supervised learning framework that performs self-distillation with exponential moving average updates . Caron et al . ( 2021 ) show that DINO ’ s class-attention can produce localized and semantically meaningful salient object segmentations . Our work shows that DINO ’ s features not only detect salient objects but can be used to extract dense and semantically meaningful correspondences between images . In STEGO , we refine the features of this pre-trained backbone to yield semantic segmentation predictions when clustered . We focus on DINO ’ s embeddings because of their quality but note that STEGO can work with any deep network features . 3 METHODS . 3.1 FEATURE CORRESPONDENCES PREDICT CLASS CO-OCCURRENCE . Recent progress in self-supervised visual feature learning has yielded methods with powerful and semantically relevant features that improve a variety of downstream tasks . Though most works aim to generate a single vector for an image , many works show that intermediate dense features are semantically relevant ( Hamilton et al. , 2021 ; Collins et al. , 2018 ; Zhou et al. , 2016 ) . To use this information , we focus on the “ correlation volume ” ( Teed & Deng , 2020 ) between the dense feature maps . For convolutional or transformer architectures , these dense feature maps can be the activation map of a specific layer . Additionally , the Q , K or V matrices in transformers can also serve as candidate features , though we find these attention tensors do not perform as well in practice . More formally , let f ∈ RCHW , g ∈ RCIJ be the feature tensors for two different images where C represents the channel dimension and ( H , W ) , ( I , J ) represent spatial dimensions . We form the feature correspondence tensor : Fhwij : = ∑ c fchw |fhw| gcij |gij | , ( 1 ) whose entries represent the cosine similarity between the feature at spatial position ( h , w ) of feature tensor f and position ( i , j ) of feature tensor g. In the special case where f = g these correspondences measure the similarity between two regions of the same image . We note that this quantity appears often as the “ cost-volume ” within the optical flow literature , and Hamilton et al . ( 2021 ) show this acts a higher-order generalization of Class Activation Maps ( Zhou et al. , 2016 ) for contrastive architectures and visual search engines . By examining slices of the correspondence tensor , F , at a given ( h , w ) we are able to visualize how two images relate according the featurizer . For example , Figure 2 shows how three different points from the source image ( shown in blue , red , and green ) are in correspondence with relevant semantic areas within the image and its K-nearest neighbors with respect to the DINO ( Caron et al. , 2021 ) as the feature extractor . This feature correspondence tensor not only allows us to visualize image correspondences but is strongly correlated with the true label co-occurrence tensor . In particular , we can form the ground truth label co-occurrence tensor given a pair of ground-truth semantic segmentation labels k ∈ CHW , l ∈ CIJ where C represents the set of possible classes : Lhwij : = { 1 , if lhw = kij 0 , if lhw 6= kij By examining how well the feature correspondences , F , predict the ground-truth label cooccurrences , L , we can measure how compatible the features are with the semantic segmentation labels . More specifically we treat the feature correspondences as a probability logit and compute the average precision when used as a classifier for L. This approach not only acts as a quick diagnostic tool to determine the efficacy of features , but also allows us to compare with other forms of supervision such as the fully connected Conditional Random Field ( CRF ) ( Krähenbühl & Koltun , 2011 ) , which uses correspondences between pixels to refine low-resolution label predictions . In Figure 3 we plot precision-recall curves for the DINO backbone , the MoCoV2 backbone , the CRF Kernel , and our trained STEGO architecture . Interestingly , we find that DINO is already a spectacular predictor of label co-occurrence within the Coco stuff dataset despite never seeing the labels . In particular , DINO recalls 50 % of true label co-occurrences with a precision of 90 % and significantly outperforms both MoCoV2 feature correspondences and the CRF kernel . One curious note is that our final trained model is a better label predictor than the supervisory signal it learns from . We attribute this to the distillation process discussed in Section 3.2 which amplifies this supervisory signal and drives consistency across the entire dataset . Finally , we stress that our comparison to ground truth labels within this section is solely to provide intuition about the quality of feature correspondences as a supervisory signal . We do not use the ground truth labels to tune any parameters of STEGO . | This paper proposed STEGO, an unsupervised approach for semantic clustering/segmentation using feature refinement on top of self-supervised neural networks. A distilled version of self-supervised features is learned segmentation specifically through a feed-forward network via SGD. The main contribution comes from an idea of distilling powerful deep features from strong self-supervised backbones to further improve their semantic discriminativeness. Though the idea is intuitive, several loss functions and regularisations are proposed to avoid trivial solutions and make the idea really work in practice. Extensive qualitative and quantitatve results are done to demonstrate the performance of STEGO and its design choices. | SP:e89dfe9b1ca26fb502b32ddf1d4564f6227b63ac |
Single-Cell Capsule Attention : an interpretable method of cell type classification for single-cell RNA-sequencing data | 1 INTRODUCTION . Genes ’ expression on celluar level provides lots of information for us to explore homogeneity and heterogeneity among different cells ( Liang et al. , 2014 ; Muraro et al. , 2016 ; Baron et al. , 2016 ) . Traditional bulk sequencing technique can only measure the average gene expression level of all cells in a sample . Compared with bulk data , single-cell RNA-sequencing ( single-cell RNA-seq ) technique gives a more accurate measurement of every cell . Though single-cell RNA-sequencing is a rising technique , it is not mature enough and still has some limitations . For example , single-cell RNA-seq data has high ’ dropout ’ . ’ Dropout ’ means zero or low read counts in the data because many genes ’ expression are hard to detect ( Huang et al. , 2018 ; Pierson & Yau , 2015 ) . Also , singlecell RNA-seq data are of high dimension because thousands of genes are in the transcriptome . These all bring challenges for studies on single-cell RNA-seq data . So far , many data mining and analysis methods have been applied on single-cell RNA-seq data to solve these challenges and explore more information on cells and genes ( Svensson et al. , 2018 ; Han et al. , 2018 ; Lopez et al. , 2018 ; Hwang et al. , 2018 ; Guo et al. , 2015 ) . Cell type classification is one of the most important tasks in single-cell RNA-seq data analysis . It helps to identify different cell types and explore the cellular heterogeneity . Previous methods on cell type classification can be categorized into methods based on statistical distribution hypotheses of gene expression and deep learning methods based on neural networks ( Abdelaal et al. , 2019 ) . These methods lay a solid foundation for cell type classification . However , they still have limitations and weaknesses . Methods based on statistical hypotheses rely on prior knowledge of marker genes or gene expression ’ s distribution assumptions . Up to now , there is no widely recognized hypotheses of gene expression . Deep neural network ( DNN ) , a fast and effective learning method , is widely used in various fields . DNN does not rely on distribution hypotheses , however , it ’ s hard to provide an reasonable explanation for the result behind the deep ’ black-box ’ ( Lin et al. , 2017 ) . ACTINN ( Ma & Pellegrini , 2020 ) , a deep neural network based method for cell type classification , uses three hidden layers in its architecture . Though its model can predict cell types , the model still lacks reliable interpretations for the result . The challenge is that DNN does not provide a recognizable pattern that is closely related to different cell types . Here , we introduce a new neural network based method scCA for cell type classification . scCA provides an reasonable interpretation for its classification result . Our model learns features from cells and provides patterns behind these features that are related to cell types . The first part is feature extraction . Given the gene expression level in every cell , We use capsules to extract different features . A capsule is a vector that represents a feature , which captures a certain type of information of the cell . Capsules also serve as dimension reduction for single cell RNA-seq data . We calculate the Pearson coefficients between every two capsules and generate the coefficient heatmap of every cell . We discover that cells from same cell type share a similar pattern in the heatmap while cells from different cell types have different patterns . These capsules capture features that are important for cell type classification . And then , we use a sequence to sequence model , bidirectional-LSTM , to generate compound features . Each compound feature combines a set of features ’ information . Furthermore , we train multiple attention weights for these compound features . Then we multiply these weights to the compound features and generate the classification result . Attention weights help to lay more emphasis on important compound features . The larger the attention weight is , the more crucial its corresponding compound feature is . we also draw heatmaps for every cells ’ attention weights . We discover that cells from same cell type have similar patterns in the heatmaps of attention weights . This leads to a better classification performance and also provides an reliable interpretation behind the network . We evaluate scCA on several datasets and compare with previous work . Our method shows high accuracy and robustness on all datasets . The main contributions of our article are as follow : We propose an interpretable neural network based method scCA for cell type classification . Through heatmaps of capsules ’ Pearson coefficients and attention weights , we find that cells from same cell type share a similar pattern . scCA achieves high accuracy and stable performances on all datasets . 2 RELATED WORK . Statistical Hypotheses Based Model Many machine learning methods with statistical hypotheses of gene expression level have been proposed for cell type classification . Some of these methods suggest distribution hypotheses of gene expression level and employ machine learning models based on these hypotheses . Some build a hierarchy with prior knowledge and statistical hypotheses . Moana takes advantage of KNN and SVM model ( Wagner & Yanai , 2018 ) . It first preprocesses data with KNN , then it uses dimension reduction method to extract useful features , at last it uses a Support Vector Machine classifier . In addition , Moana uses marker gene as a prior knowledge for cell type identification . Different from Moana , Garnett proposes a model based on tree hierarchy . It discovers the relation between cell types and subtypes with a tree structure ( Pliner et al. , 2019 ) . With the help of marker gene , it builds a linear model to classify cell types . CellAssign builds a probabilistic model based on prior knowledge to classify cell types ( Zhang et al. , 2019b ) . SCINA makes use of EM algorithm and the information of marker gene to accomplish clustering as well as cell type identification task ( Zhang et al. , 2019a ) . scPred extracts features using SVD , then with a nonlinear kernel SVM , it predicts different cell types ( Alquicira-Hernandez et al. , 2019 ) . However , these methods have their limitations . Their hypotheses such as ZINB ( Zero-inflated Negative Binomial ) or NB distribution of gene expression are not the exact reflections of gene ’ s expression levels in cells . Actually , there is not a consensus statistical distribution for single-cell RNA-seq data . Also they need prior knowledge like marker gene for cell type classification . Neural Network Based Models Deep neural network ( DNN ) based methods have been applied to cell type classification task . They are fast and robust . Neural networks don ’ t heavily rely on prior knowledge and statistical hypotheses . ACTINN uses three fully connected layers and ReLU , softmax activation function to classify different cell types ( Ma & Pellegrini , 2020 ) . scCapsNet builds a capsule network to extract features from the data and use dynamic routings among capsules to get a classification result ( Wang et al. , 2020 ) . However , neural networks lack interpretation for their result . Through training the whole network by propagation , it ’ s hard to explain every component ’ s contribution for the result . 3 PRELIMINARIES . Capsule Network and Dynamic Routing Hinton , Sabour and other co-authors first proposed Capsule network and dynamic routing in computer vision ( Sabour et al. , 2017 ) . A capsule is a set of neurons or a vector that can represent an entity . Dynamic routing is the mechanism between capsule ’ s layers . As they point out , the capsules in the layer above are parents and the capsules in the layer below are children . ( Hinton et al. , 2000 ; 2011 ) . Dynamic routing helps to send the results from capsules in children layer to its appropriate parents in the above layer . By adjusting the coupling coefficients between capsules in two layers , the network learns which parent capsule is more important for the children capsule . Here in scCA , we use capsules to capture different features of the cell . We use a vector for implementation of capsule in our model . However , different from dynamic routing , we use LSTM and attention as the architecture for cell type classification . Long Short-Term Memory and Attention Long Short-Term Memory ( LSTM ) is a model widely used in Natural Language Processing ( NLP ) ( Hochreiter & Schmidhuber , 1997 ) . It is a sequence to sequence model . The inputs are usually the word embeddings ( representations ) of a sentence and the output is a sequence of representations . Each output representation not only contains information from its corresponding input , but also information from representations in the context , especially its neighbor inputs . In our method , after extracting different features into capsules , we input these capsules into the LSTM model and obtain the output sequence . Every vector in the output sequence contains information of its corresponding input feature as well as other features . We name these vectors compound features . Some of the compound features have closer relationships with certain cell types . Inspired by Attention ( Vaswani et al. , 2017 ) mechanism , we apply attention weights to compound features in order to differentiate and emphasize those important compound features for classification . 4 SINGLE-CELL CAPSULE ATTENTION . In this section , we provide a detailed description of single-cell Capsule Attention ( scCA ) . We will also give a vivid and transparent explanation for our classification result . scCA can be divided into three parts . The figure below shows its architecture . 4.1 FEATURE EXTRACTION . The first part of scCA is feature extraction . Given thousands of genes ’ expression levels for a cell , we hope to learn simple but efficacious features for classification . Capsules not only serve as dimension reduction , but also extract a type of information of the cell . Through fully connected layers , we generate capsules to extract different features . capsulei = ReLU ( W ix+ bi ) , i ∈ [ 1 , 2 · · ·m ] ( 1 ) In this formula , m represents the number of capsules . W and b are the weight matrices of fully connected layers . ReLU is the activation function we use . Every capsule contains a feature of a cell . We hope to discover different patterns of these features for different cell types . We calculate the Pearson coefficients between every two capsules . The Pearson coefficient is an indication of relationship between two features behind capsules . For every cell , we come to a matrix where every element is a Pearson coefficient of two capsules . We then draw a heatmap of the matrix . We discover that cells from same cell type share a similar pattern while cells from different cell types differ from each other . Taking six cells from dataset GSE72056 as examples , we can see cells from same cell type have a similar pattern of capsules ’ relationship . The deeper the color is in the heatmap , the closer the relationship of two capsules is . The pattern is distinguishable among different cell types . From the figures above , we conclude that there exists some particular feature patterns for different cell types . Our feature extraction really extracts useful and discriminative features . | This manuscript describes scCA (single-cell capsule attention) for cell-type classification using single-cell RNA-seq data. scCA usesa combination of capsule networks, LSTN and attentions to undertake cell-type classifiation. Capsules extract features from single-cell data, the combination of capsules called "compound features" through LSTM and attention weights are employed. scCA is compared against a host of other techniques and performs well across 9 diverse single-cell datasets. scCA results are also robust to various parameter choices across datasets. | SP:6ba0ae7d3707640ee4018653af8e6dc179e15309 |
Single-Cell Capsule Attention : an interpretable method of cell type classification for single-cell RNA-sequencing data | 1 INTRODUCTION . Genes ’ expression on celluar level provides lots of information for us to explore homogeneity and heterogeneity among different cells ( Liang et al. , 2014 ; Muraro et al. , 2016 ; Baron et al. , 2016 ) . Traditional bulk sequencing technique can only measure the average gene expression level of all cells in a sample . Compared with bulk data , single-cell RNA-sequencing ( single-cell RNA-seq ) technique gives a more accurate measurement of every cell . Though single-cell RNA-sequencing is a rising technique , it is not mature enough and still has some limitations . For example , single-cell RNA-seq data has high ’ dropout ’ . ’ Dropout ’ means zero or low read counts in the data because many genes ’ expression are hard to detect ( Huang et al. , 2018 ; Pierson & Yau , 2015 ) . Also , singlecell RNA-seq data are of high dimension because thousands of genes are in the transcriptome . These all bring challenges for studies on single-cell RNA-seq data . So far , many data mining and analysis methods have been applied on single-cell RNA-seq data to solve these challenges and explore more information on cells and genes ( Svensson et al. , 2018 ; Han et al. , 2018 ; Lopez et al. , 2018 ; Hwang et al. , 2018 ; Guo et al. , 2015 ) . Cell type classification is one of the most important tasks in single-cell RNA-seq data analysis . It helps to identify different cell types and explore the cellular heterogeneity . Previous methods on cell type classification can be categorized into methods based on statistical distribution hypotheses of gene expression and deep learning methods based on neural networks ( Abdelaal et al. , 2019 ) . These methods lay a solid foundation for cell type classification . However , they still have limitations and weaknesses . Methods based on statistical hypotheses rely on prior knowledge of marker genes or gene expression ’ s distribution assumptions . Up to now , there is no widely recognized hypotheses of gene expression . Deep neural network ( DNN ) , a fast and effective learning method , is widely used in various fields . DNN does not rely on distribution hypotheses , however , it ’ s hard to provide an reasonable explanation for the result behind the deep ’ black-box ’ ( Lin et al. , 2017 ) . ACTINN ( Ma & Pellegrini , 2020 ) , a deep neural network based method for cell type classification , uses three hidden layers in its architecture . Though its model can predict cell types , the model still lacks reliable interpretations for the result . The challenge is that DNN does not provide a recognizable pattern that is closely related to different cell types . Here , we introduce a new neural network based method scCA for cell type classification . scCA provides an reasonable interpretation for its classification result . Our model learns features from cells and provides patterns behind these features that are related to cell types . The first part is feature extraction . Given the gene expression level in every cell , We use capsules to extract different features . A capsule is a vector that represents a feature , which captures a certain type of information of the cell . Capsules also serve as dimension reduction for single cell RNA-seq data . We calculate the Pearson coefficients between every two capsules and generate the coefficient heatmap of every cell . We discover that cells from same cell type share a similar pattern in the heatmap while cells from different cell types have different patterns . These capsules capture features that are important for cell type classification . And then , we use a sequence to sequence model , bidirectional-LSTM , to generate compound features . Each compound feature combines a set of features ’ information . Furthermore , we train multiple attention weights for these compound features . Then we multiply these weights to the compound features and generate the classification result . Attention weights help to lay more emphasis on important compound features . The larger the attention weight is , the more crucial its corresponding compound feature is . we also draw heatmaps for every cells ’ attention weights . We discover that cells from same cell type have similar patterns in the heatmaps of attention weights . This leads to a better classification performance and also provides an reliable interpretation behind the network . We evaluate scCA on several datasets and compare with previous work . Our method shows high accuracy and robustness on all datasets . The main contributions of our article are as follow : We propose an interpretable neural network based method scCA for cell type classification . Through heatmaps of capsules ’ Pearson coefficients and attention weights , we find that cells from same cell type share a similar pattern . scCA achieves high accuracy and stable performances on all datasets . 2 RELATED WORK . Statistical Hypotheses Based Model Many machine learning methods with statistical hypotheses of gene expression level have been proposed for cell type classification . Some of these methods suggest distribution hypotheses of gene expression level and employ machine learning models based on these hypotheses . Some build a hierarchy with prior knowledge and statistical hypotheses . Moana takes advantage of KNN and SVM model ( Wagner & Yanai , 2018 ) . It first preprocesses data with KNN , then it uses dimension reduction method to extract useful features , at last it uses a Support Vector Machine classifier . In addition , Moana uses marker gene as a prior knowledge for cell type identification . Different from Moana , Garnett proposes a model based on tree hierarchy . It discovers the relation between cell types and subtypes with a tree structure ( Pliner et al. , 2019 ) . With the help of marker gene , it builds a linear model to classify cell types . CellAssign builds a probabilistic model based on prior knowledge to classify cell types ( Zhang et al. , 2019b ) . SCINA makes use of EM algorithm and the information of marker gene to accomplish clustering as well as cell type identification task ( Zhang et al. , 2019a ) . scPred extracts features using SVD , then with a nonlinear kernel SVM , it predicts different cell types ( Alquicira-Hernandez et al. , 2019 ) . However , these methods have their limitations . Their hypotheses such as ZINB ( Zero-inflated Negative Binomial ) or NB distribution of gene expression are not the exact reflections of gene ’ s expression levels in cells . Actually , there is not a consensus statistical distribution for single-cell RNA-seq data . Also they need prior knowledge like marker gene for cell type classification . Neural Network Based Models Deep neural network ( DNN ) based methods have been applied to cell type classification task . They are fast and robust . Neural networks don ’ t heavily rely on prior knowledge and statistical hypotheses . ACTINN uses three fully connected layers and ReLU , softmax activation function to classify different cell types ( Ma & Pellegrini , 2020 ) . scCapsNet builds a capsule network to extract features from the data and use dynamic routings among capsules to get a classification result ( Wang et al. , 2020 ) . However , neural networks lack interpretation for their result . Through training the whole network by propagation , it ’ s hard to explain every component ’ s contribution for the result . 3 PRELIMINARIES . Capsule Network and Dynamic Routing Hinton , Sabour and other co-authors first proposed Capsule network and dynamic routing in computer vision ( Sabour et al. , 2017 ) . A capsule is a set of neurons or a vector that can represent an entity . Dynamic routing is the mechanism between capsule ’ s layers . As they point out , the capsules in the layer above are parents and the capsules in the layer below are children . ( Hinton et al. , 2000 ; 2011 ) . Dynamic routing helps to send the results from capsules in children layer to its appropriate parents in the above layer . By adjusting the coupling coefficients between capsules in two layers , the network learns which parent capsule is more important for the children capsule . Here in scCA , we use capsules to capture different features of the cell . We use a vector for implementation of capsule in our model . However , different from dynamic routing , we use LSTM and attention as the architecture for cell type classification . Long Short-Term Memory and Attention Long Short-Term Memory ( LSTM ) is a model widely used in Natural Language Processing ( NLP ) ( Hochreiter & Schmidhuber , 1997 ) . It is a sequence to sequence model . The inputs are usually the word embeddings ( representations ) of a sentence and the output is a sequence of representations . Each output representation not only contains information from its corresponding input , but also information from representations in the context , especially its neighbor inputs . In our method , after extracting different features into capsules , we input these capsules into the LSTM model and obtain the output sequence . Every vector in the output sequence contains information of its corresponding input feature as well as other features . We name these vectors compound features . Some of the compound features have closer relationships with certain cell types . Inspired by Attention ( Vaswani et al. , 2017 ) mechanism , we apply attention weights to compound features in order to differentiate and emphasize those important compound features for classification . 4 SINGLE-CELL CAPSULE ATTENTION . In this section , we provide a detailed description of single-cell Capsule Attention ( scCA ) . We will also give a vivid and transparent explanation for our classification result . scCA can be divided into three parts . The figure below shows its architecture . 4.1 FEATURE EXTRACTION . The first part of scCA is feature extraction . Given thousands of genes ’ expression levels for a cell , we hope to learn simple but efficacious features for classification . Capsules not only serve as dimension reduction , but also extract a type of information of the cell . Through fully connected layers , we generate capsules to extract different features . capsulei = ReLU ( W ix+ bi ) , i ∈ [ 1 , 2 · · ·m ] ( 1 ) In this formula , m represents the number of capsules . W and b are the weight matrices of fully connected layers . ReLU is the activation function we use . Every capsule contains a feature of a cell . We hope to discover different patterns of these features for different cell types . We calculate the Pearson coefficients between every two capsules . The Pearson coefficient is an indication of relationship between two features behind capsules . For every cell , we come to a matrix where every element is a Pearson coefficient of two capsules . We then draw a heatmap of the matrix . We discover that cells from same cell type share a similar pattern while cells from different cell types differ from each other . Taking six cells from dataset GSE72056 as examples , we can see cells from same cell type have a similar pattern of capsules ’ relationship . The deeper the color is in the heatmap , the closer the relationship of two capsules is . The pattern is distinguishable among different cell types . From the figures above , we conclude that there exists some particular feature patterns for different cell types . Our feature extraction really extracts useful and discriminative features . | The paper tackles the problem of classifying single-cell RNA-seq expression data according to their underlying cell types. To do so, they suggest a deep neural network, inspired from Capsule Networks, and making use of attention to provide interpretable results. The features extracted by the network are empirically shown to exhibit cell-type-specific patterns. In terms of performance, the suggested method compares favourably or is on par with classic, standard ML approaches. Finally, the authors show that the results seem to be relatively stable with respect to the choice of some architecture hyper-parameters. | SP:6ba0ae7d3707640ee4018653af8e6dc179e15309 |
Single-Cell Capsule Attention : an interpretable method of cell type classification for single-cell RNA-sequencing data | 1 INTRODUCTION . Genes ’ expression on celluar level provides lots of information for us to explore homogeneity and heterogeneity among different cells ( Liang et al. , 2014 ; Muraro et al. , 2016 ; Baron et al. , 2016 ) . Traditional bulk sequencing technique can only measure the average gene expression level of all cells in a sample . Compared with bulk data , single-cell RNA-sequencing ( single-cell RNA-seq ) technique gives a more accurate measurement of every cell . Though single-cell RNA-sequencing is a rising technique , it is not mature enough and still has some limitations . For example , single-cell RNA-seq data has high ’ dropout ’ . ’ Dropout ’ means zero or low read counts in the data because many genes ’ expression are hard to detect ( Huang et al. , 2018 ; Pierson & Yau , 2015 ) . Also , singlecell RNA-seq data are of high dimension because thousands of genes are in the transcriptome . These all bring challenges for studies on single-cell RNA-seq data . So far , many data mining and analysis methods have been applied on single-cell RNA-seq data to solve these challenges and explore more information on cells and genes ( Svensson et al. , 2018 ; Han et al. , 2018 ; Lopez et al. , 2018 ; Hwang et al. , 2018 ; Guo et al. , 2015 ) . Cell type classification is one of the most important tasks in single-cell RNA-seq data analysis . It helps to identify different cell types and explore the cellular heterogeneity . Previous methods on cell type classification can be categorized into methods based on statistical distribution hypotheses of gene expression and deep learning methods based on neural networks ( Abdelaal et al. , 2019 ) . These methods lay a solid foundation for cell type classification . However , they still have limitations and weaknesses . Methods based on statistical hypotheses rely on prior knowledge of marker genes or gene expression ’ s distribution assumptions . Up to now , there is no widely recognized hypotheses of gene expression . Deep neural network ( DNN ) , a fast and effective learning method , is widely used in various fields . DNN does not rely on distribution hypotheses , however , it ’ s hard to provide an reasonable explanation for the result behind the deep ’ black-box ’ ( Lin et al. , 2017 ) . ACTINN ( Ma & Pellegrini , 2020 ) , a deep neural network based method for cell type classification , uses three hidden layers in its architecture . Though its model can predict cell types , the model still lacks reliable interpretations for the result . The challenge is that DNN does not provide a recognizable pattern that is closely related to different cell types . Here , we introduce a new neural network based method scCA for cell type classification . scCA provides an reasonable interpretation for its classification result . Our model learns features from cells and provides patterns behind these features that are related to cell types . The first part is feature extraction . Given the gene expression level in every cell , We use capsules to extract different features . A capsule is a vector that represents a feature , which captures a certain type of information of the cell . Capsules also serve as dimension reduction for single cell RNA-seq data . We calculate the Pearson coefficients between every two capsules and generate the coefficient heatmap of every cell . We discover that cells from same cell type share a similar pattern in the heatmap while cells from different cell types have different patterns . These capsules capture features that are important for cell type classification . And then , we use a sequence to sequence model , bidirectional-LSTM , to generate compound features . Each compound feature combines a set of features ’ information . Furthermore , we train multiple attention weights for these compound features . Then we multiply these weights to the compound features and generate the classification result . Attention weights help to lay more emphasis on important compound features . The larger the attention weight is , the more crucial its corresponding compound feature is . we also draw heatmaps for every cells ’ attention weights . We discover that cells from same cell type have similar patterns in the heatmaps of attention weights . This leads to a better classification performance and also provides an reliable interpretation behind the network . We evaluate scCA on several datasets and compare with previous work . Our method shows high accuracy and robustness on all datasets . The main contributions of our article are as follow : We propose an interpretable neural network based method scCA for cell type classification . Through heatmaps of capsules ’ Pearson coefficients and attention weights , we find that cells from same cell type share a similar pattern . scCA achieves high accuracy and stable performances on all datasets . 2 RELATED WORK . Statistical Hypotheses Based Model Many machine learning methods with statistical hypotheses of gene expression level have been proposed for cell type classification . Some of these methods suggest distribution hypotheses of gene expression level and employ machine learning models based on these hypotheses . Some build a hierarchy with prior knowledge and statistical hypotheses . Moana takes advantage of KNN and SVM model ( Wagner & Yanai , 2018 ) . It first preprocesses data with KNN , then it uses dimension reduction method to extract useful features , at last it uses a Support Vector Machine classifier . In addition , Moana uses marker gene as a prior knowledge for cell type identification . Different from Moana , Garnett proposes a model based on tree hierarchy . It discovers the relation between cell types and subtypes with a tree structure ( Pliner et al. , 2019 ) . With the help of marker gene , it builds a linear model to classify cell types . CellAssign builds a probabilistic model based on prior knowledge to classify cell types ( Zhang et al. , 2019b ) . SCINA makes use of EM algorithm and the information of marker gene to accomplish clustering as well as cell type identification task ( Zhang et al. , 2019a ) . scPred extracts features using SVD , then with a nonlinear kernel SVM , it predicts different cell types ( Alquicira-Hernandez et al. , 2019 ) . However , these methods have their limitations . Their hypotheses such as ZINB ( Zero-inflated Negative Binomial ) or NB distribution of gene expression are not the exact reflections of gene ’ s expression levels in cells . Actually , there is not a consensus statistical distribution for single-cell RNA-seq data . Also they need prior knowledge like marker gene for cell type classification . Neural Network Based Models Deep neural network ( DNN ) based methods have been applied to cell type classification task . They are fast and robust . Neural networks don ’ t heavily rely on prior knowledge and statistical hypotheses . ACTINN uses three fully connected layers and ReLU , softmax activation function to classify different cell types ( Ma & Pellegrini , 2020 ) . scCapsNet builds a capsule network to extract features from the data and use dynamic routings among capsules to get a classification result ( Wang et al. , 2020 ) . However , neural networks lack interpretation for their result . Through training the whole network by propagation , it ’ s hard to explain every component ’ s contribution for the result . 3 PRELIMINARIES . Capsule Network and Dynamic Routing Hinton , Sabour and other co-authors first proposed Capsule network and dynamic routing in computer vision ( Sabour et al. , 2017 ) . A capsule is a set of neurons or a vector that can represent an entity . Dynamic routing is the mechanism between capsule ’ s layers . As they point out , the capsules in the layer above are parents and the capsules in the layer below are children . ( Hinton et al. , 2000 ; 2011 ) . Dynamic routing helps to send the results from capsules in children layer to its appropriate parents in the above layer . By adjusting the coupling coefficients between capsules in two layers , the network learns which parent capsule is more important for the children capsule . Here in scCA , we use capsules to capture different features of the cell . We use a vector for implementation of capsule in our model . However , different from dynamic routing , we use LSTM and attention as the architecture for cell type classification . Long Short-Term Memory and Attention Long Short-Term Memory ( LSTM ) is a model widely used in Natural Language Processing ( NLP ) ( Hochreiter & Schmidhuber , 1997 ) . It is a sequence to sequence model . The inputs are usually the word embeddings ( representations ) of a sentence and the output is a sequence of representations . Each output representation not only contains information from its corresponding input , but also information from representations in the context , especially its neighbor inputs . In our method , after extracting different features into capsules , we input these capsules into the LSTM model and obtain the output sequence . Every vector in the output sequence contains information of its corresponding input feature as well as other features . We name these vectors compound features . Some of the compound features have closer relationships with certain cell types . Inspired by Attention ( Vaswani et al. , 2017 ) mechanism , we apply attention weights to compound features in order to differentiate and emphasize those important compound features for classification . 4 SINGLE-CELL CAPSULE ATTENTION . In this section , we provide a detailed description of single-cell Capsule Attention ( scCA ) . We will also give a vivid and transparent explanation for our classification result . scCA can be divided into three parts . The figure below shows its architecture . 4.1 FEATURE EXTRACTION . The first part of scCA is feature extraction . Given thousands of genes ’ expression levels for a cell , we hope to learn simple but efficacious features for classification . Capsules not only serve as dimension reduction , but also extract a type of information of the cell . Through fully connected layers , we generate capsules to extract different features . capsulei = ReLU ( W ix+ bi ) , i ∈ [ 1 , 2 · · ·m ] ( 1 ) In this formula , m represents the number of capsules . W and b are the weight matrices of fully connected layers . ReLU is the activation function we use . Every capsule contains a feature of a cell . We hope to discover different patterns of these features for different cell types . We calculate the Pearson coefficients between every two capsules . The Pearson coefficient is an indication of relationship between two features behind capsules . For every cell , we come to a matrix where every element is a Pearson coefficient of two capsules . We then draw a heatmap of the matrix . We discover that cells from same cell type share a similar pattern while cells from different cell types differ from each other . Taking six cells from dataset GSE72056 as examples , we can see cells from same cell type have a similar pattern of capsules ’ relationship . The deeper the color is in the heatmap , the closer the relationship of two capsules is . The pattern is distinguishable among different cell types . From the figures above , we conclude that there exists some particular feature patterns for different cell types . Our feature extraction really extracts useful and discriminative features . | In this work, the authors implemented a capsule network to classify cell types for single-cell RNA-seq data. The capsule network was compared with standard methods such as SVM, LDA, and a fully connected neural network. The authors claim the capsule network provides interpretable results on cell type classification. | SP:6ba0ae7d3707640ee4018653af8e6dc179e15309 |
Fact-driven Logical Reasoning | 1 INTRODUCTION . To understand human language , deep neural networks have been widely applied and achieved impressive benchmark results ( Chen et al. , 2016b ; Sachan & Xing , 2016 ; Seo et al. , 2017 ; Dhingra et al. , 2017 ; Cui et al. , 2017 ; Song et al. , 2018 ; Hu et al. , 2019 ; Zhang et al. , 2020a ; Back et al. , 2020 ; Zhang et al. , 2020b ; Hermann et al. , 2015 ) . However , the core requirement among the natural language understanding , logic reasoning , can not be simply solved by the current design philosophy of extracting statistical patterns from data ( Shi et al. , 2020 ) . In order to solve such a problem , there emerges an interest that accounts for human intuition about the entailment of sentences and reflects the semantic relations between sentential constituents ( Iwańska , 1993 ) . In this paper , we focus on logic reasoning in the form of natural language understanding ( NLU ) as logic reasoning may be naturally embodied in such a task and natural language offers sufficient enough clues for effective logic reasoning . In detail , we concern about a logic reasoning question-answering ( QA ) task , where given passage , question , and candidate answer options , the model has to make a proper decision with its logic reasoning ability . There are examples shown in Figure 1 from logic reasoning benchmark datasets , ReClor ( Yu et al. , 2020 ) and LogiQA ( Liu et al. , 2020 ) . Recent neural models usually exploit a pretrained language model ( PrLM ) as a key encoder for effective contextualized representation . According to diagnostic tests ( Ettinger , 2020 ; Rogers et al. , 2020 ) , though PrLMs like BERT ( Devlin et al. , 2019 ) have encoded syntactic and semantic information after large-scale pre-training , they perform sensitivity to role reversal and struggles with pragmatic inference and role-based event knowledge , which are fundamental for reasoning whose major challenge is to uncover logical structures , and reason with the candidate options and questions to predict the correct answer . It is difficult for PrLMs to capture the logical structure inherent in the texts since logical supervision is rarely available during pre-training . Existing logic reasoning has shown serious dependence on knowledge-like clues . This is due to the lengthy , noisy text in human language which is though a natural carrier of knowledge but does not provide a clean , exact knowledge form . Thus , an increasing interest is in using graph networks to model the entity-aware relationships in the passages ( Yasunaga et al. , 2021 ; Ren & Leskovec , 2020 ; Huang et al. , 2021 ; Krishna et al. , 2020 ; Lv et al. , 2020 ) . However , all these methods may insufficiently capture indispensable logical units from two perspectives . First , they mostly focus on entity-aware commonsense knowledge , but pay little attention to those non-entity , non-commonsense clues ( Zhong et al. , 2021 ) . Second , when existing models extract predicate logic inside language into knowledge , they only exploit quite limited predicates like hasA and isA but ignore a broad range of predicates in real language . From either of the perspectives , the existing methods actually only concern about those `` global '' knowledge that keeps valid across the entire data , without sufficient `` local '' perception of complete facts or events in the given specific part of logic reasoning task . We argue such insufficient modeling on logic units roots from the ignorance of language itself being the complete knowledge/clue carrier . Thus , we propose extracting a kind of broad facts according to backbone constituents of a sentence to effectively cover such indispensable logic reasoning basis , filling the gap of local , non-commonsense , non-entity , or even non-knowledge clues in existing methods as shown in Figure 2 . For example , these units may reflect the facts of who did what to whom , or who is what in Figure 3 . Such groups can be defined as `` fact unit '' following Nakashole & Mitchell ( 2014 ) in Definition 1 . The fact units are further organized into a supergraph following Definition 2 . Definition 1 ( Fact Unit ) Given an triplet T = { E1 , P , E2 } , where E1 and E2 are arguments ( including entity and non-entity ) , P is the predicate , a fact unit F is the set of all entities in T and their corresponding relations . Definition 2 ( Supergraph ) A supergraph is a structure made of fact units ( regarded as subgraphs ) as the vertices , and the relations between fact units as undirected edges . As shown in Figure 2 , we regard the defined fact as the results of syntactic processing , rather than those from semantic role labeling ( SRL ) as in previous study , thus the proposed fact also extends the processing means in existing work . Correspondingly , in this work , we propose a fact-driven logical reasoning model , called FOCAL REASONER , which builds supergraphs on top of fact units as the basis for logic reasoning , to capture both global connections between facts and the local concepts or actions inside the fact . Our model FOCAL REASONER is evaluated on two challenging logic reasoning benchmarks including ReClor , LogiQA , one dialogue reasoning dataset Mutual , for verifying the effectiveness and the generalizability across different domains and question formats . 2 RELATED WORK . Machine Reading Comprehension Recent years have witnessed massive researches on Machine Reading Comprehension ( MRC ) whose goal is training machines to understand human languages , which has become one of the most important areas of NLP ( Chen et al. , 2016b ; Sachan & Xing , 2016 ; Seo et al. , 2017 ; Dhingra et al. , 2017 ; Cui et al. , 2017 ; Song et al. , 2018 ; Hu et al. , 2019 ; Zhang et al. , 2020a ; Back et al. , 2020 ; Zhang et al. , 2020b ) . Despite the success of MRC models on various datasets such as CNN/Daily Mail ( Hermann et al. , 2015 ) , SQuAD ( Rajpurkar et al. , 2016 ) , RACE ( Lai et al. , 2017 ) and so on , researchers began to rethink what extent does the problem been solved . Nowadays , there are massive researches into the reasoning ability of machines . According to ( Kaushik & Lipton , 2018 ; Zhou et al. , 2020 ; Chen et al. , 2016a ) , reasoning abilities can be broadly categorized into ( 1 ) commonsense reasoning ( Davis & Marcus , 2015 ; Bhagavatula et al. , 2019 ; Talmor et al. , 2019 ; Huang et al. , 2019 ) ; ( 2 ) numerical reasoning ( Dua et al. , 2019 ) ; ( 3 ) multi-hop reasoning ( Yang et al. , 2018 ) and ( 4 ) logic reasoning ( Yu et al. , 2020 ; Liu et al. , 2020 ) , among which logic reasoning is essential in human intelligence but has merely been delved into . Natural Language Inference ( NLI ) ( Bowman et al. , 2015 ; Williams et al. , 2018 ; Nie et al. , 2020 ) is a task closely related to logic reasoning . However , it has two obvious drawbacks in measuring logic reasoning abilities . One is that it only has three logical types which are entailment , contradiction and neutral . The other is its limitation on sentence-level reasoning . Hence , it is important to research more comprehensive and deeper logic reasoning abilities . logic reasoning in MRC There are two main kinds of features in language data that would be the necessary basis for logic reasoning : 1 ) knowledge : global facts that keep consistency regardless of the context , such as commonsense , mostly derived from named entities ; 2 ) non-knowledge : local facts or events that may be sensitive to the context , mostly derived from detailed language . Existing works have made progress in improving logic reasoning ability ( Yasunaga et al. , 2021 ; Ren & Leskovec , 2020 ; Huang et al. , 2021 ; Krishna et al. , 2020 ; Zhong et al. , 2021 ; Wang et al. , 2021 ) . However , these approaches are barely satisfactory as they mostly focus on the global facts such as typical entity or sentence-level relations , which are obviously not sufficient . In this work , we strengthen the basis for logic reasoning by unifying both types of the features as `` facts '' . Different from previous studies that focus on the knowledge components , we propose a fact-driven logic reasoning framework that builds supergraphs on top of fact units to capture both global connections between entityaware facts and the local concepts or events inside the fact . 3 METHODOLOGY . In this section , we present a fact-driven approach for logic reasoning and the overall architecture of the model is shown in Figure 4 . The framework can be divided into three steps as following . We first extract fact units from raw texts via syntactic processing to construct a supergraph . Then it performs reasoning over the supergraph along with a logical fact regularization . Finally , it aggregates the learned representation to decode for the right answer . 3.1 FACT UNIT EXTRACTION AND SUPERGRAPH CONSTRUCTION . Fact Unit Extraction . Figure 5 illustrates our method for constructing a supergraph from raw text inputs . The first step is to obtain triplets that constitute a fact unit . To keep the framework generic , we use a fairly simple fact unit extractor based on the syntactic relations . Given a context consisting of multiple sentences , we first conduct dependency parsing of each sentence . After that , we extract the subject , the predicate , and the object tokens to get the `` Argument-Predicate-Argument '' triplets corresponding to each sentence in the context . Supergraph Construction . With the obtained triplets , the fact units are organized in the form of Levi graph ( Levi , 1942 ) , which turns arguments and predicates all into nodes . An original fact unit is in the form of F = ( V , E , R ) , where V is the set of the arguments , E is the set of edges connected between arguments , and R is the relations of each edge which are predicates here . The corresponding Levi graph is denoted as Fl = ( VL , EL , RL ) where VL = V ∪ R , which makes the originally directly connected arguments be intermediately connected via relations . As for RL , previous works such as ( Marcheggiani & Titov , 2017 ; Beck et al. , 2018a ) designed three types of edges RL = { default , reverse , self } to enhance information flow . Here in our settings , we extend it into five types : default-in , default-out , reverse-in , reverse-out , self , corresponding to the directions of edges towards the predicates . Detailed description for edge types can be found in Appendix A . We construct the supergraph by making connections between fact units Fl . In particular , we take three strategies according to global information , identical concept and co-reference information . ( 1 ) We add a node Vg initialized with the question-option representation and connect it to all the fact unit nodes . The edge type is set as aggregate for better information interaction . ( 2 ) There can be identical mentions in different sentences , resulting in repeated nodes in fact units . We connect nodes corresponding to the same non-pronoun arguments by edges with edge type same . ( 3 ) We conduct co-reference resolution on context using an off-to-shelf model1 in order to identify arguments in fact units that refer to the same one . We add edges with type coref between them . The final supergraph is denoted as S = ( Fl ∪ Vg , E ) where E is the set of edges added with the previous three strategies . | The paper claims that a lot of previous work in QA-based MRC focused only on entity-aware common sense knowledge. To overcome this, the paper proposes to build a finer-grained local fact, (entity, predicate, entity) triplet, information based on dependency parsing, and then connect the triplet nodes with global information such as coreference, entity information. Architecture-wise, a passage, question, and options are passed to RoBERTa (or DeBERTa) and then this information goes through the graph-attention network (GAN), based on the graph structure described above. In the experiment section, the authors show that the proposed FOCAL REASONER outperforms the previous SOTA models and run an ablation study on how each component contributes (in Table 5,6). The main contribution of this paper, as I understand, is bringing more fine-grained local facts and connecting them with global information such as coreference and entity linking. | SP:e1ef24ecb1d18bab9ab0d706efcdc61f8bd4b4b4 |
Fact-driven Logical Reasoning | 1 INTRODUCTION . To understand human language , deep neural networks have been widely applied and achieved impressive benchmark results ( Chen et al. , 2016b ; Sachan & Xing , 2016 ; Seo et al. , 2017 ; Dhingra et al. , 2017 ; Cui et al. , 2017 ; Song et al. , 2018 ; Hu et al. , 2019 ; Zhang et al. , 2020a ; Back et al. , 2020 ; Zhang et al. , 2020b ; Hermann et al. , 2015 ) . However , the core requirement among the natural language understanding , logic reasoning , can not be simply solved by the current design philosophy of extracting statistical patterns from data ( Shi et al. , 2020 ) . In order to solve such a problem , there emerges an interest that accounts for human intuition about the entailment of sentences and reflects the semantic relations between sentential constituents ( Iwańska , 1993 ) . In this paper , we focus on logic reasoning in the form of natural language understanding ( NLU ) as logic reasoning may be naturally embodied in such a task and natural language offers sufficient enough clues for effective logic reasoning . In detail , we concern about a logic reasoning question-answering ( QA ) task , where given passage , question , and candidate answer options , the model has to make a proper decision with its logic reasoning ability . There are examples shown in Figure 1 from logic reasoning benchmark datasets , ReClor ( Yu et al. , 2020 ) and LogiQA ( Liu et al. , 2020 ) . Recent neural models usually exploit a pretrained language model ( PrLM ) as a key encoder for effective contextualized representation . According to diagnostic tests ( Ettinger , 2020 ; Rogers et al. , 2020 ) , though PrLMs like BERT ( Devlin et al. , 2019 ) have encoded syntactic and semantic information after large-scale pre-training , they perform sensitivity to role reversal and struggles with pragmatic inference and role-based event knowledge , which are fundamental for reasoning whose major challenge is to uncover logical structures , and reason with the candidate options and questions to predict the correct answer . It is difficult for PrLMs to capture the logical structure inherent in the texts since logical supervision is rarely available during pre-training . Existing logic reasoning has shown serious dependence on knowledge-like clues . This is due to the lengthy , noisy text in human language which is though a natural carrier of knowledge but does not provide a clean , exact knowledge form . Thus , an increasing interest is in using graph networks to model the entity-aware relationships in the passages ( Yasunaga et al. , 2021 ; Ren & Leskovec , 2020 ; Huang et al. , 2021 ; Krishna et al. , 2020 ; Lv et al. , 2020 ) . However , all these methods may insufficiently capture indispensable logical units from two perspectives . First , they mostly focus on entity-aware commonsense knowledge , but pay little attention to those non-entity , non-commonsense clues ( Zhong et al. , 2021 ) . Second , when existing models extract predicate logic inside language into knowledge , they only exploit quite limited predicates like hasA and isA but ignore a broad range of predicates in real language . From either of the perspectives , the existing methods actually only concern about those `` global '' knowledge that keeps valid across the entire data , without sufficient `` local '' perception of complete facts or events in the given specific part of logic reasoning task . We argue such insufficient modeling on logic units roots from the ignorance of language itself being the complete knowledge/clue carrier . Thus , we propose extracting a kind of broad facts according to backbone constituents of a sentence to effectively cover such indispensable logic reasoning basis , filling the gap of local , non-commonsense , non-entity , or even non-knowledge clues in existing methods as shown in Figure 2 . For example , these units may reflect the facts of who did what to whom , or who is what in Figure 3 . Such groups can be defined as `` fact unit '' following Nakashole & Mitchell ( 2014 ) in Definition 1 . The fact units are further organized into a supergraph following Definition 2 . Definition 1 ( Fact Unit ) Given an triplet T = { E1 , P , E2 } , where E1 and E2 are arguments ( including entity and non-entity ) , P is the predicate , a fact unit F is the set of all entities in T and their corresponding relations . Definition 2 ( Supergraph ) A supergraph is a structure made of fact units ( regarded as subgraphs ) as the vertices , and the relations between fact units as undirected edges . As shown in Figure 2 , we regard the defined fact as the results of syntactic processing , rather than those from semantic role labeling ( SRL ) as in previous study , thus the proposed fact also extends the processing means in existing work . Correspondingly , in this work , we propose a fact-driven logical reasoning model , called FOCAL REASONER , which builds supergraphs on top of fact units as the basis for logic reasoning , to capture both global connections between facts and the local concepts or actions inside the fact . Our model FOCAL REASONER is evaluated on two challenging logic reasoning benchmarks including ReClor , LogiQA , one dialogue reasoning dataset Mutual , for verifying the effectiveness and the generalizability across different domains and question formats . 2 RELATED WORK . Machine Reading Comprehension Recent years have witnessed massive researches on Machine Reading Comprehension ( MRC ) whose goal is training machines to understand human languages , which has become one of the most important areas of NLP ( Chen et al. , 2016b ; Sachan & Xing , 2016 ; Seo et al. , 2017 ; Dhingra et al. , 2017 ; Cui et al. , 2017 ; Song et al. , 2018 ; Hu et al. , 2019 ; Zhang et al. , 2020a ; Back et al. , 2020 ; Zhang et al. , 2020b ) . Despite the success of MRC models on various datasets such as CNN/Daily Mail ( Hermann et al. , 2015 ) , SQuAD ( Rajpurkar et al. , 2016 ) , RACE ( Lai et al. , 2017 ) and so on , researchers began to rethink what extent does the problem been solved . Nowadays , there are massive researches into the reasoning ability of machines . According to ( Kaushik & Lipton , 2018 ; Zhou et al. , 2020 ; Chen et al. , 2016a ) , reasoning abilities can be broadly categorized into ( 1 ) commonsense reasoning ( Davis & Marcus , 2015 ; Bhagavatula et al. , 2019 ; Talmor et al. , 2019 ; Huang et al. , 2019 ) ; ( 2 ) numerical reasoning ( Dua et al. , 2019 ) ; ( 3 ) multi-hop reasoning ( Yang et al. , 2018 ) and ( 4 ) logic reasoning ( Yu et al. , 2020 ; Liu et al. , 2020 ) , among which logic reasoning is essential in human intelligence but has merely been delved into . Natural Language Inference ( NLI ) ( Bowman et al. , 2015 ; Williams et al. , 2018 ; Nie et al. , 2020 ) is a task closely related to logic reasoning . However , it has two obvious drawbacks in measuring logic reasoning abilities . One is that it only has three logical types which are entailment , contradiction and neutral . The other is its limitation on sentence-level reasoning . Hence , it is important to research more comprehensive and deeper logic reasoning abilities . logic reasoning in MRC There are two main kinds of features in language data that would be the necessary basis for logic reasoning : 1 ) knowledge : global facts that keep consistency regardless of the context , such as commonsense , mostly derived from named entities ; 2 ) non-knowledge : local facts or events that may be sensitive to the context , mostly derived from detailed language . Existing works have made progress in improving logic reasoning ability ( Yasunaga et al. , 2021 ; Ren & Leskovec , 2020 ; Huang et al. , 2021 ; Krishna et al. , 2020 ; Zhong et al. , 2021 ; Wang et al. , 2021 ) . However , these approaches are barely satisfactory as they mostly focus on the global facts such as typical entity or sentence-level relations , which are obviously not sufficient . In this work , we strengthen the basis for logic reasoning by unifying both types of the features as `` facts '' . Different from previous studies that focus on the knowledge components , we propose a fact-driven logic reasoning framework that builds supergraphs on top of fact units to capture both global connections between entityaware facts and the local concepts or events inside the fact . 3 METHODOLOGY . In this section , we present a fact-driven approach for logic reasoning and the overall architecture of the model is shown in Figure 4 . The framework can be divided into three steps as following . We first extract fact units from raw texts via syntactic processing to construct a supergraph . Then it performs reasoning over the supergraph along with a logical fact regularization . Finally , it aggregates the learned representation to decode for the right answer . 3.1 FACT UNIT EXTRACTION AND SUPERGRAPH CONSTRUCTION . Fact Unit Extraction . Figure 5 illustrates our method for constructing a supergraph from raw text inputs . The first step is to obtain triplets that constitute a fact unit . To keep the framework generic , we use a fairly simple fact unit extractor based on the syntactic relations . Given a context consisting of multiple sentences , we first conduct dependency parsing of each sentence . After that , we extract the subject , the predicate , and the object tokens to get the `` Argument-Predicate-Argument '' triplets corresponding to each sentence in the context . Supergraph Construction . With the obtained triplets , the fact units are organized in the form of Levi graph ( Levi , 1942 ) , which turns arguments and predicates all into nodes . An original fact unit is in the form of F = ( V , E , R ) , where V is the set of the arguments , E is the set of edges connected between arguments , and R is the relations of each edge which are predicates here . The corresponding Levi graph is denoted as Fl = ( VL , EL , RL ) where VL = V ∪ R , which makes the originally directly connected arguments be intermediately connected via relations . As for RL , previous works such as ( Marcheggiani & Titov , 2017 ; Beck et al. , 2018a ) designed three types of edges RL = { default , reverse , self } to enhance information flow . Here in our settings , we extend it into five types : default-in , default-out , reverse-in , reverse-out , self , corresponding to the directions of edges towards the predicates . Detailed description for edge types can be found in Appendix A . We construct the supergraph by making connections between fact units Fl . In particular , we take three strategies according to global information , identical concept and co-reference information . ( 1 ) We add a node Vg initialized with the question-option representation and connect it to all the fact unit nodes . The edge type is set as aggregate for better information interaction . ( 2 ) There can be identical mentions in different sentences , resulting in repeated nodes in fact units . We connect nodes corresponding to the same non-pronoun arguments by edges with edge type same . ( 3 ) We conduct co-reference resolution on context using an off-to-shelf model1 in order to identify arguments in fact units that refer to the same one . We add edges with type coref between them . The final supergraph is denoted as S = ( Fl ∪ Vg , E ) where E is the set of edges added with the previous three strategies . | The authors propose a framework, called Focal Reasoner, to perform logical reasoning to answer questions. The proposed approach first extracts the facts from raw text. The authors describe a fact unit as a triple of (argument, predicate, argument). These collections of facts can be viewed as a graph with predicates as undirected edges between arguments. After forming a subgraph, reasoning is performed by a graph convolution module to predict the correct answer. The authors show that their approach outperforms existing approaches on two benchmarked datasets. The authors also perform a detailed analysis that sheds more light on the internal workings of the model. | SP:e1ef24ecb1d18bab9ab0d706efcdc61f8bd4b4b4 |
Fact-driven Logical Reasoning | 1 INTRODUCTION . To understand human language , deep neural networks have been widely applied and achieved impressive benchmark results ( Chen et al. , 2016b ; Sachan & Xing , 2016 ; Seo et al. , 2017 ; Dhingra et al. , 2017 ; Cui et al. , 2017 ; Song et al. , 2018 ; Hu et al. , 2019 ; Zhang et al. , 2020a ; Back et al. , 2020 ; Zhang et al. , 2020b ; Hermann et al. , 2015 ) . However , the core requirement among the natural language understanding , logic reasoning , can not be simply solved by the current design philosophy of extracting statistical patterns from data ( Shi et al. , 2020 ) . In order to solve such a problem , there emerges an interest that accounts for human intuition about the entailment of sentences and reflects the semantic relations between sentential constituents ( Iwańska , 1993 ) . In this paper , we focus on logic reasoning in the form of natural language understanding ( NLU ) as logic reasoning may be naturally embodied in such a task and natural language offers sufficient enough clues for effective logic reasoning . In detail , we concern about a logic reasoning question-answering ( QA ) task , where given passage , question , and candidate answer options , the model has to make a proper decision with its logic reasoning ability . There are examples shown in Figure 1 from logic reasoning benchmark datasets , ReClor ( Yu et al. , 2020 ) and LogiQA ( Liu et al. , 2020 ) . Recent neural models usually exploit a pretrained language model ( PrLM ) as a key encoder for effective contextualized representation . According to diagnostic tests ( Ettinger , 2020 ; Rogers et al. , 2020 ) , though PrLMs like BERT ( Devlin et al. , 2019 ) have encoded syntactic and semantic information after large-scale pre-training , they perform sensitivity to role reversal and struggles with pragmatic inference and role-based event knowledge , which are fundamental for reasoning whose major challenge is to uncover logical structures , and reason with the candidate options and questions to predict the correct answer . It is difficult for PrLMs to capture the logical structure inherent in the texts since logical supervision is rarely available during pre-training . Existing logic reasoning has shown serious dependence on knowledge-like clues . This is due to the lengthy , noisy text in human language which is though a natural carrier of knowledge but does not provide a clean , exact knowledge form . Thus , an increasing interest is in using graph networks to model the entity-aware relationships in the passages ( Yasunaga et al. , 2021 ; Ren & Leskovec , 2020 ; Huang et al. , 2021 ; Krishna et al. , 2020 ; Lv et al. , 2020 ) . However , all these methods may insufficiently capture indispensable logical units from two perspectives . First , they mostly focus on entity-aware commonsense knowledge , but pay little attention to those non-entity , non-commonsense clues ( Zhong et al. , 2021 ) . Second , when existing models extract predicate logic inside language into knowledge , they only exploit quite limited predicates like hasA and isA but ignore a broad range of predicates in real language . From either of the perspectives , the existing methods actually only concern about those `` global '' knowledge that keeps valid across the entire data , without sufficient `` local '' perception of complete facts or events in the given specific part of logic reasoning task . We argue such insufficient modeling on logic units roots from the ignorance of language itself being the complete knowledge/clue carrier . Thus , we propose extracting a kind of broad facts according to backbone constituents of a sentence to effectively cover such indispensable logic reasoning basis , filling the gap of local , non-commonsense , non-entity , or even non-knowledge clues in existing methods as shown in Figure 2 . For example , these units may reflect the facts of who did what to whom , or who is what in Figure 3 . Such groups can be defined as `` fact unit '' following Nakashole & Mitchell ( 2014 ) in Definition 1 . The fact units are further organized into a supergraph following Definition 2 . Definition 1 ( Fact Unit ) Given an triplet T = { E1 , P , E2 } , where E1 and E2 are arguments ( including entity and non-entity ) , P is the predicate , a fact unit F is the set of all entities in T and their corresponding relations . Definition 2 ( Supergraph ) A supergraph is a structure made of fact units ( regarded as subgraphs ) as the vertices , and the relations between fact units as undirected edges . As shown in Figure 2 , we regard the defined fact as the results of syntactic processing , rather than those from semantic role labeling ( SRL ) as in previous study , thus the proposed fact also extends the processing means in existing work . Correspondingly , in this work , we propose a fact-driven logical reasoning model , called FOCAL REASONER , which builds supergraphs on top of fact units as the basis for logic reasoning , to capture both global connections between facts and the local concepts or actions inside the fact . Our model FOCAL REASONER is evaluated on two challenging logic reasoning benchmarks including ReClor , LogiQA , one dialogue reasoning dataset Mutual , for verifying the effectiveness and the generalizability across different domains and question formats . 2 RELATED WORK . Machine Reading Comprehension Recent years have witnessed massive researches on Machine Reading Comprehension ( MRC ) whose goal is training machines to understand human languages , which has become one of the most important areas of NLP ( Chen et al. , 2016b ; Sachan & Xing , 2016 ; Seo et al. , 2017 ; Dhingra et al. , 2017 ; Cui et al. , 2017 ; Song et al. , 2018 ; Hu et al. , 2019 ; Zhang et al. , 2020a ; Back et al. , 2020 ; Zhang et al. , 2020b ) . Despite the success of MRC models on various datasets such as CNN/Daily Mail ( Hermann et al. , 2015 ) , SQuAD ( Rajpurkar et al. , 2016 ) , RACE ( Lai et al. , 2017 ) and so on , researchers began to rethink what extent does the problem been solved . Nowadays , there are massive researches into the reasoning ability of machines . According to ( Kaushik & Lipton , 2018 ; Zhou et al. , 2020 ; Chen et al. , 2016a ) , reasoning abilities can be broadly categorized into ( 1 ) commonsense reasoning ( Davis & Marcus , 2015 ; Bhagavatula et al. , 2019 ; Talmor et al. , 2019 ; Huang et al. , 2019 ) ; ( 2 ) numerical reasoning ( Dua et al. , 2019 ) ; ( 3 ) multi-hop reasoning ( Yang et al. , 2018 ) and ( 4 ) logic reasoning ( Yu et al. , 2020 ; Liu et al. , 2020 ) , among which logic reasoning is essential in human intelligence but has merely been delved into . Natural Language Inference ( NLI ) ( Bowman et al. , 2015 ; Williams et al. , 2018 ; Nie et al. , 2020 ) is a task closely related to logic reasoning . However , it has two obvious drawbacks in measuring logic reasoning abilities . One is that it only has three logical types which are entailment , contradiction and neutral . The other is its limitation on sentence-level reasoning . Hence , it is important to research more comprehensive and deeper logic reasoning abilities . logic reasoning in MRC There are two main kinds of features in language data that would be the necessary basis for logic reasoning : 1 ) knowledge : global facts that keep consistency regardless of the context , such as commonsense , mostly derived from named entities ; 2 ) non-knowledge : local facts or events that may be sensitive to the context , mostly derived from detailed language . Existing works have made progress in improving logic reasoning ability ( Yasunaga et al. , 2021 ; Ren & Leskovec , 2020 ; Huang et al. , 2021 ; Krishna et al. , 2020 ; Zhong et al. , 2021 ; Wang et al. , 2021 ) . However , these approaches are barely satisfactory as they mostly focus on the global facts such as typical entity or sentence-level relations , which are obviously not sufficient . In this work , we strengthen the basis for logic reasoning by unifying both types of the features as `` facts '' . Different from previous studies that focus on the knowledge components , we propose a fact-driven logic reasoning framework that builds supergraphs on top of fact units to capture both global connections between entityaware facts and the local concepts or events inside the fact . 3 METHODOLOGY . In this section , we present a fact-driven approach for logic reasoning and the overall architecture of the model is shown in Figure 4 . The framework can be divided into three steps as following . We first extract fact units from raw texts via syntactic processing to construct a supergraph . Then it performs reasoning over the supergraph along with a logical fact regularization . Finally , it aggregates the learned representation to decode for the right answer . 3.1 FACT UNIT EXTRACTION AND SUPERGRAPH CONSTRUCTION . Fact Unit Extraction . Figure 5 illustrates our method for constructing a supergraph from raw text inputs . The first step is to obtain triplets that constitute a fact unit . To keep the framework generic , we use a fairly simple fact unit extractor based on the syntactic relations . Given a context consisting of multiple sentences , we first conduct dependency parsing of each sentence . After that , we extract the subject , the predicate , and the object tokens to get the `` Argument-Predicate-Argument '' triplets corresponding to each sentence in the context . Supergraph Construction . With the obtained triplets , the fact units are organized in the form of Levi graph ( Levi , 1942 ) , which turns arguments and predicates all into nodes . An original fact unit is in the form of F = ( V , E , R ) , where V is the set of the arguments , E is the set of edges connected between arguments , and R is the relations of each edge which are predicates here . The corresponding Levi graph is denoted as Fl = ( VL , EL , RL ) where VL = V ∪ R , which makes the originally directly connected arguments be intermediately connected via relations . As for RL , previous works such as ( Marcheggiani & Titov , 2017 ; Beck et al. , 2018a ) designed three types of edges RL = { default , reverse , self } to enhance information flow . Here in our settings , we extend it into five types : default-in , default-out , reverse-in , reverse-out , self , corresponding to the directions of edges towards the predicates . Detailed description for edge types can be found in Appendix A . We construct the supergraph by making connections between fact units Fl . In particular , we take three strategies according to global information , identical concept and co-reference information . ( 1 ) We add a node Vg initialized with the question-option representation and connect it to all the fact unit nodes . The edge type is set as aggregate for better information interaction . ( 2 ) There can be identical mentions in different sentences , resulting in repeated nodes in fact units . We connect nodes corresponding to the same non-pronoun arguments by edges with edge type same . ( 3 ) We conduct co-reference resolution on context using an off-to-shelf model1 in order to identify arguments in fact units that refer to the same one . We add edges with type coref between them . The final supergraph is denoted as S = ( Fl ∪ Vg , E ) where E is the set of edges added with the previous three strategies . | This paper proposes a model architecture for question answering which takes advantage of shallow proposition structure and coreference links using (relational) graph convolutional networks. Subject-predicate-object triples (dubbed "fact units") are extracted from the text using dependency parsing, extra coreference links are added using a coreference system, and the resulting graph is initialized with representations from a contextualizing encoder, passed through a GCN, and recombined with the contextual representations via multi-headed attention. The resulting representations serve as input to the final classification layer for answer prediction, as well as a "logical facts regularization" step which encourages the final representation of the object in each fact unit to align (via cosine) with the sum of the subject and predicate representations. In experiments on several datasets designed to test logical reasoning, the proposed architecture scores higher than several baselines. | SP:e1ef24ecb1d18bab9ab0d706efcdc61f8bd4b4b4 |
Taming Sparsely Activated Transformer with Stochastic Experts | 1 INTRODUCTION . Large neural network models have shown to be effective in many natural language processing tasks such as machine translation ( Lewis et al. , 2020 ; Conneau & Lample , 2019 ) , natural language understanding ( Devlin et al. , 2019 ; Liu et al. , 2019 ; He et al. , 2020 ) , and natural language generation ( Radford et al. , 2019 ; Brown et al. , 2020 ) . These models are usually densely activated . That is , a model uses all its parameters to process all inputs . One drawback of these models is the prohibitive training cost . Moreover , the extreme size drastically reduces inference speed , further limiting the models ’ practicality . To address these issues , sparsely activated models ( SAMs , Shazeer et al . 2017 ) have been proposed . A SAM adaptively selects a subset of its parameters for different inputs during model training and inference . This makes it possible to train SAMs that are an order of magnitude larger than densely activated models without significant increase in computational cost . For example , the sparsely activated GShard ( Lepikhin et al. , 2020 ) consists of over 600 billion parameters and the Switch Transformer ( Fedus et al. , 2021 ) 1.5 trillion parameters , while GPT-3 ( Brown et al. , 2020 ) , which is arguably the largest densely activated model , consists of only 175 billion parameters . The building block of SAMs is the expert layer , which contains an attention mechanism and multiple feed-forward neural networks ( FFNs ) in parallel . Each FFN is referred to as an expert . During ∗Work was done during an internship at Microsoft . training , an input is routed to a fixed number of experts , such that the number of floating point operations ( FLOPs ) of one forward pass remains constant , regardless of the total number of experts . Thus , training SAMs is much more cost-efficient than training densely activated models . For example , training of Switch-large ( Fedus et al. , 2021 ) and that of T5-large ( Raffel et al. , 2019 ) require the same forward FLOPs , despite that the former is 35 times larger ( 26.3 vs. 0.74 billion parameters ) . However , SAMs have been reported to be parameter inefficient . For example , although the Switchlarge model is 35 times larger than T5-large , its performance on the GLUE benchmark ( Wang et al. , 2019a ) is only slightly better ( 88.5 vs. 87.8 ) . There are also cases where the performance of SAMs is even worse than smaller densely activated models . For example , the performance of Switchlarge is worse than T5-large on the ARC Reasoning Challenge ( 66.0 vs. 68.8 ) ( Clark et al. , 2018 ) . In another example , although GShard ( Lepikhin et al. , 2020 ) shows substantial gains over densely activated models , a diminishing return with larger number of parameters has been observed . Most on-going research has focused on improving SAMs by developing effective routing methods . Since only a subset of model parameters ( i.e. , experts ) are updated for each input during training , we need to decide which experts to be activated given an input . Existing works ( Shazeer et al. , 2017 ; Lepikhin et al. , 2020 ; Fedus et al. , 2021 ; Yang et al. , 2021 ) use a gating network for input routing . However , the gating mechanism suffers from the notorious load imbalance issue : the gate ’ s weight could collapse such that nearly all the inputs are routed to the same expert . Therefore , many methods are proposed to mitigate this issue , such as noisy gating ( Shazeer et al. , 2017 ) , expert capacity ( Lepikhin et al. , 2020 ) , load balancing loss ( Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) , and k Top-1 gating ( Yang et al. , 2021 ) . However , these routing methods have not been proved effective to make SAMs more parameter efficient . To understand why SAMs are not parameter efficient , we analyze the performance of several classic MoE models . Our analysis reveals that a SAM does not always outperform a densely activated model of a similar size , confirming the results reported in Yang et al . ( 2021 ) . Moreover , we also observe that the widely-used routing method based on the gating mechanism does not work better than randomly routing inputs to experts , Inspired by our findings , we propose a new SAM , THOR ( Transformer witH StOchastic ExpeRts ) . Unlike classic SAMs , such as the Switch Transformer , experts in THOR are randomly activated ( with no need of any gating mechanism ) for each input during training and inference . THOR models are trained by minimizing both the cross-entropy loss and a consistency regularization term , such that experts can learn not only from training data but also from other experts as teachers so that all the experts make consistent predictions . To validate the effectiveness of THOR , we have conducted extensive experiments on machine translation using three settings : low-resource , rich-resource , and multilingual . Results show that THOR models outperform state-of-the-art MoE models by an average of 2 BLEU score on twelve low-resource translation tasks . In the rich-resource setting , THOR achieves new state-of-the-art results on the two widely-used translation benchmarks , WMT ’ 16 En-De and WMT ’ 14 En-Fr . On multilingual translation tasks , the THOR model with 300 million parameters achieves 2 BLEU score improvement over a state-of-the-art MoE model of the same size . Moreover , our model achieves state-of-the-art results on these tasks — the same BLEU score that is achieved by the Z-code MoE model ( Kim et al. , 2021 ) with 5.5 billion parameters ( 18 times larger ) . 2 BACKGROUND . Transformer . The Transformer ( Vaswani et al. , 2017 ) model has demonstrated its superior performance in many sequence-to-sequence natural language processing tasks , such as neural machine translation . The model contains an encoder and a decoder . The encoder consists of multiple encoder layers , each having an identical structure . An encoder layer employs a self-attention mechanism and a feed-forward neural network ( FFN ) . The decoder is similarly constructed , except for an additional cross-attention mechanism in each decoder layer . Sparsely Activated Models . The building block of SAMs is the expert layer , which is similar to the Transformer layer . Each of these expert layers contain an attention mechanism and multiple FFNs in parallel , where each FFN is referred to as an expert . Let { Ei } Ni=1 denote the experts , and N denotes the total number of experts . A gating mechanism decides to which expert ( s ) an input should be routed . At each expert layer , given an input vector x ∈ Rd , where d is the embedding dimension , the gate value of routing x to expert Ei is pi ( x ) = [ Softmax ( Wgx ) ] i , ( 1 ) where Wg ∈ RN×d is the trainable weight matrix of the gating mechanism . Given the gate values { pi ( x ) } Ni=1 , we select the top-K experts to form an activated set of experts T ⊂ { 1 · · ·N } , where |T | = K. Then the output xout of the expert layer is xout = ∑ i∈T pi ( x ) Ei ( x ) . ( 2 ) Notice that in Eq . 2 , input x only activates K instead of N experts , where K N , e.g. , K = 2 and N = 2048 in GShard ( Lepikhin et al. , 2020 ) . This implies that the number of FLOPs required for one forward pass does not increase with the number of experts N . Therefore , SAMs can scale to an enormous size without any significant increase in training time and inference time . The gate weight matrixWg ( Eq . 1 ) is trained together with the rest of the model parameters . Because there is no constraint on the learned weights , it is possible that Wg collapses such that one row dominates , i.e. , all the inputs are routed to one expert . This problem is referred to as load imbalance . Existing works adopt various ad-hoc heuristics to mitigate this issue , e.g. , adding Gaussian noise to Eq . 1 ( noisy gating , Shazeer et al . 2017 ) , limiting the maximum number of inputs that can be routed to an expert ( expert capacity , Lepikhin et al . 2020 ) , imposing a load balancing loss ( Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) , and using linear assignment ( Lewis et al. , 2021 ) . There are other works that remove the gating mechanism such that load imbalance is no longer an issue , e.g. , by incorporating hash functions ( Roller et al. , 2021 ) . Besides the load imbalance issue , there are also heated discussions on how to construct T in Eq . 2 . For example , Shazeer et al . ( 2017 ) ; Lepikhin et al . ( 2020 ) ; Yang et al . ( 2021 ) conjecture that routing inputs to K > 1 experts is necessary , while Fedus et al . ( 2021 ) argue that using K = 1 is sufficient and more computationally efficient . 3 ANALYSIS OF SPARSELY ACTIVATED MODELS . We investigate behavior of the gating mechanism of several classic MoE models . We conduct experiments on a multilingual translation task , { De , Vi } → En . More details are presented in Appendix A . We consider two MoE models proposed in Shen et al . ( 2019 ) , referred to as MoE ( dec ) and MoE ( tok ) , respectively , and three variants of the Switch Transformer proposed in Fedus et al . ( 2021 ) . The number of experts is set to two for all the MoE models . We compare them with the Transformer ( Vaswani et al. , 2017 ) model of the same model size . Figure 1 shows the validation losses and BLEU scores of three models : Transformer , MoE ( dec ) , and MoE ( tok ) . We see that the two MoE models perform very similarly , and neither outperforms the Transformer by a significant margin . To interpret the results of Figure 1 , we examine the load of each expert and the confidence scores of routing inputs to different experts . An expert ’ s load is defined as the proportion of inputs that are assigned to it . For an input that is routed to an expert , its routing confidence score ( output of the gating mechanism ) determines the level of preference , e.g. , if the routing confidence score is 0.5 , then the gate has no preference for either expert . For each expert , we compute the average routing confidence score over all the inputs assigned to it . Figure 2 shows that after the early stage of training ( i.e. , the first 200 iterations ) , the gate weight collapses and nearly all the inputs are routed to expert 2 . Also , the average routing confidence score of expert 2 is close to 1.0 , which means that the gate strongly prefers expert 2 to expert 1 . In this case , only one of the experts is sufficiently trained . Figure 3 depicts a different scenario , where the inputs are randomly dispatched to the experts . Notice that after approximately 4000 iterations , the two experts are equally loaded , and the probabilities of assigning any input to expert 1 and expert 2 are almost identical , indicating that the gating mechanism has no preference for either expert . We have identified two behaviors of the gating mechanism : load imbalance and random routing . The former is also reported in recent papers ( Shazeer et al. , 2017 ; Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) . We further investigate the Switch Transformer ( Fedus et al. , 2021 ) , which is a state-of-theart MoE variant that incorporates various methods to resolve the load imbalance issue . In addition , because behavior of the gating mechanism in the Switch Transformer mimics random routing ( see Appendix A ) , we examine the effect of discarding the gate and randomly assigning inputs to experts . Figure 4 demonstrates the validation losses and BLEU scores of the Transformer and three variants of the Switch Transformer , where inputs are routed according to tokens ( referred to as Switch ( t ) ) , sentences ( Switch ( s ) ) , or are routed randomly ( Switch ( r ) ) . Similar to the results in Figure 1 , we see that the four models perform similarly . This shows that even after we alleviate load imbalance , model performance is not improved ( i.e. , the Switch Transformers do not outperform the vanilla Transformer ) , and the performance of the Switch Transformer does not vary much among different routing methods , including random routing . We remark that in this paper , we focus on natural language processing tasks , in particular neural machine translation . There are other works in different research fields ( e.g. , computer vision ) that draw different conclusions than ours ( Riquelme et al. , 2021 ) . We attribute this to the intrinsic differences between image classification and language generation , e.g. , each input in the former belongs to a clearly-defined category , while no such knowledge exists in the latter . In summary , the experiments reveal • A sparsely activated model does not always outperform a densely activated model of the same model size . • The widely-used routing method based on the gating mechanism does not work better than randomly routing inputs to experts . | The paper proposes a new routing mechanism for sparse models in the context of language tasks. Rather than learning a parametric router that learns how to assign tokens to experts, the proposed algorithm (THOR), randomly selects two experts per mini batch, and applies those experts independently to every input. A consistency loss is used to force experts to provide similar predictions. A number of experiments are provided suggesting the algorithm outperforms previous works. | SP:8e2d898e00e4f45d4ab48a2775334f67692785bb |
Taming Sparsely Activated Transformer with Stochastic Experts | 1 INTRODUCTION . Large neural network models have shown to be effective in many natural language processing tasks such as machine translation ( Lewis et al. , 2020 ; Conneau & Lample , 2019 ) , natural language understanding ( Devlin et al. , 2019 ; Liu et al. , 2019 ; He et al. , 2020 ) , and natural language generation ( Radford et al. , 2019 ; Brown et al. , 2020 ) . These models are usually densely activated . That is , a model uses all its parameters to process all inputs . One drawback of these models is the prohibitive training cost . Moreover , the extreme size drastically reduces inference speed , further limiting the models ’ practicality . To address these issues , sparsely activated models ( SAMs , Shazeer et al . 2017 ) have been proposed . A SAM adaptively selects a subset of its parameters for different inputs during model training and inference . This makes it possible to train SAMs that are an order of magnitude larger than densely activated models without significant increase in computational cost . For example , the sparsely activated GShard ( Lepikhin et al. , 2020 ) consists of over 600 billion parameters and the Switch Transformer ( Fedus et al. , 2021 ) 1.5 trillion parameters , while GPT-3 ( Brown et al. , 2020 ) , which is arguably the largest densely activated model , consists of only 175 billion parameters . The building block of SAMs is the expert layer , which contains an attention mechanism and multiple feed-forward neural networks ( FFNs ) in parallel . Each FFN is referred to as an expert . During ∗Work was done during an internship at Microsoft . training , an input is routed to a fixed number of experts , such that the number of floating point operations ( FLOPs ) of one forward pass remains constant , regardless of the total number of experts . Thus , training SAMs is much more cost-efficient than training densely activated models . For example , training of Switch-large ( Fedus et al. , 2021 ) and that of T5-large ( Raffel et al. , 2019 ) require the same forward FLOPs , despite that the former is 35 times larger ( 26.3 vs. 0.74 billion parameters ) . However , SAMs have been reported to be parameter inefficient . For example , although the Switchlarge model is 35 times larger than T5-large , its performance on the GLUE benchmark ( Wang et al. , 2019a ) is only slightly better ( 88.5 vs. 87.8 ) . There are also cases where the performance of SAMs is even worse than smaller densely activated models . For example , the performance of Switchlarge is worse than T5-large on the ARC Reasoning Challenge ( 66.0 vs. 68.8 ) ( Clark et al. , 2018 ) . In another example , although GShard ( Lepikhin et al. , 2020 ) shows substantial gains over densely activated models , a diminishing return with larger number of parameters has been observed . Most on-going research has focused on improving SAMs by developing effective routing methods . Since only a subset of model parameters ( i.e. , experts ) are updated for each input during training , we need to decide which experts to be activated given an input . Existing works ( Shazeer et al. , 2017 ; Lepikhin et al. , 2020 ; Fedus et al. , 2021 ; Yang et al. , 2021 ) use a gating network for input routing . However , the gating mechanism suffers from the notorious load imbalance issue : the gate ’ s weight could collapse such that nearly all the inputs are routed to the same expert . Therefore , many methods are proposed to mitigate this issue , such as noisy gating ( Shazeer et al. , 2017 ) , expert capacity ( Lepikhin et al. , 2020 ) , load balancing loss ( Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) , and k Top-1 gating ( Yang et al. , 2021 ) . However , these routing methods have not been proved effective to make SAMs more parameter efficient . To understand why SAMs are not parameter efficient , we analyze the performance of several classic MoE models . Our analysis reveals that a SAM does not always outperform a densely activated model of a similar size , confirming the results reported in Yang et al . ( 2021 ) . Moreover , we also observe that the widely-used routing method based on the gating mechanism does not work better than randomly routing inputs to experts , Inspired by our findings , we propose a new SAM , THOR ( Transformer witH StOchastic ExpeRts ) . Unlike classic SAMs , such as the Switch Transformer , experts in THOR are randomly activated ( with no need of any gating mechanism ) for each input during training and inference . THOR models are trained by minimizing both the cross-entropy loss and a consistency regularization term , such that experts can learn not only from training data but also from other experts as teachers so that all the experts make consistent predictions . To validate the effectiveness of THOR , we have conducted extensive experiments on machine translation using three settings : low-resource , rich-resource , and multilingual . Results show that THOR models outperform state-of-the-art MoE models by an average of 2 BLEU score on twelve low-resource translation tasks . In the rich-resource setting , THOR achieves new state-of-the-art results on the two widely-used translation benchmarks , WMT ’ 16 En-De and WMT ’ 14 En-Fr . On multilingual translation tasks , the THOR model with 300 million parameters achieves 2 BLEU score improvement over a state-of-the-art MoE model of the same size . Moreover , our model achieves state-of-the-art results on these tasks — the same BLEU score that is achieved by the Z-code MoE model ( Kim et al. , 2021 ) with 5.5 billion parameters ( 18 times larger ) . 2 BACKGROUND . Transformer . The Transformer ( Vaswani et al. , 2017 ) model has demonstrated its superior performance in many sequence-to-sequence natural language processing tasks , such as neural machine translation . The model contains an encoder and a decoder . The encoder consists of multiple encoder layers , each having an identical structure . An encoder layer employs a self-attention mechanism and a feed-forward neural network ( FFN ) . The decoder is similarly constructed , except for an additional cross-attention mechanism in each decoder layer . Sparsely Activated Models . The building block of SAMs is the expert layer , which is similar to the Transformer layer . Each of these expert layers contain an attention mechanism and multiple FFNs in parallel , where each FFN is referred to as an expert . Let { Ei } Ni=1 denote the experts , and N denotes the total number of experts . A gating mechanism decides to which expert ( s ) an input should be routed . At each expert layer , given an input vector x ∈ Rd , where d is the embedding dimension , the gate value of routing x to expert Ei is pi ( x ) = [ Softmax ( Wgx ) ] i , ( 1 ) where Wg ∈ RN×d is the trainable weight matrix of the gating mechanism . Given the gate values { pi ( x ) } Ni=1 , we select the top-K experts to form an activated set of experts T ⊂ { 1 · · ·N } , where |T | = K. Then the output xout of the expert layer is xout = ∑ i∈T pi ( x ) Ei ( x ) . ( 2 ) Notice that in Eq . 2 , input x only activates K instead of N experts , where K N , e.g. , K = 2 and N = 2048 in GShard ( Lepikhin et al. , 2020 ) . This implies that the number of FLOPs required for one forward pass does not increase with the number of experts N . Therefore , SAMs can scale to an enormous size without any significant increase in training time and inference time . The gate weight matrixWg ( Eq . 1 ) is trained together with the rest of the model parameters . Because there is no constraint on the learned weights , it is possible that Wg collapses such that one row dominates , i.e. , all the inputs are routed to one expert . This problem is referred to as load imbalance . Existing works adopt various ad-hoc heuristics to mitigate this issue , e.g. , adding Gaussian noise to Eq . 1 ( noisy gating , Shazeer et al . 2017 ) , limiting the maximum number of inputs that can be routed to an expert ( expert capacity , Lepikhin et al . 2020 ) , imposing a load balancing loss ( Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) , and using linear assignment ( Lewis et al. , 2021 ) . There are other works that remove the gating mechanism such that load imbalance is no longer an issue , e.g. , by incorporating hash functions ( Roller et al. , 2021 ) . Besides the load imbalance issue , there are also heated discussions on how to construct T in Eq . 2 . For example , Shazeer et al . ( 2017 ) ; Lepikhin et al . ( 2020 ) ; Yang et al . ( 2021 ) conjecture that routing inputs to K > 1 experts is necessary , while Fedus et al . ( 2021 ) argue that using K = 1 is sufficient and more computationally efficient . 3 ANALYSIS OF SPARSELY ACTIVATED MODELS . We investigate behavior of the gating mechanism of several classic MoE models . We conduct experiments on a multilingual translation task , { De , Vi } → En . More details are presented in Appendix A . We consider two MoE models proposed in Shen et al . ( 2019 ) , referred to as MoE ( dec ) and MoE ( tok ) , respectively , and three variants of the Switch Transformer proposed in Fedus et al . ( 2021 ) . The number of experts is set to two for all the MoE models . We compare them with the Transformer ( Vaswani et al. , 2017 ) model of the same model size . Figure 1 shows the validation losses and BLEU scores of three models : Transformer , MoE ( dec ) , and MoE ( tok ) . We see that the two MoE models perform very similarly , and neither outperforms the Transformer by a significant margin . To interpret the results of Figure 1 , we examine the load of each expert and the confidence scores of routing inputs to different experts . An expert ’ s load is defined as the proportion of inputs that are assigned to it . For an input that is routed to an expert , its routing confidence score ( output of the gating mechanism ) determines the level of preference , e.g. , if the routing confidence score is 0.5 , then the gate has no preference for either expert . For each expert , we compute the average routing confidence score over all the inputs assigned to it . Figure 2 shows that after the early stage of training ( i.e. , the first 200 iterations ) , the gate weight collapses and nearly all the inputs are routed to expert 2 . Also , the average routing confidence score of expert 2 is close to 1.0 , which means that the gate strongly prefers expert 2 to expert 1 . In this case , only one of the experts is sufficiently trained . Figure 3 depicts a different scenario , where the inputs are randomly dispatched to the experts . Notice that after approximately 4000 iterations , the two experts are equally loaded , and the probabilities of assigning any input to expert 1 and expert 2 are almost identical , indicating that the gating mechanism has no preference for either expert . We have identified two behaviors of the gating mechanism : load imbalance and random routing . The former is also reported in recent papers ( Shazeer et al. , 2017 ; Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) . We further investigate the Switch Transformer ( Fedus et al. , 2021 ) , which is a state-of-theart MoE variant that incorporates various methods to resolve the load imbalance issue . In addition , because behavior of the gating mechanism in the Switch Transformer mimics random routing ( see Appendix A ) , we examine the effect of discarding the gate and randomly assigning inputs to experts . Figure 4 demonstrates the validation losses and BLEU scores of the Transformer and three variants of the Switch Transformer , where inputs are routed according to tokens ( referred to as Switch ( t ) ) , sentences ( Switch ( s ) ) , or are routed randomly ( Switch ( r ) ) . Similar to the results in Figure 1 , we see that the four models perform similarly . This shows that even after we alleviate load imbalance , model performance is not improved ( i.e. , the Switch Transformers do not outperform the vanilla Transformer ) , and the performance of the Switch Transformer does not vary much among different routing methods , including random routing . We remark that in this paper , we focus on natural language processing tasks , in particular neural machine translation . There are other works in different research fields ( e.g. , computer vision ) that draw different conclusions than ours ( Riquelme et al. , 2021 ) . We attribute this to the intrinsic differences between image classification and language generation , e.g. , each input in the former belongs to a clearly-defined category , while no such knowledge exists in the latter . In summary , the experiments reveal • A sparsely activated model does not always outperform a densely activated model of the same model size . • The widely-used routing method based on the gating mechanism does not work better than randomly routing inputs to experts . | This paper proposed to use randomly selected experts for Mixture-of-Experts models instead of gating function based selection methods. To avoid large performance variance of the random selection during inference, the paper proposed to add a consistency regularization which drives the similarity between different experts. The experimental results show its superior performance over existing MoE models such as switch transformers. | SP:8e2d898e00e4f45d4ab48a2775334f67692785bb |
Taming Sparsely Activated Transformer with Stochastic Experts | 1 INTRODUCTION . Large neural network models have shown to be effective in many natural language processing tasks such as machine translation ( Lewis et al. , 2020 ; Conneau & Lample , 2019 ) , natural language understanding ( Devlin et al. , 2019 ; Liu et al. , 2019 ; He et al. , 2020 ) , and natural language generation ( Radford et al. , 2019 ; Brown et al. , 2020 ) . These models are usually densely activated . That is , a model uses all its parameters to process all inputs . One drawback of these models is the prohibitive training cost . Moreover , the extreme size drastically reduces inference speed , further limiting the models ’ practicality . To address these issues , sparsely activated models ( SAMs , Shazeer et al . 2017 ) have been proposed . A SAM adaptively selects a subset of its parameters for different inputs during model training and inference . This makes it possible to train SAMs that are an order of magnitude larger than densely activated models without significant increase in computational cost . For example , the sparsely activated GShard ( Lepikhin et al. , 2020 ) consists of over 600 billion parameters and the Switch Transformer ( Fedus et al. , 2021 ) 1.5 trillion parameters , while GPT-3 ( Brown et al. , 2020 ) , which is arguably the largest densely activated model , consists of only 175 billion parameters . The building block of SAMs is the expert layer , which contains an attention mechanism and multiple feed-forward neural networks ( FFNs ) in parallel . Each FFN is referred to as an expert . During ∗Work was done during an internship at Microsoft . training , an input is routed to a fixed number of experts , such that the number of floating point operations ( FLOPs ) of one forward pass remains constant , regardless of the total number of experts . Thus , training SAMs is much more cost-efficient than training densely activated models . For example , training of Switch-large ( Fedus et al. , 2021 ) and that of T5-large ( Raffel et al. , 2019 ) require the same forward FLOPs , despite that the former is 35 times larger ( 26.3 vs. 0.74 billion parameters ) . However , SAMs have been reported to be parameter inefficient . For example , although the Switchlarge model is 35 times larger than T5-large , its performance on the GLUE benchmark ( Wang et al. , 2019a ) is only slightly better ( 88.5 vs. 87.8 ) . There are also cases where the performance of SAMs is even worse than smaller densely activated models . For example , the performance of Switchlarge is worse than T5-large on the ARC Reasoning Challenge ( 66.0 vs. 68.8 ) ( Clark et al. , 2018 ) . In another example , although GShard ( Lepikhin et al. , 2020 ) shows substantial gains over densely activated models , a diminishing return with larger number of parameters has been observed . Most on-going research has focused on improving SAMs by developing effective routing methods . Since only a subset of model parameters ( i.e. , experts ) are updated for each input during training , we need to decide which experts to be activated given an input . Existing works ( Shazeer et al. , 2017 ; Lepikhin et al. , 2020 ; Fedus et al. , 2021 ; Yang et al. , 2021 ) use a gating network for input routing . However , the gating mechanism suffers from the notorious load imbalance issue : the gate ’ s weight could collapse such that nearly all the inputs are routed to the same expert . Therefore , many methods are proposed to mitigate this issue , such as noisy gating ( Shazeer et al. , 2017 ) , expert capacity ( Lepikhin et al. , 2020 ) , load balancing loss ( Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) , and k Top-1 gating ( Yang et al. , 2021 ) . However , these routing methods have not been proved effective to make SAMs more parameter efficient . To understand why SAMs are not parameter efficient , we analyze the performance of several classic MoE models . Our analysis reveals that a SAM does not always outperform a densely activated model of a similar size , confirming the results reported in Yang et al . ( 2021 ) . Moreover , we also observe that the widely-used routing method based on the gating mechanism does not work better than randomly routing inputs to experts , Inspired by our findings , we propose a new SAM , THOR ( Transformer witH StOchastic ExpeRts ) . Unlike classic SAMs , such as the Switch Transformer , experts in THOR are randomly activated ( with no need of any gating mechanism ) for each input during training and inference . THOR models are trained by minimizing both the cross-entropy loss and a consistency regularization term , such that experts can learn not only from training data but also from other experts as teachers so that all the experts make consistent predictions . To validate the effectiveness of THOR , we have conducted extensive experiments on machine translation using three settings : low-resource , rich-resource , and multilingual . Results show that THOR models outperform state-of-the-art MoE models by an average of 2 BLEU score on twelve low-resource translation tasks . In the rich-resource setting , THOR achieves new state-of-the-art results on the two widely-used translation benchmarks , WMT ’ 16 En-De and WMT ’ 14 En-Fr . On multilingual translation tasks , the THOR model with 300 million parameters achieves 2 BLEU score improvement over a state-of-the-art MoE model of the same size . Moreover , our model achieves state-of-the-art results on these tasks — the same BLEU score that is achieved by the Z-code MoE model ( Kim et al. , 2021 ) with 5.5 billion parameters ( 18 times larger ) . 2 BACKGROUND . Transformer . The Transformer ( Vaswani et al. , 2017 ) model has demonstrated its superior performance in many sequence-to-sequence natural language processing tasks , such as neural machine translation . The model contains an encoder and a decoder . The encoder consists of multiple encoder layers , each having an identical structure . An encoder layer employs a self-attention mechanism and a feed-forward neural network ( FFN ) . The decoder is similarly constructed , except for an additional cross-attention mechanism in each decoder layer . Sparsely Activated Models . The building block of SAMs is the expert layer , which is similar to the Transformer layer . Each of these expert layers contain an attention mechanism and multiple FFNs in parallel , where each FFN is referred to as an expert . Let { Ei } Ni=1 denote the experts , and N denotes the total number of experts . A gating mechanism decides to which expert ( s ) an input should be routed . At each expert layer , given an input vector x ∈ Rd , where d is the embedding dimension , the gate value of routing x to expert Ei is pi ( x ) = [ Softmax ( Wgx ) ] i , ( 1 ) where Wg ∈ RN×d is the trainable weight matrix of the gating mechanism . Given the gate values { pi ( x ) } Ni=1 , we select the top-K experts to form an activated set of experts T ⊂ { 1 · · ·N } , where |T | = K. Then the output xout of the expert layer is xout = ∑ i∈T pi ( x ) Ei ( x ) . ( 2 ) Notice that in Eq . 2 , input x only activates K instead of N experts , where K N , e.g. , K = 2 and N = 2048 in GShard ( Lepikhin et al. , 2020 ) . This implies that the number of FLOPs required for one forward pass does not increase with the number of experts N . Therefore , SAMs can scale to an enormous size without any significant increase in training time and inference time . The gate weight matrixWg ( Eq . 1 ) is trained together with the rest of the model parameters . Because there is no constraint on the learned weights , it is possible that Wg collapses such that one row dominates , i.e. , all the inputs are routed to one expert . This problem is referred to as load imbalance . Existing works adopt various ad-hoc heuristics to mitigate this issue , e.g. , adding Gaussian noise to Eq . 1 ( noisy gating , Shazeer et al . 2017 ) , limiting the maximum number of inputs that can be routed to an expert ( expert capacity , Lepikhin et al . 2020 ) , imposing a load balancing loss ( Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) , and using linear assignment ( Lewis et al. , 2021 ) . There are other works that remove the gating mechanism such that load imbalance is no longer an issue , e.g. , by incorporating hash functions ( Roller et al. , 2021 ) . Besides the load imbalance issue , there are also heated discussions on how to construct T in Eq . 2 . For example , Shazeer et al . ( 2017 ) ; Lepikhin et al . ( 2020 ) ; Yang et al . ( 2021 ) conjecture that routing inputs to K > 1 experts is necessary , while Fedus et al . ( 2021 ) argue that using K = 1 is sufficient and more computationally efficient . 3 ANALYSIS OF SPARSELY ACTIVATED MODELS . We investigate behavior of the gating mechanism of several classic MoE models . We conduct experiments on a multilingual translation task , { De , Vi } → En . More details are presented in Appendix A . We consider two MoE models proposed in Shen et al . ( 2019 ) , referred to as MoE ( dec ) and MoE ( tok ) , respectively , and three variants of the Switch Transformer proposed in Fedus et al . ( 2021 ) . The number of experts is set to two for all the MoE models . We compare them with the Transformer ( Vaswani et al. , 2017 ) model of the same model size . Figure 1 shows the validation losses and BLEU scores of three models : Transformer , MoE ( dec ) , and MoE ( tok ) . We see that the two MoE models perform very similarly , and neither outperforms the Transformer by a significant margin . To interpret the results of Figure 1 , we examine the load of each expert and the confidence scores of routing inputs to different experts . An expert ’ s load is defined as the proportion of inputs that are assigned to it . For an input that is routed to an expert , its routing confidence score ( output of the gating mechanism ) determines the level of preference , e.g. , if the routing confidence score is 0.5 , then the gate has no preference for either expert . For each expert , we compute the average routing confidence score over all the inputs assigned to it . Figure 2 shows that after the early stage of training ( i.e. , the first 200 iterations ) , the gate weight collapses and nearly all the inputs are routed to expert 2 . Also , the average routing confidence score of expert 2 is close to 1.0 , which means that the gate strongly prefers expert 2 to expert 1 . In this case , only one of the experts is sufficiently trained . Figure 3 depicts a different scenario , where the inputs are randomly dispatched to the experts . Notice that after approximately 4000 iterations , the two experts are equally loaded , and the probabilities of assigning any input to expert 1 and expert 2 are almost identical , indicating that the gating mechanism has no preference for either expert . We have identified two behaviors of the gating mechanism : load imbalance and random routing . The former is also reported in recent papers ( Shazeer et al. , 2017 ; Lepikhin et al. , 2020 ; Fedus et al. , 2021 ) . We further investigate the Switch Transformer ( Fedus et al. , 2021 ) , which is a state-of-theart MoE variant that incorporates various methods to resolve the load imbalance issue . In addition , because behavior of the gating mechanism in the Switch Transformer mimics random routing ( see Appendix A ) , we examine the effect of discarding the gate and randomly assigning inputs to experts . Figure 4 demonstrates the validation losses and BLEU scores of the Transformer and three variants of the Switch Transformer , where inputs are routed according to tokens ( referred to as Switch ( t ) ) , sentences ( Switch ( s ) ) , or are routed randomly ( Switch ( r ) ) . Similar to the results in Figure 1 , we see that the four models perform similarly . This shows that even after we alleviate load imbalance , model performance is not improved ( i.e. , the Switch Transformers do not outperform the vanilla Transformer ) , and the performance of the Switch Transformer does not vary much among different routing methods , including random routing . We remark that in this paper , we focus on natural language processing tasks , in particular neural machine translation . There are other works in different research fields ( e.g. , computer vision ) that draw different conclusions than ours ( Riquelme et al. , 2021 ) . We attribute this to the intrinsic differences between image classification and language generation , e.g. , each input in the former belongs to a clearly-defined category , while no such knowledge exists in the latter . In summary , the experiments reveal • A sparsely activated model does not always outperform a densely activated model of the same model size . • The widely-used routing method based on the gating mechanism does not work better than randomly routing inputs to experts . | The paper proposes THoR, an approach towards training MoE-like models (called SAMs in the paper) that have multiple internal experts which are chosen in a discrete fashion. Unlike other common methods, which generally incorporate a form of router which maps input to an specific router choice, THoR makes random, global, per-minibatch decisions on experts, and includes a cross-expert regularization term to help align experts to each other. This paper reports that THoR outperforms other methods, including the recent Switch Transformer MoE-style model, on several classes of multilingual translation tasks. | SP:8e2d898e00e4f45d4ab48a2775334f67692785bb |
Statistically Meaningful Approximation: a Theoretical Analysis for Approximating Turing Machines with Transformers | 1 INTRODUCTION . Dating back to the seminal works on universal approximation ( Cybenko , 1989 ; Hornik et al. , 1989 ; Park & Sandberg , 1991 ; Leshno et al. , 1993 ) , a common way to theoretically study neural nets has been through their expressivity , which measures the ability of neural nets to approximate well-behaved functions . This perspective has shaped how researchers perceive different types of deep learning architectures : a basic way to theoretically justify new architectures is to study their approximation capabilities . This has led to a number of analyses studying universal approximation capabilities for various widely-used architectures , such as recurrent neural nets ( RNNs ) ( Schäfer & Zimmermann , 2007 ) , graph neural nets ( Scarselli et al. , 2008 ) , convolutional networks ( Bao et al. , 2014 ; Zhou , 2020 ; Yarotsky , 2021 ) , residual networks ( Lin & Jegelka , 2018 ) , transformers ( Yun et al. , 2019 ) , and neural ODEs ( Teshima et al. , 2020 ; Zhang et al. , 2020 ) . However , approximation theoretic results often misalign with more meaningful end-to-end guarantees , because models constructed in the literature often exhibit unrealistic properties . For example , a common technique in the universal approximation literature is to rely strongly on infinite-precision weights and activations , or exponentially many parameters to encode the desired function values ( Hornik et al. , 1989 ; Cybenko , 1989 ; Leshno et al. , 1993 ; Lin & Jegelka , 2018 ; Yun et al. , 2019 ; Sannai et al. , 2019 ) . This issue even arises outside of universal approximation , e.g. , various papers demonstrate the ability of RNNs and transformers to simulate various computational models such as Turing machines and automata , but require strong reliance on arbitrary precision ( Siegelmann & Sontag , 1995 ; Pérez et al. , 2019 ; Korsky & Berwick , 2019 ; Bhattamishra et al. , 2020 ) . Infinite precision can inflate the expressivity of an architecture in a unrealistic and misleading way : for example , finite width RNNs with infinite precision can simulate Turing machines , but finite-precision , finite-width RNNs can not , as implied by streaming lower bounds ( Alon et al. , 1999 ) . As another example , Park et al . ( 2020 ) exploit infinite precision in the parameters to show that a neural net with parameter count sublinear in n can memorize n arbitrary input-label pairs . However , a simple counting argument reveals that this result can not be proven using finite precision networks – there are 2n input-labeling pairs , but only 2opnq finite precision networks with opnq parameters . More broadly , the ideal theoretical perspective should consider not only whether target functions can be expressed , but also whether the constructed networks are plausibly learnable . Learnability is important because empirical settings do not operate in the infinite data , unbounded computation regime – they require fitting the target function with access to limited number of samples from an empirical distribution . The question of studying learnability can be decomposed into studying optimization and generalization . Unfortunately , a rigorous analysis of optimization is unresolved even for simple two-layer nets ( Mei et al. , 2018 ) . Generalization is more tractable , so we propose to study expressivity and generalization together . Towards the goal of studying more meaningful notions of approximation , this work proposes the notion of statistically meaningful ( SM ) approximation . This definition requires not only the existence of an approximating network , but also that it has good statistical learnability . Consider a setting where the aim is to fit the target functionG using the approximating family F and a finite sample of training data . SM approximation requires existence of a loss whose empirical risk minimizer in F leads to a model with low approximation error in fittingG . We define the sample complexity of the approximation as the number of training samples needed to guarantee at most approximation error and study SM approximation with low sample complexity bounds . SM approximation essentially eliminates all statistical concerns for learnability ( optimization-related concerns can remain ) . We present two case studies on SM approximation . First , we demonstrate that overparameterized feedforward neural nets can SM approximate boolean circuits with a low sample complexity that depends only on the intrinsic circuit size . Though it is simple to construct neural nets to approximate boolean circuits , bounding the sample complexity of the approximation is challenging . For example , standard norm-based generalization bounds for the naive construction scale exponentially in depth ( Bartlett et al. , 2017 ) . Furthermore , VC dimension-based bounds would scale polynomially in the number of parameters in the network ( Harvey et al. , 2017 ) , which is problematic because for practical optimization concerns , neural nets are typically overparameterized in terms of width ( Zhang et al. , 2016 ) . In contrast , our sample complexity bound for SM approximation depends only on the intrinsic circuit size , up to logarithmic factors . Our second case study is on SM approximating Turing machines with transformers . We consider a class of Turing machines with bounded computation time T and construct encoder-decoder-based transformers ( Vaswani et al. , 2017 ) which SM approximate these Turing machines . The sample complexity of the approximation depends on a polynomial in logT and the sizes of the state space and the alphabet of the Turing machine . Though constructions for approximating Turing machines from prior work ( Siegelmann & Sontag , 1995 ; Pérez et al. , 2019 ; Bhattamishra et al. , 2020 ) have not been formally studied from a sample complexity perspective , existing bounds would depend at least linearly on T . Furthermore , our construction only uses loglogT precision , compared to at least logT in prior works , allowing us to achieve the exponential improvement in the sample complexity . Proving sample complexity guarantees for our statistically meaningful approximation results is nontrivial and requires additional insights , for both the constructions and the generalization analyses . To obtain our sample complexity bounds , we leverage a recent approach to bound generalization in terms of data-dependent notions of Lipschitzness ( Wei & Ma , 2019b ) . We develop theoretical tools to convert a broad class of neural nets , with possibly large Lipschitzness , into ones with small Lipschitzness on the training data , by introducing a number of new layers that is linear in depth . Our result applies to neural nets where each entry in the hidden representations on the training data takes values from a finite set ( e.g. , binary entries ) , and may be of independent interest . In summary , our contributions are : 1 ) we propose a new notion of statistically meaningful approximation , intended to provide more meaningful approximation guarantees by requiring that the approximating family have good statistical learnability ; 2 ) we prove that feedforward neural nets can meaningfully approximate boolean circuits with sample complexity that depends polynomially on the width and depth of the circuit ; and 3 ) we show that transformers can meaningfully approximate Turing machines with sample complexity logarithmic in the computation time . 1.1 RELATED WORKS . Classifical approximation theory for neural networks has a long history . Hornik et al . ( 1989 ) ; Cybenko ( 1989 ) , and Leshno et al . ( 1993 ) show that neural nets with one hidden layer are universal approximators but require the hidden layer size to grow exponentially in input dimension . Barron ( 1993 ) uses the Fourier transform to write target functions as infinite-width networks and subsamples neurons to obtain widths which depend only on target function properties . Lee et al . ( 2017 ) ; Ji et al . ( 2020 ) prove recent related developments in this direction of universal approximation . Many works study benefits of deep networks over shallow ones ( Bengio & Delalleau , 2011 ; Arora et al. , 2016 ; Telgarsky , 2016 ; Eldan & Shamir , 2016 ; Daniely , 2017 ; Chatziafratis et al. , 2020 ; 2019 ) . Bengio & Delalleau ( 2011 ) show separation for exact representation , whereas Telgarsky ( 2016 ) shows separation for approximate representations with univariate inputs . Eldan & Shamir ( 2016 ) demonstrate high-dimensional functions that can be approximated by two-layer polynomial-sized neural networks , but can not be approximated by one-layer neural nets with subexponential hidden units . Via reduction to certain complexity theo- retic questions , Vardi & Shamir ( 2020 ) show that proving constant depth separations may be hard . Malach et al . ( 2021 ) analyze the relationship between optimization and approximability , showing in various settings that deeper networks can not be optimized if shallow networks can not approximate them . This demonstrates that depth separation results ( Telgarsky , 2016 ) from approximation theory can be misleading in the sense that gradient descent anyways can not optimize the deep networks used to construct the approximation . Another area of study is on the ability of deep networks to memorize training data ( Zhang et al. , 2016 ; Yun et al. , 2018 ; Park et al. , 2020 ; Vershynin , 2020 ) . Yun et al . ( 2018 ) show that Θpnq parameters are sufficient to memorize Θpnq training points for ReLU nets with at least 3 layers , and Park et al . ( 2020 ) reduce the parameter requirement to sublinear in n. Similar results have been proven for residual architectures ( Hardt & Ma , 2016 ) and convolutional nets ( Nguyen & Hein , 2018 ) . Bartlett et al . ( 2019 ) analyze the VC-dimension of neural nets , leading to upper and lower bounds on the parameter count needed to fit training data . Other works study expressivity via connections to tensor approximation and sum-product networks ( Cohen & Shashua , 2016 ; Cohen et al. , 2016 ) . There is a long line of work on studying the ability of neural nets to recognize and represent formal languages . The seminal work of Siegelmann & Sontag ( 1995 ) shows that RNNs are Turing complete but leverages infinite precision in the hidden activations . Chen et al . ( 2018 ) extend this result to ReLU activations and study implications in language modeling . Many variants of transformers are shown to be Turing-complete , but these constructions also rely on arbitrary precision ( Pérez et al. , 2019 ; Bhattamishra et al. , 2020 ) . A number of recent works have also proven results for generating or recognizing formal languages with finite-precision neural nets ( Weiss et al. , 2018 ; Korsky & Berwick , 2019 ; Hewitt et al. , 2020 ) , but these results do not consider Turing machines or analyze statistical properties of their constructions . Bounding the sample complexity of SM approximation requires additional complications in both the construction and statistical analysis . 1.2 NOTATION . Let f˝g denote the composition of functions f and g. For a family of functions G , let f˝Gfitf˝g : gPGu denote the family of compositions between f and functions in G. For a set S and function f : SÑY , let fpSq denote the set tfpsq : sPSuĎY . We use 1d to denote the all-one ’ s vector in d dimensions , with the subscripted omitted if clear . For iPrds , we let 1dpiq denote the one-hot embedding in d-dimensions , which is 1 at index i and 0 everywhere else . We use the notation rOp¨q to hide poly-logarithmic factors in the argument . The notationÀ , Á indicates the existence of a constant factor such that the inequality holds . — denotes that theÁ andÀ relations simultaneously hold . We use polyp¨q to indicate the existence of a polynomial in the argument which makes the equation true . For a setA ( e.g. , the set of alphabet symbols for a Turing machine ) let A˚ denote the set of all sequences of elements of A , where sequence length can vary . Let P denote a distribution over a space of inputs X . Let ξ1 , ... , ξn be n i.i.d . Rademacher variables sampled from t´1 , ` 1u . The expected n-sample Rademacher complexity of F on P is as follows : Radn , P pFqfi E pxiqni “ 1 i.i.d „ P “ Eξ1 , ... , ξn “ supFPF 1 n řn i “ 1ξiFpxiq ‰‰ , where pxiqni “ 1 denotes n i.i.d . samples from P . | This paper studies the approximation of boolean circuits and Turing machines using neural networks. To this end, the authors propose a new notion of approximation criteria -- statistically meaningful (SM) approximation -- that specifies that the function class is both learnable and has a small bias in the sense of classical approximation theory. For both the boolean circuits and Turing machines, the authors explicitly construct two classes of neural networks, based on feedforward network and transformers respectively, that achieve SM approximation. | SP:0c0f4862966a6374c299012d302bb95e7dc88dea |
Statistically Meaningful Approximation: a Theoretical Analysis for Approximating Turing Machines with Transformers | 1 INTRODUCTION . Dating back to the seminal works on universal approximation ( Cybenko , 1989 ; Hornik et al. , 1989 ; Park & Sandberg , 1991 ; Leshno et al. , 1993 ) , a common way to theoretically study neural nets has been through their expressivity , which measures the ability of neural nets to approximate well-behaved functions . This perspective has shaped how researchers perceive different types of deep learning architectures : a basic way to theoretically justify new architectures is to study their approximation capabilities . This has led to a number of analyses studying universal approximation capabilities for various widely-used architectures , such as recurrent neural nets ( RNNs ) ( Schäfer & Zimmermann , 2007 ) , graph neural nets ( Scarselli et al. , 2008 ) , convolutional networks ( Bao et al. , 2014 ; Zhou , 2020 ; Yarotsky , 2021 ) , residual networks ( Lin & Jegelka , 2018 ) , transformers ( Yun et al. , 2019 ) , and neural ODEs ( Teshima et al. , 2020 ; Zhang et al. , 2020 ) . However , approximation theoretic results often misalign with more meaningful end-to-end guarantees , because models constructed in the literature often exhibit unrealistic properties . For example , a common technique in the universal approximation literature is to rely strongly on infinite-precision weights and activations , or exponentially many parameters to encode the desired function values ( Hornik et al. , 1989 ; Cybenko , 1989 ; Leshno et al. , 1993 ; Lin & Jegelka , 2018 ; Yun et al. , 2019 ; Sannai et al. , 2019 ) . This issue even arises outside of universal approximation , e.g. , various papers demonstrate the ability of RNNs and transformers to simulate various computational models such as Turing machines and automata , but require strong reliance on arbitrary precision ( Siegelmann & Sontag , 1995 ; Pérez et al. , 2019 ; Korsky & Berwick , 2019 ; Bhattamishra et al. , 2020 ) . Infinite precision can inflate the expressivity of an architecture in a unrealistic and misleading way : for example , finite width RNNs with infinite precision can simulate Turing machines , but finite-precision , finite-width RNNs can not , as implied by streaming lower bounds ( Alon et al. , 1999 ) . As another example , Park et al . ( 2020 ) exploit infinite precision in the parameters to show that a neural net with parameter count sublinear in n can memorize n arbitrary input-label pairs . However , a simple counting argument reveals that this result can not be proven using finite precision networks – there are 2n input-labeling pairs , but only 2opnq finite precision networks with opnq parameters . More broadly , the ideal theoretical perspective should consider not only whether target functions can be expressed , but also whether the constructed networks are plausibly learnable . Learnability is important because empirical settings do not operate in the infinite data , unbounded computation regime – they require fitting the target function with access to limited number of samples from an empirical distribution . The question of studying learnability can be decomposed into studying optimization and generalization . Unfortunately , a rigorous analysis of optimization is unresolved even for simple two-layer nets ( Mei et al. , 2018 ) . Generalization is more tractable , so we propose to study expressivity and generalization together . Towards the goal of studying more meaningful notions of approximation , this work proposes the notion of statistically meaningful ( SM ) approximation . This definition requires not only the existence of an approximating network , but also that it has good statistical learnability . Consider a setting where the aim is to fit the target functionG using the approximating family F and a finite sample of training data . SM approximation requires existence of a loss whose empirical risk minimizer in F leads to a model with low approximation error in fittingG . We define the sample complexity of the approximation as the number of training samples needed to guarantee at most approximation error and study SM approximation with low sample complexity bounds . SM approximation essentially eliminates all statistical concerns for learnability ( optimization-related concerns can remain ) . We present two case studies on SM approximation . First , we demonstrate that overparameterized feedforward neural nets can SM approximate boolean circuits with a low sample complexity that depends only on the intrinsic circuit size . Though it is simple to construct neural nets to approximate boolean circuits , bounding the sample complexity of the approximation is challenging . For example , standard norm-based generalization bounds for the naive construction scale exponentially in depth ( Bartlett et al. , 2017 ) . Furthermore , VC dimension-based bounds would scale polynomially in the number of parameters in the network ( Harvey et al. , 2017 ) , which is problematic because for practical optimization concerns , neural nets are typically overparameterized in terms of width ( Zhang et al. , 2016 ) . In contrast , our sample complexity bound for SM approximation depends only on the intrinsic circuit size , up to logarithmic factors . Our second case study is on SM approximating Turing machines with transformers . We consider a class of Turing machines with bounded computation time T and construct encoder-decoder-based transformers ( Vaswani et al. , 2017 ) which SM approximate these Turing machines . The sample complexity of the approximation depends on a polynomial in logT and the sizes of the state space and the alphabet of the Turing machine . Though constructions for approximating Turing machines from prior work ( Siegelmann & Sontag , 1995 ; Pérez et al. , 2019 ; Bhattamishra et al. , 2020 ) have not been formally studied from a sample complexity perspective , existing bounds would depend at least linearly on T . Furthermore , our construction only uses loglogT precision , compared to at least logT in prior works , allowing us to achieve the exponential improvement in the sample complexity . Proving sample complexity guarantees for our statistically meaningful approximation results is nontrivial and requires additional insights , for both the constructions and the generalization analyses . To obtain our sample complexity bounds , we leverage a recent approach to bound generalization in terms of data-dependent notions of Lipschitzness ( Wei & Ma , 2019b ) . We develop theoretical tools to convert a broad class of neural nets , with possibly large Lipschitzness , into ones with small Lipschitzness on the training data , by introducing a number of new layers that is linear in depth . Our result applies to neural nets where each entry in the hidden representations on the training data takes values from a finite set ( e.g. , binary entries ) , and may be of independent interest . In summary , our contributions are : 1 ) we propose a new notion of statistically meaningful approximation , intended to provide more meaningful approximation guarantees by requiring that the approximating family have good statistical learnability ; 2 ) we prove that feedforward neural nets can meaningfully approximate boolean circuits with sample complexity that depends polynomially on the width and depth of the circuit ; and 3 ) we show that transformers can meaningfully approximate Turing machines with sample complexity logarithmic in the computation time . 1.1 RELATED WORKS . Classifical approximation theory for neural networks has a long history . Hornik et al . ( 1989 ) ; Cybenko ( 1989 ) , and Leshno et al . ( 1993 ) show that neural nets with one hidden layer are universal approximators but require the hidden layer size to grow exponentially in input dimension . Barron ( 1993 ) uses the Fourier transform to write target functions as infinite-width networks and subsamples neurons to obtain widths which depend only on target function properties . Lee et al . ( 2017 ) ; Ji et al . ( 2020 ) prove recent related developments in this direction of universal approximation . Many works study benefits of deep networks over shallow ones ( Bengio & Delalleau , 2011 ; Arora et al. , 2016 ; Telgarsky , 2016 ; Eldan & Shamir , 2016 ; Daniely , 2017 ; Chatziafratis et al. , 2020 ; 2019 ) . Bengio & Delalleau ( 2011 ) show separation for exact representation , whereas Telgarsky ( 2016 ) shows separation for approximate representations with univariate inputs . Eldan & Shamir ( 2016 ) demonstrate high-dimensional functions that can be approximated by two-layer polynomial-sized neural networks , but can not be approximated by one-layer neural nets with subexponential hidden units . Via reduction to certain complexity theo- retic questions , Vardi & Shamir ( 2020 ) show that proving constant depth separations may be hard . Malach et al . ( 2021 ) analyze the relationship between optimization and approximability , showing in various settings that deeper networks can not be optimized if shallow networks can not approximate them . This demonstrates that depth separation results ( Telgarsky , 2016 ) from approximation theory can be misleading in the sense that gradient descent anyways can not optimize the deep networks used to construct the approximation . Another area of study is on the ability of deep networks to memorize training data ( Zhang et al. , 2016 ; Yun et al. , 2018 ; Park et al. , 2020 ; Vershynin , 2020 ) . Yun et al . ( 2018 ) show that Θpnq parameters are sufficient to memorize Θpnq training points for ReLU nets with at least 3 layers , and Park et al . ( 2020 ) reduce the parameter requirement to sublinear in n. Similar results have been proven for residual architectures ( Hardt & Ma , 2016 ) and convolutional nets ( Nguyen & Hein , 2018 ) . Bartlett et al . ( 2019 ) analyze the VC-dimension of neural nets , leading to upper and lower bounds on the parameter count needed to fit training data . Other works study expressivity via connections to tensor approximation and sum-product networks ( Cohen & Shashua , 2016 ; Cohen et al. , 2016 ) . There is a long line of work on studying the ability of neural nets to recognize and represent formal languages . The seminal work of Siegelmann & Sontag ( 1995 ) shows that RNNs are Turing complete but leverages infinite precision in the hidden activations . Chen et al . ( 2018 ) extend this result to ReLU activations and study implications in language modeling . Many variants of transformers are shown to be Turing-complete , but these constructions also rely on arbitrary precision ( Pérez et al. , 2019 ; Bhattamishra et al. , 2020 ) . A number of recent works have also proven results for generating or recognizing formal languages with finite-precision neural nets ( Weiss et al. , 2018 ; Korsky & Berwick , 2019 ; Hewitt et al. , 2020 ) , but these results do not consider Turing machines or analyze statistical properties of their constructions . Bounding the sample complexity of SM approximation requires additional complications in both the construction and statistical analysis . 1.2 NOTATION . Let f˝g denote the composition of functions f and g. For a family of functions G , let f˝Gfitf˝g : gPGu denote the family of compositions between f and functions in G. For a set S and function f : SÑY , let fpSq denote the set tfpsq : sPSuĎY . We use 1d to denote the all-one ’ s vector in d dimensions , with the subscripted omitted if clear . For iPrds , we let 1dpiq denote the one-hot embedding in d-dimensions , which is 1 at index i and 0 everywhere else . We use the notation rOp¨q to hide poly-logarithmic factors in the argument . The notationÀ , Á indicates the existence of a constant factor such that the inequality holds . — denotes that theÁ andÀ relations simultaneously hold . We use polyp¨q to indicate the existence of a polynomial in the argument which makes the equation true . For a setA ( e.g. , the set of alphabet symbols for a Turing machine ) let A˚ denote the set of all sequences of elements of A , where sequence length can vary . Let P denote a distribution over a space of inputs X . Let ξ1 , ... , ξn be n i.i.d . Rademacher variables sampled from t´1 , ` 1u . The expected n-sample Rademacher complexity of F on P is as follows : Radn , P pFqfi E pxiqni “ 1 i.i.d „ P “ Eξ1 , ... , ξn “ supFPF 1 n řn i “ 1ξiFpxiq ‰‰ , where pxiqni “ 1 denotes n i.i.d . samples from P . | This paper proposes a definition of "statistically meaningful approximation" or SM-approximation which combines learnability, generalization ability and expressivity (approximation accuracy) in a straightforward way. Two theorems on SM-approximation are formulated and proven, one for approximating Boolean circuits and one for approximating a general Turing machine computation with a transformer network. It is argued that the new definition has technical advantages. | SP:0c0f4862966a6374c299012d302bb95e7dc88dea |
Statistically Meaningful Approximation: a Theoretical Analysis for Approximating Turing Machines with Transformers | 1 INTRODUCTION . Dating back to the seminal works on universal approximation ( Cybenko , 1989 ; Hornik et al. , 1989 ; Park & Sandberg , 1991 ; Leshno et al. , 1993 ) , a common way to theoretically study neural nets has been through their expressivity , which measures the ability of neural nets to approximate well-behaved functions . This perspective has shaped how researchers perceive different types of deep learning architectures : a basic way to theoretically justify new architectures is to study their approximation capabilities . This has led to a number of analyses studying universal approximation capabilities for various widely-used architectures , such as recurrent neural nets ( RNNs ) ( Schäfer & Zimmermann , 2007 ) , graph neural nets ( Scarselli et al. , 2008 ) , convolutional networks ( Bao et al. , 2014 ; Zhou , 2020 ; Yarotsky , 2021 ) , residual networks ( Lin & Jegelka , 2018 ) , transformers ( Yun et al. , 2019 ) , and neural ODEs ( Teshima et al. , 2020 ; Zhang et al. , 2020 ) . However , approximation theoretic results often misalign with more meaningful end-to-end guarantees , because models constructed in the literature often exhibit unrealistic properties . For example , a common technique in the universal approximation literature is to rely strongly on infinite-precision weights and activations , or exponentially many parameters to encode the desired function values ( Hornik et al. , 1989 ; Cybenko , 1989 ; Leshno et al. , 1993 ; Lin & Jegelka , 2018 ; Yun et al. , 2019 ; Sannai et al. , 2019 ) . This issue even arises outside of universal approximation , e.g. , various papers demonstrate the ability of RNNs and transformers to simulate various computational models such as Turing machines and automata , but require strong reliance on arbitrary precision ( Siegelmann & Sontag , 1995 ; Pérez et al. , 2019 ; Korsky & Berwick , 2019 ; Bhattamishra et al. , 2020 ) . Infinite precision can inflate the expressivity of an architecture in a unrealistic and misleading way : for example , finite width RNNs with infinite precision can simulate Turing machines , but finite-precision , finite-width RNNs can not , as implied by streaming lower bounds ( Alon et al. , 1999 ) . As another example , Park et al . ( 2020 ) exploit infinite precision in the parameters to show that a neural net with parameter count sublinear in n can memorize n arbitrary input-label pairs . However , a simple counting argument reveals that this result can not be proven using finite precision networks – there are 2n input-labeling pairs , but only 2opnq finite precision networks with opnq parameters . More broadly , the ideal theoretical perspective should consider not only whether target functions can be expressed , but also whether the constructed networks are plausibly learnable . Learnability is important because empirical settings do not operate in the infinite data , unbounded computation regime – they require fitting the target function with access to limited number of samples from an empirical distribution . The question of studying learnability can be decomposed into studying optimization and generalization . Unfortunately , a rigorous analysis of optimization is unresolved even for simple two-layer nets ( Mei et al. , 2018 ) . Generalization is more tractable , so we propose to study expressivity and generalization together . Towards the goal of studying more meaningful notions of approximation , this work proposes the notion of statistically meaningful ( SM ) approximation . This definition requires not only the existence of an approximating network , but also that it has good statistical learnability . Consider a setting where the aim is to fit the target functionG using the approximating family F and a finite sample of training data . SM approximation requires existence of a loss whose empirical risk minimizer in F leads to a model with low approximation error in fittingG . We define the sample complexity of the approximation as the number of training samples needed to guarantee at most approximation error and study SM approximation with low sample complexity bounds . SM approximation essentially eliminates all statistical concerns for learnability ( optimization-related concerns can remain ) . We present two case studies on SM approximation . First , we demonstrate that overparameterized feedforward neural nets can SM approximate boolean circuits with a low sample complexity that depends only on the intrinsic circuit size . Though it is simple to construct neural nets to approximate boolean circuits , bounding the sample complexity of the approximation is challenging . For example , standard norm-based generalization bounds for the naive construction scale exponentially in depth ( Bartlett et al. , 2017 ) . Furthermore , VC dimension-based bounds would scale polynomially in the number of parameters in the network ( Harvey et al. , 2017 ) , which is problematic because for practical optimization concerns , neural nets are typically overparameterized in terms of width ( Zhang et al. , 2016 ) . In contrast , our sample complexity bound for SM approximation depends only on the intrinsic circuit size , up to logarithmic factors . Our second case study is on SM approximating Turing machines with transformers . We consider a class of Turing machines with bounded computation time T and construct encoder-decoder-based transformers ( Vaswani et al. , 2017 ) which SM approximate these Turing machines . The sample complexity of the approximation depends on a polynomial in logT and the sizes of the state space and the alphabet of the Turing machine . Though constructions for approximating Turing machines from prior work ( Siegelmann & Sontag , 1995 ; Pérez et al. , 2019 ; Bhattamishra et al. , 2020 ) have not been formally studied from a sample complexity perspective , existing bounds would depend at least linearly on T . Furthermore , our construction only uses loglogT precision , compared to at least logT in prior works , allowing us to achieve the exponential improvement in the sample complexity . Proving sample complexity guarantees for our statistically meaningful approximation results is nontrivial and requires additional insights , for both the constructions and the generalization analyses . To obtain our sample complexity bounds , we leverage a recent approach to bound generalization in terms of data-dependent notions of Lipschitzness ( Wei & Ma , 2019b ) . We develop theoretical tools to convert a broad class of neural nets , with possibly large Lipschitzness , into ones with small Lipschitzness on the training data , by introducing a number of new layers that is linear in depth . Our result applies to neural nets where each entry in the hidden representations on the training data takes values from a finite set ( e.g. , binary entries ) , and may be of independent interest . In summary , our contributions are : 1 ) we propose a new notion of statistically meaningful approximation , intended to provide more meaningful approximation guarantees by requiring that the approximating family have good statistical learnability ; 2 ) we prove that feedforward neural nets can meaningfully approximate boolean circuits with sample complexity that depends polynomially on the width and depth of the circuit ; and 3 ) we show that transformers can meaningfully approximate Turing machines with sample complexity logarithmic in the computation time . 1.1 RELATED WORKS . Classifical approximation theory for neural networks has a long history . Hornik et al . ( 1989 ) ; Cybenko ( 1989 ) , and Leshno et al . ( 1993 ) show that neural nets with one hidden layer are universal approximators but require the hidden layer size to grow exponentially in input dimension . Barron ( 1993 ) uses the Fourier transform to write target functions as infinite-width networks and subsamples neurons to obtain widths which depend only on target function properties . Lee et al . ( 2017 ) ; Ji et al . ( 2020 ) prove recent related developments in this direction of universal approximation . Many works study benefits of deep networks over shallow ones ( Bengio & Delalleau , 2011 ; Arora et al. , 2016 ; Telgarsky , 2016 ; Eldan & Shamir , 2016 ; Daniely , 2017 ; Chatziafratis et al. , 2020 ; 2019 ) . Bengio & Delalleau ( 2011 ) show separation for exact representation , whereas Telgarsky ( 2016 ) shows separation for approximate representations with univariate inputs . Eldan & Shamir ( 2016 ) demonstrate high-dimensional functions that can be approximated by two-layer polynomial-sized neural networks , but can not be approximated by one-layer neural nets with subexponential hidden units . Via reduction to certain complexity theo- retic questions , Vardi & Shamir ( 2020 ) show that proving constant depth separations may be hard . Malach et al . ( 2021 ) analyze the relationship between optimization and approximability , showing in various settings that deeper networks can not be optimized if shallow networks can not approximate them . This demonstrates that depth separation results ( Telgarsky , 2016 ) from approximation theory can be misleading in the sense that gradient descent anyways can not optimize the deep networks used to construct the approximation . Another area of study is on the ability of deep networks to memorize training data ( Zhang et al. , 2016 ; Yun et al. , 2018 ; Park et al. , 2020 ; Vershynin , 2020 ) . Yun et al . ( 2018 ) show that Θpnq parameters are sufficient to memorize Θpnq training points for ReLU nets with at least 3 layers , and Park et al . ( 2020 ) reduce the parameter requirement to sublinear in n. Similar results have been proven for residual architectures ( Hardt & Ma , 2016 ) and convolutional nets ( Nguyen & Hein , 2018 ) . Bartlett et al . ( 2019 ) analyze the VC-dimension of neural nets , leading to upper and lower bounds on the parameter count needed to fit training data . Other works study expressivity via connections to tensor approximation and sum-product networks ( Cohen & Shashua , 2016 ; Cohen et al. , 2016 ) . There is a long line of work on studying the ability of neural nets to recognize and represent formal languages . The seminal work of Siegelmann & Sontag ( 1995 ) shows that RNNs are Turing complete but leverages infinite precision in the hidden activations . Chen et al . ( 2018 ) extend this result to ReLU activations and study implications in language modeling . Many variants of transformers are shown to be Turing-complete , but these constructions also rely on arbitrary precision ( Pérez et al. , 2019 ; Bhattamishra et al. , 2020 ) . A number of recent works have also proven results for generating or recognizing formal languages with finite-precision neural nets ( Weiss et al. , 2018 ; Korsky & Berwick , 2019 ; Hewitt et al. , 2020 ) , but these results do not consider Turing machines or analyze statistical properties of their constructions . Bounding the sample complexity of SM approximation requires additional complications in both the construction and statistical analysis . 1.2 NOTATION . Let f˝g denote the composition of functions f and g. For a family of functions G , let f˝Gfitf˝g : gPGu denote the family of compositions between f and functions in G. For a set S and function f : SÑY , let fpSq denote the set tfpsq : sPSuĎY . We use 1d to denote the all-one ’ s vector in d dimensions , with the subscripted omitted if clear . For iPrds , we let 1dpiq denote the one-hot embedding in d-dimensions , which is 1 at index i and 0 everywhere else . We use the notation rOp¨q to hide poly-logarithmic factors in the argument . The notationÀ , Á indicates the existence of a constant factor such that the inequality holds . — denotes that theÁ andÀ relations simultaneously hold . We use polyp¨q to indicate the existence of a polynomial in the argument which makes the equation true . For a setA ( e.g. , the set of alphabet symbols for a Turing machine ) let A˚ denote the set of all sequences of elements of A , where sequence length can vary . Let P denote a distribution over a space of inputs X . Let ξ1 , ... , ξn be n i.i.d . Rademacher variables sampled from t´1 , ` 1u . The expected n-sample Rademacher complexity of F on P is as follows : Radn , P pFqfi E pxiqni “ 1 i.i.d „ P “ Eξ1 , ... , ξn “ supFPF 1 n řn i “ 1ξiFpxiq ‰‰ , where pxiqni “ 1 denotes n i.i.d . samples from P . | The paper proposes a novel concept called statistically meaningful approximation. Roughly speaking, SM approximation means that performing empirical risk minimization over some properly chosen surrogate loss would result in a hypothesis with small population risk. Then the paper develops necessary tools and proves two results: (1). fully-connected neural networks can SM approximate boolean circuits with a sample size that is poly in the number of gates and $\log$ in the width and depth of the circuit; (2). transformers can SM approximate Turing machines with a sample size that is poly in the number of states, alphabet size, and the log of the computation steps. | SP:0c0f4862966a6374c299012d302bb95e7dc88dea |
Mastering Visual Continuous Control: Improved Data-Augmented Reinforcement Learning | We present DrQ-v2 , a model-free reinforcement learning ( RL ) algorithm for visual continuous control . DrQ-v2 builds on DrQ , an off-policy actor-critic approach that uses data augmentation to learn directly from pixels . We introduce several improvements that yield state-of-the-art results on the DeepMind Control Suite . Notably , DrQ-v2 is able to solve complex humanoid locomotion tasks directly from pixel observations , previously unattained by model-free RL . DrQ-v2 is conceptually simple , easy to implement , and provides significantly better computational footprint compared to prior work , with the majority of tasks taking just 8 hours to train on a single GPU . Finally , DrQ-v2 ’ s implementation is publicly released to provide RL practitioners with a strong and computationally efficient baseline . 1 INTRODUCTION . Creating sample-efficient continuous control methods that observe high-dimensional images has been a long standing challenge in reinforcement learning ( RL ) . Over the last three years , the RL community has made significant headway on this problem , improving sample-efficiency significantly . The key insight to solving visual control is the learning of better low-dimensional representations , either through autoencoders ( Yarats et al. , 2019 ; Finn et al. , 2015 ) , variational inference ( Hafner et al. , 2018 ; 2019 ; Lee et al. , 2019 ) , contrastive learning ( Srinivas et al. , 2020 ; Yarats et al. , 2021a ) , self-prediction ( Schwarzer et al. , 2020b ) , or data augmentations ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) . However , current state-of-the-art model-free methods are still limited in three ways . First , they are unable to solve the more challenging visual control problems such as quadruped and humanoid locomotion . Second , they often require significant computational resources , i.e . lengthy training times using distributed multi-GPU infrastructure . Lastly , it is often unclear how different design choices affect overall system performance . In this paper we present DrQ-v2 , a simple model-free algorithm that builds on the idea of using data augmentations ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) to solve hard visual control problems . Most notably , it is the first model-free method that solves complex humanoid tasks directly from pixels . Compared to previous state-of-the-art model-free methods , DrQ-v2 provides significant improvements in sample efficiency across tasks from the DeepMind Control Suite ( Tassa et al. , 2018 ) . Conceptually simple , DrQ-v2 is also computationally efficient , which allows solving most tasks in DeepMind Control Suite in just 8 hours on a single GPU ( see Figure 1 ) . Recently , a model-based method , DreamerV2 ( Hafner et al. , 2020 ) was also shown to solve visual continuous control problems and it was first to solve the humanoid locomotion problem from pixels . While our model-free DrQ-v2 matches DreamerV2 in terms sample efficiency and performance , it does so 4⇥ faster in terms of wall-clock time to train . We believe this makes DrQ-v2 a more accessible approach to support research in visual continuous control and it reinforces the question on whether model-free or model-based is the more suitable approach to solve this type of tasks . DrQ-v2 , which is detailed in Section 3 , improves upon DrQ ( Yarats et al. , 2021b ) by making several algorithmic changes : ( i ) switching the base RL algorithm from SAC ( Haarnoja et al. , 2018a ) to DDPG ( Lillicrap et al. , 2015a ) with clipped double Q-learning from TD3 ( Fujimoto et al. , 2018 ) , ( ii ) this allows us straightforwardly incorporating multi-step return , ( iii ) adding bilinear interpolation to the random shift image augmentation , ( iv ) introducing an exploration schedule , ( v ) selecting better hyper-parameters including a larger capacity of the replay buffer . A careful ablation study of these design choices is presented in Section 4.4 . Furthermore , we re-examine the original implementation of DrQ and identify several computational bottlenecks such as replay buffer management , data augmentation processing , batch size , and frequency of learning updates ( see Section 3.2 ) . To remedy these , we have developed a new implementation that both achieves better performance and trains around 3.5 times faster with respect to wall-clock time than the previous implementation on the same hardware with an increase in environment frame throughput ( FPS ) from 28 to 96 ( i.e. , it takes 106/96/3600 ⇡ 2.9 hours to train for 1M environment steps ) . DrQ-v2 ’ s implementation is available at https : //anonymous.4open.science/r/drqv2 . 2 BACKGROUND . 2.1 REINFORCEMENT LEARNING FROM IMAGES . We formulate image-based control as an infinite-horizon Markov Decision Process ( MDP ) ( Bellman , 1957 ) . Generally , in such a setting , an image rendering of the system is not sufficient to perfectly describe the system ’ s underlying state . To this end and per common practice ( Mnih et al. , 2013 ) , we approximate the current state of the system by stacking three consecutive prior observations . With this in mind , such MDP can be described as a tuple ( X , A , P , R , , d0 ) , where X is the state space ( a three-stack of image observations ) , A is the action space , P : X ⇥A ! ( X ) is the transition function1 that defines a probability distribution over the next state given the current state and action , R : X ⇥ A ! [ 0 , 1 ] is the reward function , 2 [ 0 , 1 ) is a discount factor , and d0 2 ( X ) is the distribution of the initial state x0 . The goal is to find a policy ⇡ : X ! ( A ) that maximizes the expected discounted sum of rewards E⇡ [ P1 t=0 trt ] , where x0 ⇠ d0 , and 8t we have at ⇠ ⇡ ( ·|xt ) , xt+1 ⇠ P ( ·|xt , at ) , and rt = R ( xt , at ) . 2.2 DEEP DETERMINISTIC POLICY GRADIENT . Deep Deterministic Policy Gradient ( DDPG ) ( Lillicrap et al. , 2015a ) is an actor-critic algorithm for continuous control that concurrently learns a Q-function Q✓ and a deterministic policy ⇡ . For this , DDPG uses Q-learning ( Watkins and Dayan , 1992 ) to learn Q✓ by minimizing the one-step Bellman residual J✓ ( D ) = E ( xt , at , rt , xt+1 ) ⇠D [ ( Q✓ ( xt , at ) rt Q✓̄ ( xt+1 , ⇡ ( xt+1 ) ) 2 ] . The policy ⇡ is learned by employing Deterministic Policy Gradient ( DPG ) ( Silver et al. , 2014 ) and maximizing J ( D ) = Ext⇠D [ Q✓ ( xt , ⇡ ( xt ) ) ] , so ⇡ ( xt ) approximates argmaxaQ✓ ( xt , a ) . Here , D is a replay buffer of environment transitions and ✓̄ is an exponential moving average of the weights . DDPG is amenable to incorporate n-step returns ( Watkins , 1989 ; eng and Williams , 1996 ) when estimating TD error beyond a single step ( Barth-Maron et al. , 2018 ) . In practice , n-step returns allow for faster 1Here , ( X ) denotes a distribution over the state space X . reward propagation and has been previously used in policy gradient and Q-learning methods ( Mnih et al. , 2016b ; Barth-Maron et al. , 2018 ; Hessel et al. , 2017 ) . 2.3 DATA AUGMENTATION IN REINFORCEMENT LEARNING . Recently , it has been shown that data augmentation techniques , commonplace in Computer Vision , are also important for achieving the state-of-the-art performance in image-based RL ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) . For example , the state-of-the-art algorithm for visual RL , DrQ ( Yarats et al. , 2021b ) builds on top of Soft Actor-Critic ( Haarnoja et al. , 2018a ) , a model-free actor-critic algorithm , by adding a convolutional encoder and data augmentation in the form of random shifts . The use of such data augmentations now forms an essential component of several recent visual RL algorithms ( Srinivas et al. , 2020 ; Raileanu et al. , 2020 ; Yarats et al. , 2021a ; Stooke et al. , 2020 ; Hansen and Wang , 2021 ; Schwarzer et al. , 2020b ) . 3 DRQ-V2 : IMPROVED DATA-AUGMENTED REINFORCEMENT LEARNING . In this section , we describe DrQ-v2 , a simple model-free actor-critic RL algorithm for image-based continuous control , that builds upon DrQ . 3.1 ALGORITHMIC DETAILS . Image Augmentation As in DrQ we apply random shifts image augmentation to pixel observations of the environment . In the settings of visual continuous control by DMC , this augmentation can be instantiated by first padding each side of 84 ⇥ 84 observation rendering by 4 pixels ( by repeating boundary pixels ) , and then selecting a random 84⇥ 84 crop , yielding the original image shifted by ±4 pixels . We also find it useful to apply bilinear interpolation on top of the shifted image ( i.e , we replace each pixel value with the average of the four nearest pixel values ) . In our experiments , this modification provides an additional performance boost across the board . Image Encoder The augmented image observation is then embedded into a low-dimensional latent vector by applying a convolutional encoder . We use the same encoder architecture as in DrQ , which first was introduced introduced in SAC-AE ( Yarats et al. , 2019 ) . This process can be succinctly summarized as h = f⇠ ( aug ( x ) ) , where f⇠ is the encoder , aug is the random shifts augmentation , and x is the original image observation . Actor-Critic Algorithm We use DDPG ( Lillicrap et al. , 2015a ) as a backbone actor-critic RL algorithm and , similarly to Barth-Maron et al . ( 2018 ) , augment it with n-step returns to estimate TD error . This results into faster reward propagation and overall learning progress ( Mnih et al. , 2016a ) . While some methods ( Hafner et al. , 2020 ) employ more sophisticated techniques such as TD ( ) or Retrace ( ) ( Munos et al. , 2016 ) , they are often computationally demanding when n is large . We find that using simple n-step returns , without an importance sampling correction , strikes a good balance between performance and efficiency . We also employ clipped double Q-learning ( Fujimoto et al. , 2018 ) to reduce overestimation bias in the target value . Practically , this requires training two Qfunctions Q✓1 and Q✓2 . For this , we sample a mini-batch of transitions ⌧ = ( xt , at , rt : t+n 1 , xt+n ) from the replay buffer D and compute the following two losses : L✓k , ⇠ ( D ) = E⌧⇠D ⇥ ( Q✓k ( ht , at ) y ) 2 ⇤ 8k 2 { 1 , 2 } , ( 1 ) with the TD target y defined as : y = n 1X i=0 irt+i + n min k=1,2 Q✓̄k ( ht+n , at+n ) , where ht = f⇠ ( aug ( xt ) ) , ht+n = f⇠ ( aug ( xt+n ) ) , at+n = ⇡ ( ht+n ) + ✏ , ✓̄1 and ✓̄2 are the slowmoving weights for the Q target networks . We note , that in contrast to DrQ , we do not employ a target network for the encoder f⇠ and always use the most recent weights ⇠ to embed xt and xt+n . The exploration noise ✏ is sampled from clip ( N ( 0 , 2 ) , c , c ) similar to TD3 ( Fujimoto et al. , 2018 ) , with the exception of decaying , which we describe below . Finally , we train the deterministic actor ⇡ using DPG with the following loss : L ( D ) = Ext⇠D ⇥ min k=1,2 Q✓k ( ht , at ) ⇤ , ( 2 ) where ht = f⇠ ( aug ( xt ) ) , at = ⇡ ( ht ) + ✏ , and ✏ ⇠ clip ( N ( 0 , 2 ) , c , c ) . Similar to DrQ , we do not use actor ’ s gradients to update the encoder ’ s parameters ⇠ . Scheduled Exploration Noise Empirically , we observe that it is helpful to have different levels of exploration at different stages of learning . At the beginning of training we want the agent to be more stochastic and explore the environment more effectively , while at the later stages of training , when the agent has already identified promising behaviors , it is better to be more deterministic and master those behaviors . Similar to Amos et al . ( 2020 ) , we instantiate this idea by using linear decay ( t ) for the variance 2 of the exploration noise defined as : ( t ) = init + ( 1 min ( t T , 1 ) ) ( final init ) , ( 3 ) where init and final are the initial and final values for standard deviation , and T is the decay horizon . Key Hyper-Parameters We conduct an extensive hyper-parameter search and identify several hyper-parameter changes compared to DrQ . The three most important hyper-parameters are : ( i ) the size of the replay buffer , ( ii ) mini-batch size , and ( iii ) learning rate . Specifically , we use a 10 times larger replay buffer than DrQ . We also use a smaller mini-batch size of 256 without any noticeable performance degradation . This is in contrast to CURL ( Srinivas et al. , 2020 ) and DrQ ( Yarats et al. , 2021b ) that both use a larger batch size of 512 to attain more stable training in the expense of computational efficiency . Finally , we find that using smaller learning rate of 1⇥ 10 4 , rather than DrQ ’ s learning rate of 1⇥ 10 3 , results into more stable training without any loss in learning speed . | This paper introduces DrQ-v2, an improvement over DrQ, a model free off-policy actor-critic approach which uses SAC+data augmentation to learn directly from pixels. DrQ-v2, on the other hand, switches SAC with DDPG and proposes a series of algorithmic, hyperparameter choices and implementation improvements. These improvements make it both faster computationally and better empirically as it is able to solve the Humanoid environment from pixel. | SP:81d1638fa1056786415f8630b6a4f44ec75556c4 |
Mastering Visual Continuous Control: Improved Data-Augmented Reinforcement Learning | We present DrQ-v2 , a model-free reinforcement learning ( RL ) algorithm for visual continuous control . DrQ-v2 builds on DrQ , an off-policy actor-critic approach that uses data augmentation to learn directly from pixels . We introduce several improvements that yield state-of-the-art results on the DeepMind Control Suite . Notably , DrQ-v2 is able to solve complex humanoid locomotion tasks directly from pixel observations , previously unattained by model-free RL . DrQ-v2 is conceptually simple , easy to implement , and provides significantly better computational footprint compared to prior work , with the majority of tasks taking just 8 hours to train on a single GPU . Finally , DrQ-v2 ’ s implementation is publicly released to provide RL practitioners with a strong and computationally efficient baseline . 1 INTRODUCTION . Creating sample-efficient continuous control methods that observe high-dimensional images has been a long standing challenge in reinforcement learning ( RL ) . Over the last three years , the RL community has made significant headway on this problem , improving sample-efficiency significantly . The key insight to solving visual control is the learning of better low-dimensional representations , either through autoencoders ( Yarats et al. , 2019 ; Finn et al. , 2015 ) , variational inference ( Hafner et al. , 2018 ; 2019 ; Lee et al. , 2019 ) , contrastive learning ( Srinivas et al. , 2020 ; Yarats et al. , 2021a ) , self-prediction ( Schwarzer et al. , 2020b ) , or data augmentations ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) . However , current state-of-the-art model-free methods are still limited in three ways . First , they are unable to solve the more challenging visual control problems such as quadruped and humanoid locomotion . Second , they often require significant computational resources , i.e . lengthy training times using distributed multi-GPU infrastructure . Lastly , it is often unclear how different design choices affect overall system performance . In this paper we present DrQ-v2 , a simple model-free algorithm that builds on the idea of using data augmentations ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) to solve hard visual control problems . Most notably , it is the first model-free method that solves complex humanoid tasks directly from pixels . Compared to previous state-of-the-art model-free methods , DrQ-v2 provides significant improvements in sample efficiency across tasks from the DeepMind Control Suite ( Tassa et al. , 2018 ) . Conceptually simple , DrQ-v2 is also computationally efficient , which allows solving most tasks in DeepMind Control Suite in just 8 hours on a single GPU ( see Figure 1 ) . Recently , a model-based method , DreamerV2 ( Hafner et al. , 2020 ) was also shown to solve visual continuous control problems and it was first to solve the humanoid locomotion problem from pixels . While our model-free DrQ-v2 matches DreamerV2 in terms sample efficiency and performance , it does so 4⇥ faster in terms of wall-clock time to train . We believe this makes DrQ-v2 a more accessible approach to support research in visual continuous control and it reinforces the question on whether model-free or model-based is the more suitable approach to solve this type of tasks . DrQ-v2 , which is detailed in Section 3 , improves upon DrQ ( Yarats et al. , 2021b ) by making several algorithmic changes : ( i ) switching the base RL algorithm from SAC ( Haarnoja et al. , 2018a ) to DDPG ( Lillicrap et al. , 2015a ) with clipped double Q-learning from TD3 ( Fujimoto et al. , 2018 ) , ( ii ) this allows us straightforwardly incorporating multi-step return , ( iii ) adding bilinear interpolation to the random shift image augmentation , ( iv ) introducing an exploration schedule , ( v ) selecting better hyper-parameters including a larger capacity of the replay buffer . A careful ablation study of these design choices is presented in Section 4.4 . Furthermore , we re-examine the original implementation of DrQ and identify several computational bottlenecks such as replay buffer management , data augmentation processing , batch size , and frequency of learning updates ( see Section 3.2 ) . To remedy these , we have developed a new implementation that both achieves better performance and trains around 3.5 times faster with respect to wall-clock time than the previous implementation on the same hardware with an increase in environment frame throughput ( FPS ) from 28 to 96 ( i.e. , it takes 106/96/3600 ⇡ 2.9 hours to train for 1M environment steps ) . DrQ-v2 ’ s implementation is available at https : //anonymous.4open.science/r/drqv2 . 2 BACKGROUND . 2.1 REINFORCEMENT LEARNING FROM IMAGES . We formulate image-based control as an infinite-horizon Markov Decision Process ( MDP ) ( Bellman , 1957 ) . Generally , in such a setting , an image rendering of the system is not sufficient to perfectly describe the system ’ s underlying state . To this end and per common practice ( Mnih et al. , 2013 ) , we approximate the current state of the system by stacking three consecutive prior observations . With this in mind , such MDP can be described as a tuple ( X , A , P , R , , d0 ) , where X is the state space ( a three-stack of image observations ) , A is the action space , P : X ⇥A ! ( X ) is the transition function1 that defines a probability distribution over the next state given the current state and action , R : X ⇥ A ! [ 0 , 1 ] is the reward function , 2 [ 0 , 1 ) is a discount factor , and d0 2 ( X ) is the distribution of the initial state x0 . The goal is to find a policy ⇡ : X ! ( A ) that maximizes the expected discounted sum of rewards E⇡ [ P1 t=0 trt ] , where x0 ⇠ d0 , and 8t we have at ⇠ ⇡ ( ·|xt ) , xt+1 ⇠ P ( ·|xt , at ) , and rt = R ( xt , at ) . 2.2 DEEP DETERMINISTIC POLICY GRADIENT . Deep Deterministic Policy Gradient ( DDPG ) ( Lillicrap et al. , 2015a ) is an actor-critic algorithm for continuous control that concurrently learns a Q-function Q✓ and a deterministic policy ⇡ . For this , DDPG uses Q-learning ( Watkins and Dayan , 1992 ) to learn Q✓ by minimizing the one-step Bellman residual J✓ ( D ) = E ( xt , at , rt , xt+1 ) ⇠D [ ( Q✓ ( xt , at ) rt Q✓̄ ( xt+1 , ⇡ ( xt+1 ) ) 2 ] . The policy ⇡ is learned by employing Deterministic Policy Gradient ( DPG ) ( Silver et al. , 2014 ) and maximizing J ( D ) = Ext⇠D [ Q✓ ( xt , ⇡ ( xt ) ) ] , so ⇡ ( xt ) approximates argmaxaQ✓ ( xt , a ) . Here , D is a replay buffer of environment transitions and ✓̄ is an exponential moving average of the weights . DDPG is amenable to incorporate n-step returns ( Watkins , 1989 ; eng and Williams , 1996 ) when estimating TD error beyond a single step ( Barth-Maron et al. , 2018 ) . In practice , n-step returns allow for faster 1Here , ( X ) denotes a distribution over the state space X . reward propagation and has been previously used in policy gradient and Q-learning methods ( Mnih et al. , 2016b ; Barth-Maron et al. , 2018 ; Hessel et al. , 2017 ) . 2.3 DATA AUGMENTATION IN REINFORCEMENT LEARNING . Recently , it has been shown that data augmentation techniques , commonplace in Computer Vision , are also important for achieving the state-of-the-art performance in image-based RL ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) . For example , the state-of-the-art algorithm for visual RL , DrQ ( Yarats et al. , 2021b ) builds on top of Soft Actor-Critic ( Haarnoja et al. , 2018a ) , a model-free actor-critic algorithm , by adding a convolutional encoder and data augmentation in the form of random shifts . The use of such data augmentations now forms an essential component of several recent visual RL algorithms ( Srinivas et al. , 2020 ; Raileanu et al. , 2020 ; Yarats et al. , 2021a ; Stooke et al. , 2020 ; Hansen and Wang , 2021 ; Schwarzer et al. , 2020b ) . 3 DRQ-V2 : IMPROVED DATA-AUGMENTED REINFORCEMENT LEARNING . In this section , we describe DrQ-v2 , a simple model-free actor-critic RL algorithm for image-based continuous control , that builds upon DrQ . 3.1 ALGORITHMIC DETAILS . Image Augmentation As in DrQ we apply random shifts image augmentation to pixel observations of the environment . In the settings of visual continuous control by DMC , this augmentation can be instantiated by first padding each side of 84 ⇥ 84 observation rendering by 4 pixels ( by repeating boundary pixels ) , and then selecting a random 84⇥ 84 crop , yielding the original image shifted by ±4 pixels . We also find it useful to apply bilinear interpolation on top of the shifted image ( i.e , we replace each pixel value with the average of the four nearest pixel values ) . In our experiments , this modification provides an additional performance boost across the board . Image Encoder The augmented image observation is then embedded into a low-dimensional latent vector by applying a convolutional encoder . We use the same encoder architecture as in DrQ , which first was introduced introduced in SAC-AE ( Yarats et al. , 2019 ) . This process can be succinctly summarized as h = f⇠ ( aug ( x ) ) , where f⇠ is the encoder , aug is the random shifts augmentation , and x is the original image observation . Actor-Critic Algorithm We use DDPG ( Lillicrap et al. , 2015a ) as a backbone actor-critic RL algorithm and , similarly to Barth-Maron et al . ( 2018 ) , augment it with n-step returns to estimate TD error . This results into faster reward propagation and overall learning progress ( Mnih et al. , 2016a ) . While some methods ( Hafner et al. , 2020 ) employ more sophisticated techniques such as TD ( ) or Retrace ( ) ( Munos et al. , 2016 ) , they are often computationally demanding when n is large . We find that using simple n-step returns , without an importance sampling correction , strikes a good balance between performance and efficiency . We also employ clipped double Q-learning ( Fujimoto et al. , 2018 ) to reduce overestimation bias in the target value . Practically , this requires training two Qfunctions Q✓1 and Q✓2 . For this , we sample a mini-batch of transitions ⌧ = ( xt , at , rt : t+n 1 , xt+n ) from the replay buffer D and compute the following two losses : L✓k , ⇠ ( D ) = E⌧⇠D ⇥ ( Q✓k ( ht , at ) y ) 2 ⇤ 8k 2 { 1 , 2 } , ( 1 ) with the TD target y defined as : y = n 1X i=0 irt+i + n min k=1,2 Q✓̄k ( ht+n , at+n ) , where ht = f⇠ ( aug ( xt ) ) , ht+n = f⇠ ( aug ( xt+n ) ) , at+n = ⇡ ( ht+n ) + ✏ , ✓̄1 and ✓̄2 are the slowmoving weights for the Q target networks . We note , that in contrast to DrQ , we do not employ a target network for the encoder f⇠ and always use the most recent weights ⇠ to embed xt and xt+n . The exploration noise ✏ is sampled from clip ( N ( 0 , 2 ) , c , c ) similar to TD3 ( Fujimoto et al. , 2018 ) , with the exception of decaying , which we describe below . Finally , we train the deterministic actor ⇡ using DPG with the following loss : L ( D ) = Ext⇠D ⇥ min k=1,2 Q✓k ( ht , at ) ⇤ , ( 2 ) where ht = f⇠ ( aug ( xt ) ) , at = ⇡ ( ht ) + ✏ , and ✏ ⇠ clip ( N ( 0 , 2 ) , c , c ) . Similar to DrQ , we do not use actor ’ s gradients to update the encoder ’ s parameters ⇠ . Scheduled Exploration Noise Empirically , we observe that it is helpful to have different levels of exploration at different stages of learning . At the beginning of training we want the agent to be more stochastic and explore the environment more effectively , while at the later stages of training , when the agent has already identified promising behaviors , it is better to be more deterministic and master those behaviors . Similar to Amos et al . ( 2020 ) , we instantiate this idea by using linear decay ( t ) for the variance 2 of the exploration noise defined as : ( t ) = init + ( 1 min ( t T , 1 ) ) ( final init ) , ( 3 ) where init and final are the initial and final values for standard deviation , and T is the decay horizon . Key Hyper-Parameters We conduct an extensive hyper-parameter search and identify several hyper-parameter changes compared to DrQ . The three most important hyper-parameters are : ( i ) the size of the replay buffer , ( ii ) mini-batch size , and ( iii ) learning rate . Specifically , we use a 10 times larger replay buffer than DrQ . We also use a smaller mini-batch size of 256 without any noticeable performance degradation . This is in contrast to CURL ( Srinivas et al. , 2020 ) and DrQ ( Yarats et al. , 2021b ) that both use a larger batch size of 512 to attain more stable training in the expense of computational efficiency . Finally , we find that using smaller learning rate of 1⇥ 10 4 , rather than DrQ ’ s learning rate of 1⇥ 10 3 , results into more stable training without any loss in learning speed . | This paper presents DrQ-v2, an improvement over DrQ, for solving visual-based RL problems. Key components include (1) DDPG over SAC (2) n-step returns for the critic (3) replay buffer size and (4) decaying exploration. Implementation improvement is also discussed. Notably, DrQ-v2 is able to solve the humanoid task effectively compared to recent work in this domain. | SP:81d1638fa1056786415f8630b6a4f44ec75556c4 |
Mastering Visual Continuous Control: Improved Data-Augmented Reinforcement Learning | We present DrQ-v2 , a model-free reinforcement learning ( RL ) algorithm for visual continuous control . DrQ-v2 builds on DrQ , an off-policy actor-critic approach that uses data augmentation to learn directly from pixels . We introduce several improvements that yield state-of-the-art results on the DeepMind Control Suite . Notably , DrQ-v2 is able to solve complex humanoid locomotion tasks directly from pixel observations , previously unattained by model-free RL . DrQ-v2 is conceptually simple , easy to implement , and provides significantly better computational footprint compared to prior work , with the majority of tasks taking just 8 hours to train on a single GPU . Finally , DrQ-v2 ’ s implementation is publicly released to provide RL practitioners with a strong and computationally efficient baseline . 1 INTRODUCTION . Creating sample-efficient continuous control methods that observe high-dimensional images has been a long standing challenge in reinforcement learning ( RL ) . Over the last three years , the RL community has made significant headway on this problem , improving sample-efficiency significantly . The key insight to solving visual control is the learning of better low-dimensional representations , either through autoencoders ( Yarats et al. , 2019 ; Finn et al. , 2015 ) , variational inference ( Hafner et al. , 2018 ; 2019 ; Lee et al. , 2019 ) , contrastive learning ( Srinivas et al. , 2020 ; Yarats et al. , 2021a ) , self-prediction ( Schwarzer et al. , 2020b ) , or data augmentations ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) . However , current state-of-the-art model-free methods are still limited in three ways . First , they are unable to solve the more challenging visual control problems such as quadruped and humanoid locomotion . Second , they often require significant computational resources , i.e . lengthy training times using distributed multi-GPU infrastructure . Lastly , it is often unclear how different design choices affect overall system performance . In this paper we present DrQ-v2 , a simple model-free algorithm that builds on the idea of using data augmentations ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) to solve hard visual control problems . Most notably , it is the first model-free method that solves complex humanoid tasks directly from pixels . Compared to previous state-of-the-art model-free methods , DrQ-v2 provides significant improvements in sample efficiency across tasks from the DeepMind Control Suite ( Tassa et al. , 2018 ) . Conceptually simple , DrQ-v2 is also computationally efficient , which allows solving most tasks in DeepMind Control Suite in just 8 hours on a single GPU ( see Figure 1 ) . Recently , a model-based method , DreamerV2 ( Hafner et al. , 2020 ) was also shown to solve visual continuous control problems and it was first to solve the humanoid locomotion problem from pixels . While our model-free DrQ-v2 matches DreamerV2 in terms sample efficiency and performance , it does so 4⇥ faster in terms of wall-clock time to train . We believe this makes DrQ-v2 a more accessible approach to support research in visual continuous control and it reinforces the question on whether model-free or model-based is the more suitable approach to solve this type of tasks . DrQ-v2 , which is detailed in Section 3 , improves upon DrQ ( Yarats et al. , 2021b ) by making several algorithmic changes : ( i ) switching the base RL algorithm from SAC ( Haarnoja et al. , 2018a ) to DDPG ( Lillicrap et al. , 2015a ) with clipped double Q-learning from TD3 ( Fujimoto et al. , 2018 ) , ( ii ) this allows us straightforwardly incorporating multi-step return , ( iii ) adding bilinear interpolation to the random shift image augmentation , ( iv ) introducing an exploration schedule , ( v ) selecting better hyper-parameters including a larger capacity of the replay buffer . A careful ablation study of these design choices is presented in Section 4.4 . Furthermore , we re-examine the original implementation of DrQ and identify several computational bottlenecks such as replay buffer management , data augmentation processing , batch size , and frequency of learning updates ( see Section 3.2 ) . To remedy these , we have developed a new implementation that both achieves better performance and trains around 3.5 times faster with respect to wall-clock time than the previous implementation on the same hardware with an increase in environment frame throughput ( FPS ) from 28 to 96 ( i.e. , it takes 106/96/3600 ⇡ 2.9 hours to train for 1M environment steps ) . DrQ-v2 ’ s implementation is available at https : //anonymous.4open.science/r/drqv2 . 2 BACKGROUND . 2.1 REINFORCEMENT LEARNING FROM IMAGES . We formulate image-based control as an infinite-horizon Markov Decision Process ( MDP ) ( Bellman , 1957 ) . Generally , in such a setting , an image rendering of the system is not sufficient to perfectly describe the system ’ s underlying state . To this end and per common practice ( Mnih et al. , 2013 ) , we approximate the current state of the system by stacking three consecutive prior observations . With this in mind , such MDP can be described as a tuple ( X , A , P , R , , d0 ) , where X is the state space ( a three-stack of image observations ) , A is the action space , P : X ⇥A ! ( X ) is the transition function1 that defines a probability distribution over the next state given the current state and action , R : X ⇥ A ! [ 0 , 1 ] is the reward function , 2 [ 0 , 1 ) is a discount factor , and d0 2 ( X ) is the distribution of the initial state x0 . The goal is to find a policy ⇡ : X ! ( A ) that maximizes the expected discounted sum of rewards E⇡ [ P1 t=0 trt ] , where x0 ⇠ d0 , and 8t we have at ⇠ ⇡ ( ·|xt ) , xt+1 ⇠ P ( ·|xt , at ) , and rt = R ( xt , at ) . 2.2 DEEP DETERMINISTIC POLICY GRADIENT . Deep Deterministic Policy Gradient ( DDPG ) ( Lillicrap et al. , 2015a ) is an actor-critic algorithm for continuous control that concurrently learns a Q-function Q✓ and a deterministic policy ⇡ . For this , DDPG uses Q-learning ( Watkins and Dayan , 1992 ) to learn Q✓ by minimizing the one-step Bellman residual J✓ ( D ) = E ( xt , at , rt , xt+1 ) ⇠D [ ( Q✓ ( xt , at ) rt Q✓̄ ( xt+1 , ⇡ ( xt+1 ) ) 2 ] . The policy ⇡ is learned by employing Deterministic Policy Gradient ( DPG ) ( Silver et al. , 2014 ) and maximizing J ( D ) = Ext⇠D [ Q✓ ( xt , ⇡ ( xt ) ) ] , so ⇡ ( xt ) approximates argmaxaQ✓ ( xt , a ) . Here , D is a replay buffer of environment transitions and ✓̄ is an exponential moving average of the weights . DDPG is amenable to incorporate n-step returns ( Watkins , 1989 ; eng and Williams , 1996 ) when estimating TD error beyond a single step ( Barth-Maron et al. , 2018 ) . In practice , n-step returns allow for faster 1Here , ( X ) denotes a distribution over the state space X . reward propagation and has been previously used in policy gradient and Q-learning methods ( Mnih et al. , 2016b ; Barth-Maron et al. , 2018 ; Hessel et al. , 2017 ) . 2.3 DATA AUGMENTATION IN REINFORCEMENT LEARNING . Recently , it has been shown that data augmentation techniques , commonplace in Computer Vision , are also important for achieving the state-of-the-art performance in image-based RL ( Yarats et al. , 2021b ; Laskin et al. , 2020 ) . For example , the state-of-the-art algorithm for visual RL , DrQ ( Yarats et al. , 2021b ) builds on top of Soft Actor-Critic ( Haarnoja et al. , 2018a ) , a model-free actor-critic algorithm , by adding a convolutional encoder and data augmentation in the form of random shifts . The use of such data augmentations now forms an essential component of several recent visual RL algorithms ( Srinivas et al. , 2020 ; Raileanu et al. , 2020 ; Yarats et al. , 2021a ; Stooke et al. , 2020 ; Hansen and Wang , 2021 ; Schwarzer et al. , 2020b ) . 3 DRQ-V2 : IMPROVED DATA-AUGMENTED REINFORCEMENT LEARNING . In this section , we describe DrQ-v2 , a simple model-free actor-critic RL algorithm for image-based continuous control , that builds upon DrQ . 3.1 ALGORITHMIC DETAILS . Image Augmentation As in DrQ we apply random shifts image augmentation to pixel observations of the environment . In the settings of visual continuous control by DMC , this augmentation can be instantiated by first padding each side of 84 ⇥ 84 observation rendering by 4 pixels ( by repeating boundary pixels ) , and then selecting a random 84⇥ 84 crop , yielding the original image shifted by ±4 pixels . We also find it useful to apply bilinear interpolation on top of the shifted image ( i.e , we replace each pixel value with the average of the four nearest pixel values ) . In our experiments , this modification provides an additional performance boost across the board . Image Encoder The augmented image observation is then embedded into a low-dimensional latent vector by applying a convolutional encoder . We use the same encoder architecture as in DrQ , which first was introduced introduced in SAC-AE ( Yarats et al. , 2019 ) . This process can be succinctly summarized as h = f⇠ ( aug ( x ) ) , where f⇠ is the encoder , aug is the random shifts augmentation , and x is the original image observation . Actor-Critic Algorithm We use DDPG ( Lillicrap et al. , 2015a ) as a backbone actor-critic RL algorithm and , similarly to Barth-Maron et al . ( 2018 ) , augment it with n-step returns to estimate TD error . This results into faster reward propagation and overall learning progress ( Mnih et al. , 2016a ) . While some methods ( Hafner et al. , 2020 ) employ more sophisticated techniques such as TD ( ) or Retrace ( ) ( Munos et al. , 2016 ) , they are often computationally demanding when n is large . We find that using simple n-step returns , without an importance sampling correction , strikes a good balance between performance and efficiency . We also employ clipped double Q-learning ( Fujimoto et al. , 2018 ) to reduce overestimation bias in the target value . Practically , this requires training two Qfunctions Q✓1 and Q✓2 . For this , we sample a mini-batch of transitions ⌧ = ( xt , at , rt : t+n 1 , xt+n ) from the replay buffer D and compute the following two losses : L✓k , ⇠ ( D ) = E⌧⇠D ⇥ ( Q✓k ( ht , at ) y ) 2 ⇤ 8k 2 { 1 , 2 } , ( 1 ) with the TD target y defined as : y = n 1X i=0 irt+i + n min k=1,2 Q✓̄k ( ht+n , at+n ) , where ht = f⇠ ( aug ( xt ) ) , ht+n = f⇠ ( aug ( xt+n ) ) , at+n = ⇡ ( ht+n ) + ✏ , ✓̄1 and ✓̄2 are the slowmoving weights for the Q target networks . We note , that in contrast to DrQ , we do not employ a target network for the encoder f⇠ and always use the most recent weights ⇠ to embed xt and xt+n . The exploration noise ✏ is sampled from clip ( N ( 0 , 2 ) , c , c ) similar to TD3 ( Fujimoto et al. , 2018 ) , with the exception of decaying , which we describe below . Finally , we train the deterministic actor ⇡ using DPG with the following loss : L ( D ) = Ext⇠D ⇥ min k=1,2 Q✓k ( ht , at ) ⇤ , ( 2 ) where ht = f⇠ ( aug ( xt ) ) , at = ⇡ ( ht ) + ✏ , and ✏ ⇠ clip ( N ( 0 , 2 ) , c , c ) . Similar to DrQ , we do not use actor ’ s gradients to update the encoder ’ s parameters ⇠ . Scheduled Exploration Noise Empirically , we observe that it is helpful to have different levels of exploration at different stages of learning . At the beginning of training we want the agent to be more stochastic and explore the environment more effectively , while at the later stages of training , when the agent has already identified promising behaviors , it is better to be more deterministic and master those behaviors . Similar to Amos et al . ( 2020 ) , we instantiate this idea by using linear decay ( t ) for the variance 2 of the exploration noise defined as : ( t ) = init + ( 1 min ( t T , 1 ) ) ( final init ) , ( 3 ) where init and final are the initial and final values for standard deviation , and T is the decay horizon . Key Hyper-Parameters We conduct an extensive hyper-parameter search and identify several hyper-parameter changes compared to DrQ . The three most important hyper-parameters are : ( i ) the size of the replay buffer , ( ii ) mini-batch size , and ( iii ) learning rate . Specifically , we use a 10 times larger replay buffer than DrQ . We also use a smaller mini-batch size of 256 without any noticeable performance degradation . This is in contrast to CURL ( Srinivas et al. , 2020 ) and DrQ ( Yarats et al. , 2021b ) that both use a larger batch size of 512 to attain more stable training in the expense of computational efficiency . Finally , we find that using smaller learning rate of 1⇥ 10 4 , rather than DrQ ’ s learning rate of 1⇥ 10 3 , results into more stable training without any loss in learning speed . | The paper presents DrQ-v2, a model-free reinforcement learning (RL) algorithm for visual continuous control. The algorithm is tested on the Deepmind Control Suite and it is proven to learn visual tasks in a remarkably short wall-clock time and with an extremely high computational efficiency (i.e. small number of frames). The paper is an improvement over a previous algorithm DrQ. Improvements are achieved with: (1) a new learning backbone (DDPG vs SAC); (2) the introduction of n-steps returns; (3) a bigger reply buffer and (4) a better scheduled exploration strategy. | SP:81d1638fa1056786415f8630b6a4f44ec75556c4 |
Beyond Faithfulness: A Framework to Characterize and Compare Saliency Methods | 1 INTRODUCTION . As machine learning ( ML ) systems are increasingly deployed into real-world contexts , stakeholder interviews ( Tonekaboni et al. , 2019 ; Bhatt et al. , 2020 ) , design best practices ( Amershi et al. , 2019 ) , and legal frameworks ( European Commission , 2018 ) have underscored the need for explainability . Saliency methods — a class of attribution methods that aim to identify features in an input that were important to a trained ML prediction — are frequently used to provide explanations . However , each method operates differently and , thus , multiple methods can produce seemingly varying explanations for the same model and input . How , then , should we reason about choosing and comparing methods for a particular application ? Prior work has suggested that “ faithfulness , ” or how accurately a saliency result reflects the underlying model , is a desideratum for these methods ( Li et al. , 2021 ; Ding & Koehn , 2021 ; Tomsett et al. , 2020 ; Adebayo et al. , 2018 ) . In this paper , however , we argue that faithfulness is not a productive goal for saliency methods — by design , they can not offer a complete and accurate reflection of a model ’ s behavior , akin to a printout of model weights . Rather , we frame saliency methods as abstractions of model behavior that selectively preserve and necessarily sacrifice information in service of human-centric goals such as simplicity and understandability . With this framing , we propose a nine-dimensional framework to characterize and compare saliency methods . These dimensions fall into three categories , corresponding to different parts of the interpretation process : methodology , or how the saliency is computed ; sensitivity , or relationships between the saliency and the underlying model or input ; and perceptibility , or how an end-user perceives the saliency . These dimensions decompose a singular notion of faithfulness into more granular units that can be reasoned about individually : by situating methods along these dimensions , we can surface their relative strengths , limitations , and differences to better understand trade offs that are latent in their design . In doing so , we demonstrate how our framework allows us to develop “ saliency 0 cards ” ( akin to “ model cards ” ( Mitchell et al. , 2019 ) and “ datasheets ” ( Gebru et al. , 2018 ) ) to document individual methods and to better contextualize saliency results . Moreover , using a concrete example of ML-based radiology diagnostic systems , we show how downstream stakeholders can use our framework to weigh tradeoffs and choose a task-appropriate method . Finally , we show how our framework identifies compelling opportunities for future work including exploring understudied dimensions and developing new metrics that target particular dimensions . 2 RELATED WORK : SALIENCY METHODS AND THEIR EVALUATIONS . Saliency methods ( sometimes referred to as feature attribution methods ) produce explanations for an ML model ’ s output . Given an input , saliency methods compute an importance score for each input feature describing its influence on the model ’ s output . Existing categorizations for saliency methods have focused on algorithmic properties ( gradient or perturbation-based , path-attribution or gradientonly ) Molnar ( 2019 ) . Our framework aims to categorize a broader range of important characteristics . We apply our framework to a variety of common saliency methods listed in Table 1 . Evaluations of saliency methods have primarily focused on how accurately their results represent model behavior , often referred to as faithfulness . A growing body of work has identified failures of some methods such as susceptibility to adversarial perturbations ( Ghorbani et al. , 2019 ) , lack of neuron discriminativity ( Mahendran & Vedaldi , 2016 ) , and a predisposition towards input recovery ( Nie et al. , 2018 ; Adebayo et al. , 2018 ) . Other work has proposed proxy tests that measure different aspects of faithfulness . Adebayo et al . ( 2018 ) recommend model randomization and data label randomization tests , and Kindermans et al . ( 2019 ) test whether constant input shifts affect saliency results . Samek et al . ( 2016 ) judge saliency methods by iteratively replacing features that have high importance values with random noise and measuring how much the output changes . While these tests quantitatively analyze saliency methods , Tomsett et al . ( 2020 ) found they can produce inconsistent rankings . An alternate line of work has looked to break faithfulness down into measurable axioms . Sundararajan et al . ( 2017 ) propose implementation invariance ( a saliency method should produce the same output on functionally equivalent models ) , sensitivity ( a saliency method should give importance to a feature if and only if changing it leads to a different output ) , and linearity ( if a model is the composition of two sub-models , the saliency method ’ s output for the model should be the weighted sum of its outputs for each sub-model ) . Sundararajan et al . ( 2017 ) and Shrikumar et al . ( 2017 ) both posit a saliency method should exhibit completeness — a saliency method ’ s attributions should sum to the difference between the model ’ s output on the input and the model ’ s output on a neutral input . Shrikumar et al . ( 2017 ) and Montavon et al . ( 2018 ) claim saliency methods should output a continuous function . Montavon et al . ( 2018 ) also propose the axiom of selectivity — a saliency method should distribute importance to features that have the greatest impact on the model ’ s output . Finally , Kindermans et al . ( 2019 ) state that a saliency method should be invariant to constant transforms . While axioms are presented as constraints that all saliency methods should attain , the axes in our framework describe attributes of saliency methods that can be traded off to select the best method for different use cases . To situate saliency methods along each axis , we utilize existing tests ( Kindermans et al. , 2019 ; Adebayo et al. , 2018 ) ; however , our framework surfaces the need for additional evaluations , since existing tests do not fully describe each axis and have not assessed all methods . 3 NINE DIMENSIONS TO CHARACTERIZE & COMPARE SALIENCY METHODS . To distill a language to characterize and compare saliency methods , we treat saliency methods as abstractions of the underlying model behavior . That is , to explain model behavior in a humanunderstandable format ( e.g. , a feature attribution heatmap ) , we must necessarily abstract away some detail and concreteness . A saliency method , by design , is not as precise a reflection of model behavior as a printout of the weights , but it is much more human-interpretable . The idea of abstraction recurs in other parts of the machine learning pipeline . For example , “ test accuracy ” is an abstraction of model performance : it does not capture all aspects of model performance but is nevertheless a convenient approximation . Abstractions selectively preserve information and can be combined to arrive at a more complete picture of the underlying phenomenon — i.e. , combining test accuracy with AUROC to more fully understand model performance . This framing reveals that a single saliency method can not fully explain model behavior . Instead , it is necessary to understand different methods ’ strengths and limitations to choose an appropriate method for the model , domain , and task . Our framework defines nine dimensions that describe what information saliency methods abstract , and how this abstraction is computed . We group these dimensions into three categories that map to different stages of the interpretability process ( see Figure 1 ) . Methodology covers the model , input , and saliency method , and dimensions under this category describe how the saliency method operates . Sensitivity goes one stage further , with dimensions that describe the relationship between the saliency output and the model or input . Perceptibility covers the final stages of the process , with dimensions that describe how users perceive the saliency output and any associated visualizations . These dimensions allow us to decompose faithfulness into aspects that can be reasoned about individually . For instance , Figure 2 demonstrates how our dimensions allow us to situate saliency methods in relation to one another to understand their strengths , limitations , and differences . In the following subsections , we describe and provide an illustrative example of each dimension . 3.1 METHODOLOGY : HOW SALIENCY METHODS OPERATE . Determinism Some saliency techniques are non-deterministic , and running them with different random seeds can produce significantly different outputs . For instance , methods such as RISE or SHAP rely on randomly-generated masks or coalitions . The output from a non-deterministic method represents only one example from a potentially high-variance distribution , and may be afforded authority that does not account for its inherent uncertainty . Although we may never explicitly choose a method for its non-determinism , we may nevertheless accept non-determinism in service of other dimensions such as increased model agnosticism ( e.g. , RISE ) , perceptual correspondence ( e.g. , SHAP ) , or minimality ( e.g. , MP ) . Example . Consider a model assisting dermatologists diagnosing skin cancer . In Figure 3 ( a ) , we show that two runs of a non-deterministic method ( LIME ) on a melanoma prediction model result in different saliency maps . These variations may have significant consequences given small areas of the lesion or surrounding skin may be integral to the diagnosis . Visualizing and interpreting multiple runs together is time-consuming and confusing , but only looking at one run could skew a clinician ’ s judgment . Thus , in this case , it might make sense to prioritize a method with deterministic outputs . Hyperparameter Dependence Hyperparameter dependence captures how many hyperparameters or design decisions the user must specify to run a particular method , and how sensitive it is to them . If there are not sufficient resources or expertise to devote to hyperparameter optimization , simply using default values can lead to misleading results . Similarly , misleading or confusing results might arise if the hyperparameters were chosen based on a particular dataset but deployed in a setting in which there is a significant distribution shift . In situations like this , it makes sense to prioritize methods with low hyperparameter dependence . In other cases , where developers have dedicated appropriate time and resources to tuning , it might be preferable to use a method dependent on hyperparameters ( e.g. , IG , SmoothGrad ) because it improves other dimensions ( e.g. , minimality ) . Example . IG computes feature importances by interpolating between a “ meaningless ” baseline input and the actual input , accumulating the gradients at each step . A common practice is to use a baseline value of all zeroes ; however , in some cases , a zero baseline can be misleading and potentially harmful . Take a model trained to predict bone fractures from x-ray images . Fractures often appear as a black line in the bone , but , since they have the same pixel value as the baseline ( zero ) , the IG result will indicate that they are not important . In Figure 3 ( b ) , we show a similar example where the choice of baseline ( black , white ) has a strong effect on the saliency map . Choosing an appropriate baseline requires time and a deep understanding of the data and method ; if this is not available in a particular application , it is better to prioritize a saliency method without important hyperparameters . Model Agnosticism Model agnostic methods ( i.e. , SHAP ) treat the underlying model as a blackbox , relying only on its input and output . On the other hand , model-dependent methods need access to various model internals , and may have specific requirements , such as differentiability ( i.e. , | The paper describes a new framework for characterizing and analyzing saliency methods. Nine dimensions are defined, grouped into three high-level categories, with some form of quantification in each dimension to enable comparative measures. The dimensions are used to formulate saliency scorecards, concise 1-page summaries of the saliency attributes of a particular algorithm that should help a user determine which algorithm is best for their needs. A set of ten recent algorithms are analyzed and compared using the framework, revealing that some have gaps in their assessments that could be address with future work. Compelling, illustrative examples are used throughout to convey understanding of the dimensions and the differences between them. | SP:d8c1b4c5169c6e2675d7b211c8beee638a256a55 |
Beyond Faithfulness: A Framework to Characterize and Compare Saliency Methods | 1 INTRODUCTION . As machine learning ( ML ) systems are increasingly deployed into real-world contexts , stakeholder interviews ( Tonekaboni et al. , 2019 ; Bhatt et al. , 2020 ) , design best practices ( Amershi et al. , 2019 ) , and legal frameworks ( European Commission , 2018 ) have underscored the need for explainability . Saliency methods — a class of attribution methods that aim to identify features in an input that were important to a trained ML prediction — are frequently used to provide explanations . However , each method operates differently and , thus , multiple methods can produce seemingly varying explanations for the same model and input . How , then , should we reason about choosing and comparing methods for a particular application ? Prior work has suggested that “ faithfulness , ” or how accurately a saliency result reflects the underlying model , is a desideratum for these methods ( Li et al. , 2021 ; Ding & Koehn , 2021 ; Tomsett et al. , 2020 ; Adebayo et al. , 2018 ) . In this paper , however , we argue that faithfulness is not a productive goal for saliency methods — by design , they can not offer a complete and accurate reflection of a model ’ s behavior , akin to a printout of model weights . Rather , we frame saliency methods as abstractions of model behavior that selectively preserve and necessarily sacrifice information in service of human-centric goals such as simplicity and understandability . With this framing , we propose a nine-dimensional framework to characterize and compare saliency methods . These dimensions fall into three categories , corresponding to different parts of the interpretation process : methodology , or how the saliency is computed ; sensitivity , or relationships between the saliency and the underlying model or input ; and perceptibility , or how an end-user perceives the saliency . These dimensions decompose a singular notion of faithfulness into more granular units that can be reasoned about individually : by situating methods along these dimensions , we can surface their relative strengths , limitations , and differences to better understand trade offs that are latent in their design . In doing so , we demonstrate how our framework allows us to develop “ saliency 0 cards ” ( akin to “ model cards ” ( Mitchell et al. , 2019 ) and “ datasheets ” ( Gebru et al. , 2018 ) ) to document individual methods and to better contextualize saliency results . Moreover , using a concrete example of ML-based radiology diagnostic systems , we show how downstream stakeholders can use our framework to weigh tradeoffs and choose a task-appropriate method . Finally , we show how our framework identifies compelling opportunities for future work including exploring understudied dimensions and developing new metrics that target particular dimensions . 2 RELATED WORK : SALIENCY METHODS AND THEIR EVALUATIONS . Saliency methods ( sometimes referred to as feature attribution methods ) produce explanations for an ML model ’ s output . Given an input , saliency methods compute an importance score for each input feature describing its influence on the model ’ s output . Existing categorizations for saliency methods have focused on algorithmic properties ( gradient or perturbation-based , path-attribution or gradientonly ) Molnar ( 2019 ) . Our framework aims to categorize a broader range of important characteristics . We apply our framework to a variety of common saliency methods listed in Table 1 . Evaluations of saliency methods have primarily focused on how accurately their results represent model behavior , often referred to as faithfulness . A growing body of work has identified failures of some methods such as susceptibility to adversarial perturbations ( Ghorbani et al. , 2019 ) , lack of neuron discriminativity ( Mahendran & Vedaldi , 2016 ) , and a predisposition towards input recovery ( Nie et al. , 2018 ; Adebayo et al. , 2018 ) . Other work has proposed proxy tests that measure different aspects of faithfulness . Adebayo et al . ( 2018 ) recommend model randomization and data label randomization tests , and Kindermans et al . ( 2019 ) test whether constant input shifts affect saliency results . Samek et al . ( 2016 ) judge saliency methods by iteratively replacing features that have high importance values with random noise and measuring how much the output changes . While these tests quantitatively analyze saliency methods , Tomsett et al . ( 2020 ) found they can produce inconsistent rankings . An alternate line of work has looked to break faithfulness down into measurable axioms . Sundararajan et al . ( 2017 ) propose implementation invariance ( a saliency method should produce the same output on functionally equivalent models ) , sensitivity ( a saliency method should give importance to a feature if and only if changing it leads to a different output ) , and linearity ( if a model is the composition of two sub-models , the saliency method ’ s output for the model should be the weighted sum of its outputs for each sub-model ) . Sundararajan et al . ( 2017 ) and Shrikumar et al . ( 2017 ) both posit a saliency method should exhibit completeness — a saliency method ’ s attributions should sum to the difference between the model ’ s output on the input and the model ’ s output on a neutral input . Shrikumar et al . ( 2017 ) and Montavon et al . ( 2018 ) claim saliency methods should output a continuous function . Montavon et al . ( 2018 ) also propose the axiom of selectivity — a saliency method should distribute importance to features that have the greatest impact on the model ’ s output . Finally , Kindermans et al . ( 2019 ) state that a saliency method should be invariant to constant transforms . While axioms are presented as constraints that all saliency methods should attain , the axes in our framework describe attributes of saliency methods that can be traded off to select the best method for different use cases . To situate saliency methods along each axis , we utilize existing tests ( Kindermans et al. , 2019 ; Adebayo et al. , 2018 ) ; however , our framework surfaces the need for additional evaluations , since existing tests do not fully describe each axis and have not assessed all methods . 3 NINE DIMENSIONS TO CHARACTERIZE & COMPARE SALIENCY METHODS . To distill a language to characterize and compare saliency methods , we treat saliency methods as abstractions of the underlying model behavior . That is , to explain model behavior in a humanunderstandable format ( e.g. , a feature attribution heatmap ) , we must necessarily abstract away some detail and concreteness . A saliency method , by design , is not as precise a reflection of model behavior as a printout of the weights , but it is much more human-interpretable . The idea of abstraction recurs in other parts of the machine learning pipeline . For example , “ test accuracy ” is an abstraction of model performance : it does not capture all aspects of model performance but is nevertheless a convenient approximation . Abstractions selectively preserve information and can be combined to arrive at a more complete picture of the underlying phenomenon — i.e. , combining test accuracy with AUROC to more fully understand model performance . This framing reveals that a single saliency method can not fully explain model behavior . Instead , it is necessary to understand different methods ’ strengths and limitations to choose an appropriate method for the model , domain , and task . Our framework defines nine dimensions that describe what information saliency methods abstract , and how this abstraction is computed . We group these dimensions into three categories that map to different stages of the interpretability process ( see Figure 1 ) . Methodology covers the model , input , and saliency method , and dimensions under this category describe how the saliency method operates . Sensitivity goes one stage further , with dimensions that describe the relationship between the saliency output and the model or input . Perceptibility covers the final stages of the process , with dimensions that describe how users perceive the saliency output and any associated visualizations . These dimensions allow us to decompose faithfulness into aspects that can be reasoned about individually . For instance , Figure 2 demonstrates how our dimensions allow us to situate saliency methods in relation to one another to understand their strengths , limitations , and differences . In the following subsections , we describe and provide an illustrative example of each dimension . 3.1 METHODOLOGY : HOW SALIENCY METHODS OPERATE . Determinism Some saliency techniques are non-deterministic , and running them with different random seeds can produce significantly different outputs . For instance , methods such as RISE or SHAP rely on randomly-generated masks or coalitions . The output from a non-deterministic method represents only one example from a potentially high-variance distribution , and may be afforded authority that does not account for its inherent uncertainty . Although we may never explicitly choose a method for its non-determinism , we may nevertheless accept non-determinism in service of other dimensions such as increased model agnosticism ( e.g. , RISE ) , perceptual correspondence ( e.g. , SHAP ) , or minimality ( e.g. , MP ) . Example . Consider a model assisting dermatologists diagnosing skin cancer . In Figure 3 ( a ) , we show that two runs of a non-deterministic method ( LIME ) on a melanoma prediction model result in different saliency maps . These variations may have significant consequences given small areas of the lesion or surrounding skin may be integral to the diagnosis . Visualizing and interpreting multiple runs together is time-consuming and confusing , but only looking at one run could skew a clinician ’ s judgment . Thus , in this case , it might make sense to prioritize a method with deterministic outputs . Hyperparameter Dependence Hyperparameter dependence captures how many hyperparameters or design decisions the user must specify to run a particular method , and how sensitive it is to them . If there are not sufficient resources or expertise to devote to hyperparameter optimization , simply using default values can lead to misleading results . Similarly , misleading or confusing results might arise if the hyperparameters were chosen based on a particular dataset but deployed in a setting in which there is a significant distribution shift . In situations like this , it makes sense to prioritize methods with low hyperparameter dependence . In other cases , where developers have dedicated appropriate time and resources to tuning , it might be preferable to use a method dependent on hyperparameters ( e.g. , IG , SmoothGrad ) because it improves other dimensions ( e.g. , minimality ) . Example . IG computes feature importances by interpolating between a “ meaningless ” baseline input and the actual input , accumulating the gradients at each step . A common practice is to use a baseline value of all zeroes ; however , in some cases , a zero baseline can be misleading and potentially harmful . Take a model trained to predict bone fractures from x-ray images . Fractures often appear as a black line in the bone , but , since they have the same pixel value as the baseline ( zero ) , the IG result will indicate that they are not important . In Figure 3 ( b ) , we show a similar example where the choice of baseline ( black , white ) has a strong effect on the saliency map . Choosing an appropriate baseline requires time and a deep understanding of the data and method ; if this is not available in a particular application , it is better to prioritize a saliency method without important hyperparameters . Model Agnosticism Model agnostic methods ( i.e. , SHAP ) treat the underlying model as a blackbox , relying only on its input and output . On the other hand , model-dependent methods need access to various model internals , and may have specific requirements , such as differentiability ( i.e. , | This article proposes a review of "saliency" methods in deep neural networks and rates them according to 9 dimensions: Computational efficiency, determinism, hyperparameters, model agnosticism, input sensitivity, minimality, perceptual correspondence and semantic directness. The author's claim is that this new taxonomy provides a better way for choosing between saliency approaches for the end-user. Note: What is meant by saliency in this paper is methods, such as Grad-CAM to determine which inputs were more prevalent in a network's decision (the word has different meanings in other sub-fields). | SP:d8c1b4c5169c6e2675d7b211c8beee638a256a55 |
Beyond Faithfulness: A Framework to Characterize and Compare Saliency Methods | 1 INTRODUCTION . As machine learning ( ML ) systems are increasingly deployed into real-world contexts , stakeholder interviews ( Tonekaboni et al. , 2019 ; Bhatt et al. , 2020 ) , design best practices ( Amershi et al. , 2019 ) , and legal frameworks ( European Commission , 2018 ) have underscored the need for explainability . Saliency methods — a class of attribution methods that aim to identify features in an input that were important to a trained ML prediction — are frequently used to provide explanations . However , each method operates differently and , thus , multiple methods can produce seemingly varying explanations for the same model and input . How , then , should we reason about choosing and comparing methods for a particular application ? Prior work has suggested that “ faithfulness , ” or how accurately a saliency result reflects the underlying model , is a desideratum for these methods ( Li et al. , 2021 ; Ding & Koehn , 2021 ; Tomsett et al. , 2020 ; Adebayo et al. , 2018 ) . In this paper , however , we argue that faithfulness is not a productive goal for saliency methods — by design , they can not offer a complete and accurate reflection of a model ’ s behavior , akin to a printout of model weights . Rather , we frame saliency methods as abstractions of model behavior that selectively preserve and necessarily sacrifice information in service of human-centric goals such as simplicity and understandability . With this framing , we propose a nine-dimensional framework to characterize and compare saliency methods . These dimensions fall into three categories , corresponding to different parts of the interpretation process : methodology , or how the saliency is computed ; sensitivity , or relationships between the saliency and the underlying model or input ; and perceptibility , or how an end-user perceives the saliency . These dimensions decompose a singular notion of faithfulness into more granular units that can be reasoned about individually : by situating methods along these dimensions , we can surface their relative strengths , limitations , and differences to better understand trade offs that are latent in their design . In doing so , we demonstrate how our framework allows us to develop “ saliency 0 cards ” ( akin to “ model cards ” ( Mitchell et al. , 2019 ) and “ datasheets ” ( Gebru et al. , 2018 ) ) to document individual methods and to better contextualize saliency results . Moreover , using a concrete example of ML-based radiology diagnostic systems , we show how downstream stakeholders can use our framework to weigh tradeoffs and choose a task-appropriate method . Finally , we show how our framework identifies compelling opportunities for future work including exploring understudied dimensions and developing new metrics that target particular dimensions . 2 RELATED WORK : SALIENCY METHODS AND THEIR EVALUATIONS . Saliency methods ( sometimes referred to as feature attribution methods ) produce explanations for an ML model ’ s output . Given an input , saliency methods compute an importance score for each input feature describing its influence on the model ’ s output . Existing categorizations for saliency methods have focused on algorithmic properties ( gradient or perturbation-based , path-attribution or gradientonly ) Molnar ( 2019 ) . Our framework aims to categorize a broader range of important characteristics . We apply our framework to a variety of common saliency methods listed in Table 1 . Evaluations of saliency methods have primarily focused on how accurately their results represent model behavior , often referred to as faithfulness . A growing body of work has identified failures of some methods such as susceptibility to adversarial perturbations ( Ghorbani et al. , 2019 ) , lack of neuron discriminativity ( Mahendran & Vedaldi , 2016 ) , and a predisposition towards input recovery ( Nie et al. , 2018 ; Adebayo et al. , 2018 ) . Other work has proposed proxy tests that measure different aspects of faithfulness . Adebayo et al . ( 2018 ) recommend model randomization and data label randomization tests , and Kindermans et al . ( 2019 ) test whether constant input shifts affect saliency results . Samek et al . ( 2016 ) judge saliency methods by iteratively replacing features that have high importance values with random noise and measuring how much the output changes . While these tests quantitatively analyze saliency methods , Tomsett et al . ( 2020 ) found they can produce inconsistent rankings . An alternate line of work has looked to break faithfulness down into measurable axioms . Sundararajan et al . ( 2017 ) propose implementation invariance ( a saliency method should produce the same output on functionally equivalent models ) , sensitivity ( a saliency method should give importance to a feature if and only if changing it leads to a different output ) , and linearity ( if a model is the composition of two sub-models , the saliency method ’ s output for the model should be the weighted sum of its outputs for each sub-model ) . Sundararajan et al . ( 2017 ) and Shrikumar et al . ( 2017 ) both posit a saliency method should exhibit completeness — a saliency method ’ s attributions should sum to the difference between the model ’ s output on the input and the model ’ s output on a neutral input . Shrikumar et al . ( 2017 ) and Montavon et al . ( 2018 ) claim saliency methods should output a continuous function . Montavon et al . ( 2018 ) also propose the axiom of selectivity — a saliency method should distribute importance to features that have the greatest impact on the model ’ s output . Finally , Kindermans et al . ( 2019 ) state that a saliency method should be invariant to constant transforms . While axioms are presented as constraints that all saliency methods should attain , the axes in our framework describe attributes of saliency methods that can be traded off to select the best method for different use cases . To situate saliency methods along each axis , we utilize existing tests ( Kindermans et al. , 2019 ; Adebayo et al. , 2018 ) ; however , our framework surfaces the need for additional evaluations , since existing tests do not fully describe each axis and have not assessed all methods . 3 NINE DIMENSIONS TO CHARACTERIZE & COMPARE SALIENCY METHODS . To distill a language to characterize and compare saliency methods , we treat saliency methods as abstractions of the underlying model behavior . That is , to explain model behavior in a humanunderstandable format ( e.g. , a feature attribution heatmap ) , we must necessarily abstract away some detail and concreteness . A saliency method , by design , is not as precise a reflection of model behavior as a printout of the weights , but it is much more human-interpretable . The idea of abstraction recurs in other parts of the machine learning pipeline . For example , “ test accuracy ” is an abstraction of model performance : it does not capture all aspects of model performance but is nevertheless a convenient approximation . Abstractions selectively preserve information and can be combined to arrive at a more complete picture of the underlying phenomenon — i.e. , combining test accuracy with AUROC to more fully understand model performance . This framing reveals that a single saliency method can not fully explain model behavior . Instead , it is necessary to understand different methods ’ strengths and limitations to choose an appropriate method for the model , domain , and task . Our framework defines nine dimensions that describe what information saliency methods abstract , and how this abstraction is computed . We group these dimensions into three categories that map to different stages of the interpretability process ( see Figure 1 ) . Methodology covers the model , input , and saliency method , and dimensions under this category describe how the saliency method operates . Sensitivity goes one stage further , with dimensions that describe the relationship between the saliency output and the model or input . Perceptibility covers the final stages of the process , with dimensions that describe how users perceive the saliency output and any associated visualizations . These dimensions allow us to decompose faithfulness into aspects that can be reasoned about individually . For instance , Figure 2 demonstrates how our dimensions allow us to situate saliency methods in relation to one another to understand their strengths , limitations , and differences . In the following subsections , we describe and provide an illustrative example of each dimension . 3.1 METHODOLOGY : HOW SALIENCY METHODS OPERATE . Determinism Some saliency techniques are non-deterministic , and running them with different random seeds can produce significantly different outputs . For instance , methods such as RISE or SHAP rely on randomly-generated masks or coalitions . The output from a non-deterministic method represents only one example from a potentially high-variance distribution , and may be afforded authority that does not account for its inherent uncertainty . Although we may never explicitly choose a method for its non-determinism , we may nevertheless accept non-determinism in service of other dimensions such as increased model agnosticism ( e.g. , RISE ) , perceptual correspondence ( e.g. , SHAP ) , or minimality ( e.g. , MP ) . Example . Consider a model assisting dermatologists diagnosing skin cancer . In Figure 3 ( a ) , we show that two runs of a non-deterministic method ( LIME ) on a melanoma prediction model result in different saliency maps . These variations may have significant consequences given small areas of the lesion or surrounding skin may be integral to the diagnosis . Visualizing and interpreting multiple runs together is time-consuming and confusing , but only looking at one run could skew a clinician ’ s judgment . Thus , in this case , it might make sense to prioritize a method with deterministic outputs . Hyperparameter Dependence Hyperparameter dependence captures how many hyperparameters or design decisions the user must specify to run a particular method , and how sensitive it is to them . If there are not sufficient resources or expertise to devote to hyperparameter optimization , simply using default values can lead to misleading results . Similarly , misleading or confusing results might arise if the hyperparameters were chosen based on a particular dataset but deployed in a setting in which there is a significant distribution shift . In situations like this , it makes sense to prioritize methods with low hyperparameter dependence . In other cases , where developers have dedicated appropriate time and resources to tuning , it might be preferable to use a method dependent on hyperparameters ( e.g. , IG , SmoothGrad ) because it improves other dimensions ( e.g. , minimality ) . Example . IG computes feature importances by interpolating between a “ meaningless ” baseline input and the actual input , accumulating the gradients at each step . A common practice is to use a baseline value of all zeroes ; however , in some cases , a zero baseline can be misleading and potentially harmful . Take a model trained to predict bone fractures from x-ray images . Fractures often appear as a black line in the bone , but , since they have the same pixel value as the baseline ( zero ) , the IG result will indicate that they are not important . In Figure 3 ( b ) , we show a similar example where the choice of baseline ( black , white ) has a strong effect on the saliency map . Choosing an appropriate baseline requires time and a deep understanding of the data and method ; if this is not available in a particular application , it is better to prioritize a saliency method without important hyperparameters . Model Agnosticism Model agnostic methods ( i.e. , SHAP ) treat the underlying model as a blackbox , relying only on its input and output . On the other hand , model-dependent methods need access to various model internals , and may have specific requirements , such as differentiability ( i.e. , | This paper proposes a framework to evaluate saliency-based explanations for neural network decisions. The framework evaluates each method along a number of different dimensions (for example, reporting whether the saliency map output is fixed or stochastic, or the degree to which the output depends on hyperparameters). The proposed framework is not actually implemented or tested in this paper. | SP:d8c1b4c5169c6e2675d7b211c8beee638a256a55 |
How does BERT address polysemy of Korean adverbial postpositions -ey, -eyse, and -(u)lo? | 1 INTRODUCTION . Polysemy , one type of ambiguity , occurs when one form delivers multiple , and yet related , meanings/functions and vice versa ( Glynn & Robinson , 2014 ) . Traditional word-embedding models showed an unsatisfactory level of performance in polysemy interpretation . This is due to the technical nature of these models : they are static in that a single vector is assigned to each word ( Desagulier , 2019 ; Ethayarajh , 2019 ; Liu et al. , 2019a ) . To overcome this issue , recent studies have proposed a contextualized word-embedding model which considers neighborhood information about a polysemous word on the basis of sequences of words around the target word . Various models have been suggested for this task , such as Embeddings from Language Models ( Peters et al. , 2018 ) , Generative Pre-Training ( Radford et al. , 2018 ) , and Bidirectional Encoder Representations from Transformer ( BERT ; Devlin et al. , 2018 ) . Among these models , BERT shows the best performance in many language tasks such as translation , classification , and question-answering ( e.g. , Devlin et al. , 2018 ; Tang et al. , 2019 ) . Despite a good deal of research on BERT in English , very few studies have investigated BERT-based polysemy interpretation in languages that are typologically different from English . We turn our attention to Korean , an agglutinative Subject–Object–Verb language in which multiple postpositions or affixes with dedicated forms and meanings are attached to the stem of nominals or predicates . A postposition is a function word providing grammatical information to words it is attached ( Sohn , 1999 ) . It normally involves many-to-many associations between form and function ; as such , a postposition is polysemous ( Choo & Kwak , 2008 ) . Several studies have used word-embedding models to capture and tease apart the different meanings/functions of Korean postpositions ( e.g. , Bae et al. , 2014 ; 2015 ; Kim & Ock , 2016 ; Lee et al. , 2015 ; Mun & Shin , 2020 ; Shin et al. , 2005 ) . However , the model performance reported in the previous studies is unsatisfactory , with the accuracy ranging from 0.621 ( Bae et al. , 2014 ) to 0.837 ( Kim & Ock , 2016 ) . One possible reason for this unsatisfactory performance is that they did not consider contextual information . Against this background , the current study employs BERT for the same kind of classification task for Korean postpositions . BERT produces contextual embeddings , and this characteristic may help us to create a better classification system for postpositions . Still unclear is the particular reason for BERT ’ s superior performance over the others . In order to further understand how BERT recognizes the word-level polysemy , we propose a BERT-based visualization system in addressing polysemy interpretation of three adverbial postpositions , -ey , -eyse , and - ( u ) lo , which are frequently used and documented in the previous studies ( e.g. , Cho & Kim , 1996 ; Jeong , 2010 ; Nam , 1993 ; Park , 1999 ; Song , 2014 ) . 2 KOREAN ADVERBIAL POSTPOSITIONS : -ey , -eyse , AND - ( u ) lo In order to determine the number of functions of each postposition , this study considers the major functions of these postpositions which are frequently attested in the Sejong dictionary : eight for -ey , two for -eyse , and six for - ( u ) lo ( Shin , 2008 ) . -ey involves the following functions : agent ( AGT ) , criterion ( CRT ) , effector ( EFF ) , final state ( FNS ) , goal ( GOL ) , instrument ( INS ) , location ( LOC ) , and theme ( THM ) . ( 1 ) -ey as AGT ( agent ) 가두 katwu street 진출이 cinchwul-i go.out-NOM 경찰에 kyengchal-ey police-AGT 저지되었다 . ceci-toy-ess-ta . stop-PSV-PST-DECL ‘ By going out to the street was stopped by the police. ’ ( 2 ) -ey as CRT ( criterion ) 영호는 Yenghuy-nun Yenghuy-TOP 20만원에 20manwen-ey 200,000 won-CRT 모니터를 monithe-lul moniter-ACC 낙찰했다 . nakchalhay-ss-ta . sell-PST-DECL ‘ Yenghuy sold the monitor ( to a bidder ) for 200,000 won. ’ ( 3 ) -ey as EFF ( effector ) 문들이 mwun-tul-i door-PL-NOM 거센 keseyn strong 바람에 palam-ey wind-EFF 모두 motwu all 건들댄다 . kentultay-n-ta . sway-PRS-DECL ‘ The doors all sway by the strong wind. ’ ( 4 ) -ey as FNS ( final state ) 김교수는 kimkyoswu-nun professor.Kim-TOP 조교에 cokyo-ey assistant-FNS 박군을 park-kwun-ul Park-Mr-ACC 추천했다 . chwuchenhay-ss-ta . recommend-PST-DECL ‘ Professor Kim recommended Park as an assistant. ’ ( 5 ) -ey as GOL ( goal ) 철수가 Chelswu-ka Chelswu-TOP 던진 tenc-i-n throw-CST-PRS 칼이 khal-i knife-NOM 땅바닥에 ttangpatak-ey ground-GOL 내리꽂혔다 . naylyekkoc-hi-ess-ta . stick-PSV-PST-DECL ‘ The knife thrown by Chelswu stuck to the ground. ’ ( 6 ) -ey as INS ( instrument ) 그 ku That 어린 eli-n young-REL 소년은 sonye-nun boy-TOP 화롯불에 hwalospwul-ey fire-INS 손을 son-ul hand-ACC 녹이고 nok-i-ko melt-CST-and 있었다 . iss-ess-ta . be-PST-DECL ‘ The young boy was using the fire to warm his hands. ’ ( 7 ) -ey as LOC ( location ) 그는 ku-nun He-TOP 온종일 oncongil all day 서재에 secay-ey study.room-LOC 파묻혀 phamwut-hi-e bury.in-PSV-PRS 지낸다 . cinay-n-ta . be-PRS-DECL ‘ He is buried in his study room all day. ’ ( 8 ) -ey as THM ( theme ) 현대인들은 hyentayin-tul-un modern.people-PL-TOP 모두 motwu all 참된 chamtoy-n true-REL 지식에 cisik-ey knowledge-THM 허기져있다 . hekicye-iss-ta . hungry-PRS-DECL ‘ All modern people are hungry for true knowledge. ’ -eyse has only two functions , location ( LOC ) ( 9 ) and source ( SRC ) ( 10 ) . -eyse manifests fewer functions than -ey ( Choo & Kwak , 2008 ) . However , frequency is equally high compared to that of -ey ( e.g. , Cho & Kim , 1996 ; Song , 2014 ) . ( 9 ) -eyse as LOC ( location ) 철수는 Chelswu-nun Chelswu-TOP 서울에서 sewul-eyse seoul-LOC 태어났다 . thayena-ss-ta . born-PST-DECL ‘ Chelswu was born in Seoul. ’ ( 10 ) -eyse as SRC ( source ) 광부들이 kwangpwutul-i miner-PL-NOM 바다에서 pata-eyse sea-SRC 석유를 sekyu-lul oil-ACC 뽑아올린다 . ppopaoll-i-n-ta . pull-CST-PRS-DECL ‘ Miners pull oil from the sea. ’ - ( u ) lo engages in six functions : criterion ( CRT ) , direction ( DIR ) , effector ( EFF ) , final state ( FNS ) , instrument ( INS ) , and location ( LOC ) ( Shin , 2008 ) . ( 11 ) - ( u ) lo as CRT ( criterion ) 적당한 cektangha-n appropriate-REL 시간 sikan time 간격으로 kankyek-ulo interval-CRT 배차되었다 . paycha-toy-ess-ta . arrange-PSV-PST-DECL ‘ It was arranged at appropriate time intervals. ’ ( 12 ) - ( u ) lo as DIR ( direction ) 범인은 pemin-un criminal-NOM 어두운 etwuwun dark 골목으로 kolmok-ulo alley-DIR 달아났다 . talana-ss-ta . flee-PST-DECL ‘ The criminal fled into a dark alley. ’ ( 13 ) - ( u ) lo as EFF ( effector ) 환자가 hwanca-ka patient-NOM 위암으로 wiam-ulo stomach.cancer-EFF 매우 maywu very 괴로워하고 koyloweha-ko suffer-and 있습니다 . iss-supni-ta . be-HON-DECL ‘ The patient is suffering greatly due to stomach cancer. ’ ( 14 ) - ( u ) lo as FNS ( final state ) 그는 ku-nun He-TOP 대표 tayphyo representative 강사로 kangsa-lo lecturer-FNS 초빙되었다 . choping-toy-ess-ta . invite-PSV-PST-DECL ‘ He was invited as a representative lecturer. ’ ( 15 ) - ( u ) lo as INS ( instrument ) 전선이 censen-i wire-NOM 연줄로 yencwul-lo connection.wire-INS 감겼다 . kam-ki-ess-ta . wind-PSV-PST-DECL ‘ The wire wound around with the connection wire. ’ ( 16 ) - ( u ) lo as LOC ( location ) 경찰이 kyengchal-i police-NOM 피해자를 phiuyca-lul suspect-ACC 검찰로 kemchal-lo prosecution-LOC 압송했다 . apsonghay-ss-ta . transport.do-PST-DECL ‘ The police transported the suspect to the prosecution . ’ 3 METHODS . 3.1 CREATING INPUT CORPUS . The Sejong primary corpus1 , the representative corpus in Korean , does not code the information about the functions of postpositions directly in each sentence ( which is necessary for model training ) . We thus annotated a portion of the original corpus data manually . For this purpose , we extracted sentences that have only one postposition and predicate . This also allowed us to control for additional confounding factors which might have interfered with the performance of our model . We then extracted 5,000 sentences randomly for each postposition from the initial dataset . 1Sejong corpus is available at : https : //www.korean.go.kr Three native speakers of Korean annotated each postposition for its function in this 15,000-sentence corpus . Fleiss ’ kappa scores showed that the annotators ’ outcomes were almost identical : 0.948 ( - ey ) , 0.928 ( -eyse ) , and 0.947 ( - ( u ) lo ) . We further excluded instances which showed disagreement among the annotators . The final corpus consisted of 4,715 sentences for -ey , 4,853 sentences for -eyse , and 4,708 sentences for - ( u ) lo . Table 1 presents the detailed by-function frequency list of the three postpositions 2 . 3.2 CREATING TRAINING AND TEST SETS . We pre-processed the data in consideration of how BERT works ( we used the original BERT model for this task ) . First , we added [ CLS ] ( ‘ classification ’ ; indicating the start of a sentence ) before a sentence and [ SEP ] ( ‘ separation ’ ; indicating the end of a sentence ) after a sentence to indicate where the sentence starts and ends . These indicators made it possible for the BERT model to recognize a sentence boundary in a text , allowing the model to learn word meaning while considering intersentential variations . Second , we made a separate column ( ‘ Label ’ ) to indicate the intended function of each postposition in each sentence ( Figure 1 ) . We then split the corpus into two sub-sets , one with 90 per cent of the corpus for the training and with the remaining 10 per cent of the corpus for the testing . 3.3 DEVELOPING BERT CLASSIFICATION MODEL . We set the parameters related to BERT training such as batch size ( 32 ) , epoch ( 50 ) , seed ( 42 ) , epsilon ( 0.00000008 ) , and learning rate ( 0.00002 ) , as advised by McCormick ( 2019 ) . We then employed a pre-trained language model in order to obtain high accuracy of outcomes ; for this purpose , we used a Korean BERT model ( KoBERT ; Jeon et al. , 2019 ) . Before the actual BERT training , we transformed the input data into three embedding types—token embeddings , position embeddings , and segment embeddings ( cf. , Devlin et al. , 2018 ) —in the following ways . First , for the token embedding , we used KoBertTokenizer for the sentence tokenization ( the maximum number of tokens for each sentence was set to 128 ) . Second , we converted each token into numeric values indicating unique indices of the tokens in the vocabulary of KoBERT for the position embedding . Third , for the segment embedding , we converted the number of tokens of each sentence into 128 numeric values using 0 ( i.e. , not existed ) or 1 ( i.e. , existed ) . The labels of the data indicating the intended function of each postposition in the sentence were stored separately . After this transformation step , we proceeded to the model training as follows . We first loaded KoBERT through the function BertForSequenceClassification from transformers ( Wolf et al. , 2019 ) . 2Our corpus is available at : we will propose a URL Next , we fine-tuned the pre-trained model by using the training set , with a view to reducing loss values and updating the learning rate for better classification accuracy of the model . We then loaded the testing set to evaluate whether the fine-tuned model successfully recognized the intended functions of each postposition in each sentence . In this part , the rates of accuracy for each function and the total accuracy rate were calculated by comparing the intended function of each postposition in each test sentence with the classified function of each postposition via the BERT model . Lastly , we employed t-distributed Stochastic Neighbor Embedding ( t-SNE ; Maaten & Hinton , 2008 ) for dimension reduction of classification embeddings from the postposition per epoch . In addition , to statistically confirm the changes of sentence-level embedding outcomes by each epoch , we performed densitybased clustering ( Sander et al. , 1998 ) . These outcomes were fed into the visualization system , which we outline next . | This paper builds a BERT classifier for word-level polysemy of Korean language. Adverbial positions such as -ey, eyes, and –(u)lo are involved in the investigation. There are three findings during the experiments, if there are more functions then the classification accuracy is lower, corpus size is sensitive and more epochs are preferred. | SP:623af8b95ab193ee419da1a8d2962faadc82e2d8 |
How does BERT address polysemy of Korean adverbial postpositions -ey, -eyse, and -(u)lo? | 1 INTRODUCTION . Polysemy , one type of ambiguity , occurs when one form delivers multiple , and yet related , meanings/functions and vice versa ( Glynn & Robinson , 2014 ) . Traditional word-embedding models showed an unsatisfactory level of performance in polysemy interpretation . This is due to the technical nature of these models : they are static in that a single vector is assigned to each word ( Desagulier , 2019 ; Ethayarajh , 2019 ; Liu et al. , 2019a ) . To overcome this issue , recent studies have proposed a contextualized word-embedding model which considers neighborhood information about a polysemous word on the basis of sequences of words around the target word . Various models have been suggested for this task , such as Embeddings from Language Models ( Peters et al. , 2018 ) , Generative Pre-Training ( Radford et al. , 2018 ) , and Bidirectional Encoder Representations from Transformer ( BERT ; Devlin et al. , 2018 ) . Among these models , BERT shows the best performance in many language tasks such as translation , classification , and question-answering ( e.g. , Devlin et al. , 2018 ; Tang et al. , 2019 ) . Despite a good deal of research on BERT in English , very few studies have investigated BERT-based polysemy interpretation in languages that are typologically different from English . We turn our attention to Korean , an agglutinative Subject–Object–Verb language in which multiple postpositions or affixes with dedicated forms and meanings are attached to the stem of nominals or predicates . A postposition is a function word providing grammatical information to words it is attached ( Sohn , 1999 ) . It normally involves many-to-many associations between form and function ; as such , a postposition is polysemous ( Choo & Kwak , 2008 ) . Several studies have used word-embedding models to capture and tease apart the different meanings/functions of Korean postpositions ( e.g. , Bae et al. , 2014 ; 2015 ; Kim & Ock , 2016 ; Lee et al. , 2015 ; Mun & Shin , 2020 ; Shin et al. , 2005 ) . However , the model performance reported in the previous studies is unsatisfactory , with the accuracy ranging from 0.621 ( Bae et al. , 2014 ) to 0.837 ( Kim & Ock , 2016 ) . One possible reason for this unsatisfactory performance is that they did not consider contextual information . Against this background , the current study employs BERT for the same kind of classification task for Korean postpositions . BERT produces contextual embeddings , and this characteristic may help us to create a better classification system for postpositions . Still unclear is the particular reason for BERT ’ s superior performance over the others . In order to further understand how BERT recognizes the word-level polysemy , we propose a BERT-based visualization system in addressing polysemy interpretation of three adverbial postpositions , -ey , -eyse , and - ( u ) lo , which are frequently used and documented in the previous studies ( e.g. , Cho & Kim , 1996 ; Jeong , 2010 ; Nam , 1993 ; Park , 1999 ; Song , 2014 ) . 2 KOREAN ADVERBIAL POSTPOSITIONS : -ey , -eyse , AND - ( u ) lo In order to determine the number of functions of each postposition , this study considers the major functions of these postpositions which are frequently attested in the Sejong dictionary : eight for -ey , two for -eyse , and six for - ( u ) lo ( Shin , 2008 ) . -ey involves the following functions : agent ( AGT ) , criterion ( CRT ) , effector ( EFF ) , final state ( FNS ) , goal ( GOL ) , instrument ( INS ) , location ( LOC ) , and theme ( THM ) . ( 1 ) -ey as AGT ( agent ) 가두 katwu street 진출이 cinchwul-i go.out-NOM 경찰에 kyengchal-ey police-AGT 저지되었다 . ceci-toy-ess-ta . stop-PSV-PST-DECL ‘ By going out to the street was stopped by the police. ’ ( 2 ) -ey as CRT ( criterion ) 영호는 Yenghuy-nun Yenghuy-TOP 20만원에 20manwen-ey 200,000 won-CRT 모니터를 monithe-lul moniter-ACC 낙찰했다 . nakchalhay-ss-ta . sell-PST-DECL ‘ Yenghuy sold the monitor ( to a bidder ) for 200,000 won. ’ ( 3 ) -ey as EFF ( effector ) 문들이 mwun-tul-i door-PL-NOM 거센 keseyn strong 바람에 palam-ey wind-EFF 모두 motwu all 건들댄다 . kentultay-n-ta . sway-PRS-DECL ‘ The doors all sway by the strong wind. ’ ( 4 ) -ey as FNS ( final state ) 김교수는 kimkyoswu-nun professor.Kim-TOP 조교에 cokyo-ey assistant-FNS 박군을 park-kwun-ul Park-Mr-ACC 추천했다 . chwuchenhay-ss-ta . recommend-PST-DECL ‘ Professor Kim recommended Park as an assistant. ’ ( 5 ) -ey as GOL ( goal ) 철수가 Chelswu-ka Chelswu-TOP 던진 tenc-i-n throw-CST-PRS 칼이 khal-i knife-NOM 땅바닥에 ttangpatak-ey ground-GOL 내리꽂혔다 . naylyekkoc-hi-ess-ta . stick-PSV-PST-DECL ‘ The knife thrown by Chelswu stuck to the ground. ’ ( 6 ) -ey as INS ( instrument ) 그 ku That 어린 eli-n young-REL 소년은 sonye-nun boy-TOP 화롯불에 hwalospwul-ey fire-INS 손을 son-ul hand-ACC 녹이고 nok-i-ko melt-CST-and 있었다 . iss-ess-ta . be-PST-DECL ‘ The young boy was using the fire to warm his hands. ’ ( 7 ) -ey as LOC ( location ) 그는 ku-nun He-TOP 온종일 oncongil all day 서재에 secay-ey study.room-LOC 파묻혀 phamwut-hi-e bury.in-PSV-PRS 지낸다 . cinay-n-ta . be-PRS-DECL ‘ He is buried in his study room all day. ’ ( 8 ) -ey as THM ( theme ) 현대인들은 hyentayin-tul-un modern.people-PL-TOP 모두 motwu all 참된 chamtoy-n true-REL 지식에 cisik-ey knowledge-THM 허기져있다 . hekicye-iss-ta . hungry-PRS-DECL ‘ All modern people are hungry for true knowledge. ’ -eyse has only two functions , location ( LOC ) ( 9 ) and source ( SRC ) ( 10 ) . -eyse manifests fewer functions than -ey ( Choo & Kwak , 2008 ) . However , frequency is equally high compared to that of -ey ( e.g. , Cho & Kim , 1996 ; Song , 2014 ) . ( 9 ) -eyse as LOC ( location ) 철수는 Chelswu-nun Chelswu-TOP 서울에서 sewul-eyse seoul-LOC 태어났다 . thayena-ss-ta . born-PST-DECL ‘ Chelswu was born in Seoul. ’ ( 10 ) -eyse as SRC ( source ) 광부들이 kwangpwutul-i miner-PL-NOM 바다에서 pata-eyse sea-SRC 석유를 sekyu-lul oil-ACC 뽑아올린다 . ppopaoll-i-n-ta . pull-CST-PRS-DECL ‘ Miners pull oil from the sea. ’ - ( u ) lo engages in six functions : criterion ( CRT ) , direction ( DIR ) , effector ( EFF ) , final state ( FNS ) , instrument ( INS ) , and location ( LOC ) ( Shin , 2008 ) . ( 11 ) - ( u ) lo as CRT ( criterion ) 적당한 cektangha-n appropriate-REL 시간 sikan time 간격으로 kankyek-ulo interval-CRT 배차되었다 . paycha-toy-ess-ta . arrange-PSV-PST-DECL ‘ It was arranged at appropriate time intervals. ’ ( 12 ) - ( u ) lo as DIR ( direction ) 범인은 pemin-un criminal-NOM 어두운 etwuwun dark 골목으로 kolmok-ulo alley-DIR 달아났다 . talana-ss-ta . flee-PST-DECL ‘ The criminal fled into a dark alley. ’ ( 13 ) - ( u ) lo as EFF ( effector ) 환자가 hwanca-ka patient-NOM 위암으로 wiam-ulo stomach.cancer-EFF 매우 maywu very 괴로워하고 koyloweha-ko suffer-and 있습니다 . iss-supni-ta . be-HON-DECL ‘ The patient is suffering greatly due to stomach cancer. ’ ( 14 ) - ( u ) lo as FNS ( final state ) 그는 ku-nun He-TOP 대표 tayphyo representative 강사로 kangsa-lo lecturer-FNS 초빙되었다 . choping-toy-ess-ta . invite-PSV-PST-DECL ‘ He was invited as a representative lecturer. ’ ( 15 ) - ( u ) lo as INS ( instrument ) 전선이 censen-i wire-NOM 연줄로 yencwul-lo connection.wire-INS 감겼다 . kam-ki-ess-ta . wind-PSV-PST-DECL ‘ The wire wound around with the connection wire. ’ ( 16 ) - ( u ) lo as LOC ( location ) 경찰이 kyengchal-i police-NOM 피해자를 phiuyca-lul suspect-ACC 검찰로 kemchal-lo prosecution-LOC 압송했다 . apsonghay-ss-ta . transport.do-PST-DECL ‘ The police transported the suspect to the prosecution . ’ 3 METHODS . 3.1 CREATING INPUT CORPUS . The Sejong primary corpus1 , the representative corpus in Korean , does not code the information about the functions of postpositions directly in each sentence ( which is necessary for model training ) . We thus annotated a portion of the original corpus data manually . For this purpose , we extracted sentences that have only one postposition and predicate . This also allowed us to control for additional confounding factors which might have interfered with the performance of our model . We then extracted 5,000 sentences randomly for each postposition from the initial dataset . 1Sejong corpus is available at : https : //www.korean.go.kr Three native speakers of Korean annotated each postposition for its function in this 15,000-sentence corpus . Fleiss ’ kappa scores showed that the annotators ’ outcomes were almost identical : 0.948 ( - ey ) , 0.928 ( -eyse ) , and 0.947 ( - ( u ) lo ) . We further excluded instances which showed disagreement among the annotators . The final corpus consisted of 4,715 sentences for -ey , 4,853 sentences for -eyse , and 4,708 sentences for - ( u ) lo . Table 1 presents the detailed by-function frequency list of the three postpositions 2 . 3.2 CREATING TRAINING AND TEST SETS . We pre-processed the data in consideration of how BERT works ( we used the original BERT model for this task ) . First , we added [ CLS ] ( ‘ classification ’ ; indicating the start of a sentence ) before a sentence and [ SEP ] ( ‘ separation ’ ; indicating the end of a sentence ) after a sentence to indicate where the sentence starts and ends . These indicators made it possible for the BERT model to recognize a sentence boundary in a text , allowing the model to learn word meaning while considering intersentential variations . Second , we made a separate column ( ‘ Label ’ ) to indicate the intended function of each postposition in each sentence ( Figure 1 ) . We then split the corpus into two sub-sets , one with 90 per cent of the corpus for the training and with the remaining 10 per cent of the corpus for the testing . 3.3 DEVELOPING BERT CLASSIFICATION MODEL . We set the parameters related to BERT training such as batch size ( 32 ) , epoch ( 50 ) , seed ( 42 ) , epsilon ( 0.00000008 ) , and learning rate ( 0.00002 ) , as advised by McCormick ( 2019 ) . We then employed a pre-trained language model in order to obtain high accuracy of outcomes ; for this purpose , we used a Korean BERT model ( KoBERT ; Jeon et al. , 2019 ) . Before the actual BERT training , we transformed the input data into three embedding types—token embeddings , position embeddings , and segment embeddings ( cf. , Devlin et al. , 2018 ) —in the following ways . First , for the token embedding , we used KoBertTokenizer for the sentence tokenization ( the maximum number of tokens for each sentence was set to 128 ) . Second , we converted each token into numeric values indicating unique indices of the tokens in the vocabulary of KoBERT for the position embedding . Third , for the segment embedding , we converted the number of tokens of each sentence into 128 numeric values using 0 ( i.e. , not existed ) or 1 ( i.e. , existed ) . The labels of the data indicating the intended function of each postposition in the sentence were stored separately . After this transformation step , we proceeded to the model training as follows . We first loaded KoBERT through the function BertForSequenceClassification from transformers ( Wolf et al. , 2019 ) . 2Our corpus is available at : we will propose a URL Next , we fine-tuned the pre-trained model by using the training set , with a view to reducing loss values and updating the learning rate for better classification accuracy of the model . We then loaded the testing set to evaluate whether the fine-tuned model successfully recognized the intended functions of each postposition in each sentence . In this part , the rates of accuracy for each function and the total accuracy rate were calculated by comparing the intended function of each postposition in each test sentence with the classified function of each postposition via the BERT model . Lastly , we employed t-distributed Stochastic Neighbor Embedding ( t-SNE ; Maaten & Hinton , 2008 ) for dimension reduction of classification embeddings from the postposition per epoch . In addition , to statistically confirm the changes of sentence-level embedding outcomes by each epoch , we performed densitybased clustering ( Sander et al. , 1998 ) . These outcomes were fed into the visualization system , which we outline next . | The paper presents a study of training a BERT-style model on the task of function identification for 3 different Korean adverbial postpositions(ey, eyse and (u)lo). The paper presents a new dataset for the task, show that training a BERT model performs well and also introduce a method to visualise BERT embeddings as the training proceeds. The authors show that the model performs better for postpositions which have fewer associated functions (classes for classification), performs better for classes that occur more frequently in data and that model performance improves with additional epochs. | SP:623af8b95ab193ee419da1a8d2962faadc82e2d8 |
How does BERT address polysemy of Korean adverbial postpositions -ey, -eyse, and -(u)lo? | 1 INTRODUCTION . Polysemy , one type of ambiguity , occurs when one form delivers multiple , and yet related , meanings/functions and vice versa ( Glynn & Robinson , 2014 ) . Traditional word-embedding models showed an unsatisfactory level of performance in polysemy interpretation . This is due to the technical nature of these models : they are static in that a single vector is assigned to each word ( Desagulier , 2019 ; Ethayarajh , 2019 ; Liu et al. , 2019a ) . To overcome this issue , recent studies have proposed a contextualized word-embedding model which considers neighborhood information about a polysemous word on the basis of sequences of words around the target word . Various models have been suggested for this task , such as Embeddings from Language Models ( Peters et al. , 2018 ) , Generative Pre-Training ( Radford et al. , 2018 ) , and Bidirectional Encoder Representations from Transformer ( BERT ; Devlin et al. , 2018 ) . Among these models , BERT shows the best performance in many language tasks such as translation , classification , and question-answering ( e.g. , Devlin et al. , 2018 ; Tang et al. , 2019 ) . Despite a good deal of research on BERT in English , very few studies have investigated BERT-based polysemy interpretation in languages that are typologically different from English . We turn our attention to Korean , an agglutinative Subject–Object–Verb language in which multiple postpositions or affixes with dedicated forms and meanings are attached to the stem of nominals or predicates . A postposition is a function word providing grammatical information to words it is attached ( Sohn , 1999 ) . It normally involves many-to-many associations between form and function ; as such , a postposition is polysemous ( Choo & Kwak , 2008 ) . Several studies have used word-embedding models to capture and tease apart the different meanings/functions of Korean postpositions ( e.g. , Bae et al. , 2014 ; 2015 ; Kim & Ock , 2016 ; Lee et al. , 2015 ; Mun & Shin , 2020 ; Shin et al. , 2005 ) . However , the model performance reported in the previous studies is unsatisfactory , with the accuracy ranging from 0.621 ( Bae et al. , 2014 ) to 0.837 ( Kim & Ock , 2016 ) . One possible reason for this unsatisfactory performance is that they did not consider contextual information . Against this background , the current study employs BERT for the same kind of classification task for Korean postpositions . BERT produces contextual embeddings , and this characteristic may help us to create a better classification system for postpositions . Still unclear is the particular reason for BERT ’ s superior performance over the others . In order to further understand how BERT recognizes the word-level polysemy , we propose a BERT-based visualization system in addressing polysemy interpretation of three adverbial postpositions , -ey , -eyse , and - ( u ) lo , which are frequently used and documented in the previous studies ( e.g. , Cho & Kim , 1996 ; Jeong , 2010 ; Nam , 1993 ; Park , 1999 ; Song , 2014 ) . 2 KOREAN ADVERBIAL POSTPOSITIONS : -ey , -eyse , AND - ( u ) lo In order to determine the number of functions of each postposition , this study considers the major functions of these postpositions which are frequently attested in the Sejong dictionary : eight for -ey , two for -eyse , and six for - ( u ) lo ( Shin , 2008 ) . -ey involves the following functions : agent ( AGT ) , criterion ( CRT ) , effector ( EFF ) , final state ( FNS ) , goal ( GOL ) , instrument ( INS ) , location ( LOC ) , and theme ( THM ) . ( 1 ) -ey as AGT ( agent ) 가두 katwu street 진출이 cinchwul-i go.out-NOM 경찰에 kyengchal-ey police-AGT 저지되었다 . ceci-toy-ess-ta . stop-PSV-PST-DECL ‘ By going out to the street was stopped by the police. ’ ( 2 ) -ey as CRT ( criterion ) 영호는 Yenghuy-nun Yenghuy-TOP 20만원에 20manwen-ey 200,000 won-CRT 모니터를 monithe-lul moniter-ACC 낙찰했다 . nakchalhay-ss-ta . sell-PST-DECL ‘ Yenghuy sold the monitor ( to a bidder ) for 200,000 won. ’ ( 3 ) -ey as EFF ( effector ) 문들이 mwun-tul-i door-PL-NOM 거센 keseyn strong 바람에 palam-ey wind-EFF 모두 motwu all 건들댄다 . kentultay-n-ta . sway-PRS-DECL ‘ The doors all sway by the strong wind. ’ ( 4 ) -ey as FNS ( final state ) 김교수는 kimkyoswu-nun professor.Kim-TOP 조교에 cokyo-ey assistant-FNS 박군을 park-kwun-ul Park-Mr-ACC 추천했다 . chwuchenhay-ss-ta . recommend-PST-DECL ‘ Professor Kim recommended Park as an assistant. ’ ( 5 ) -ey as GOL ( goal ) 철수가 Chelswu-ka Chelswu-TOP 던진 tenc-i-n throw-CST-PRS 칼이 khal-i knife-NOM 땅바닥에 ttangpatak-ey ground-GOL 내리꽂혔다 . naylyekkoc-hi-ess-ta . stick-PSV-PST-DECL ‘ The knife thrown by Chelswu stuck to the ground. ’ ( 6 ) -ey as INS ( instrument ) 그 ku That 어린 eli-n young-REL 소년은 sonye-nun boy-TOP 화롯불에 hwalospwul-ey fire-INS 손을 son-ul hand-ACC 녹이고 nok-i-ko melt-CST-and 있었다 . iss-ess-ta . be-PST-DECL ‘ The young boy was using the fire to warm his hands. ’ ( 7 ) -ey as LOC ( location ) 그는 ku-nun He-TOP 온종일 oncongil all day 서재에 secay-ey study.room-LOC 파묻혀 phamwut-hi-e bury.in-PSV-PRS 지낸다 . cinay-n-ta . be-PRS-DECL ‘ He is buried in his study room all day. ’ ( 8 ) -ey as THM ( theme ) 현대인들은 hyentayin-tul-un modern.people-PL-TOP 모두 motwu all 참된 chamtoy-n true-REL 지식에 cisik-ey knowledge-THM 허기져있다 . hekicye-iss-ta . hungry-PRS-DECL ‘ All modern people are hungry for true knowledge. ’ -eyse has only two functions , location ( LOC ) ( 9 ) and source ( SRC ) ( 10 ) . -eyse manifests fewer functions than -ey ( Choo & Kwak , 2008 ) . However , frequency is equally high compared to that of -ey ( e.g. , Cho & Kim , 1996 ; Song , 2014 ) . ( 9 ) -eyse as LOC ( location ) 철수는 Chelswu-nun Chelswu-TOP 서울에서 sewul-eyse seoul-LOC 태어났다 . thayena-ss-ta . born-PST-DECL ‘ Chelswu was born in Seoul. ’ ( 10 ) -eyse as SRC ( source ) 광부들이 kwangpwutul-i miner-PL-NOM 바다에서 pata-eyse sea-SRC 석유를 sekyu-lul oil-ACC 뽑아올린다 . ppopaoll-i-n-ta . pull-CST-PRS-DECL ‘ Miners pull oil from the sea. ’ - ( u ) lo engages in six functions : criterion ( CRT ) , direction ( DIR ) , effector ( EFF ) , final state ( FNS ) , instrument ( INS ) , and location ( LOC ) ( Shin , 2008 ) . ( 11 ) - ( u ) lo as CRT ( criterion ) 적당한 cektangha-n appropriate-REL 시간 sikan time 간격으로 kankyek-ulo interval-CRT 배차되었다 . paycha-toy-ess-ta . arrange-PSV-PST-DECL ‘ It was arranged at appropriate time intervals. ’ ( 12 ) - ( u ) lo as DIR ( direction ) 범인은 pemin-un criminal-NOM 어두운 etwuwun dark 골목으로 kolmok-ulo alley-DIR 달아났다 . talana-ss-ta . flee-PST-DECL ‘ The criminal fled into a dark alley. ’ ( 13 ) - ( u ) lo as EFF ( effector ) 환자가 hwanca-ka patient-NOM 위암으로 wiam-ulo stomach.cancer-EFF 매우 maywu very 괴로워하고 koyloweha-ko suffer-and 있습니다 . iss-supni-ta . be-HON-DECL ‘ The patient is suffering greatly due to stomach cancer. ’ ( 14 ) - ( u ) lo as FNS ( final state ) 그는 ku-nun He-TOP 대표 tayphyo representative 강사로 kangsa-lo lecturer-FNS 초빙되었다 . choping-toy-ess-ta . invite-PSV-PST-DECL ‘ He was invited as a representative lecturer. ’ ( 15 ) - ( u ) lo as INS ( instrument ) 전선이 censen-i wire-NOM 연줄로 yencwul-lo connection.wire-INS 감겼다 . kam-ki-ess-ta . wind-PSV-PST-DECL ‘ The wire wound around with the connection wire. ’ ( 16 ) - ( u ) lo as LOC ( location ) 경찰이 kyengchal-i police-NOM 피해자를 phiuyca-lul suspect-ACC 검찰로 kemchal-lo prosecution-LOC 압송했다 . apsonghay-ss-ta . transport.do-PST-DECL ‘ The police transported the suspect to the prosecution . ’ 3 METHODS . 3.1 CREATING INPUT CORPUS . The Sejong primary corpus1 , the representative corpus in Korean , does not code the information about the functions of postpositions directly in each sentence ( which is necessary for model training ) . We thus annotated a portion of the original corpus data manually . For this purpose , we extracted sentences that have only one postposition and predicate . This also allowed us to control for additional confounding factors which might have interfered with the performance of our model . We then extracted 5,000 sentences randomly for each postposition from the initial dataset . 1Sejong corpus is available at : https : //www.korean.go.kr Three native speakers of Korean annotated each postposition for its function in this 15,000-sentence corpus . Fleiss ’ kappa scores showed that the annotators ’ outcomes were almost identical : 0.948 ( - ey ) , 0.928 ( -eyse ) , and 0.947 ( - ( u ) lo ) . We further excluded instances which showed disagreement among the annotators . The final corpus consisted of 4,715 sentences for -ey , 4,853 sentences for -eyse , and 4,708 sentences for - ( u ) lo . Table 1 presents the detailed by-function frequency list of the three postpositions 2 . 3.2 CREATING TRAINING AND TEST SETS . We pre-processed the data in consideration of how BERT works ( we used the original BERT model for this task ) . First , we added [ CLS ] ( ‘ classification ’ ; indicating the start of a sentence ) before a sentence and [ SEP ] ( ‘ separation ’ ; indicating the end of a sentence ) after a sentence to indicate where the sentence starts and ends . These indicators made it possible for the BERT model to recognize a sentence boundary in a text , allowing the model to learn word meaning while considering intersentential variations . Second , we made a separate column ( ‘ Label ’ ) to indicate the intended function of each postposition in each sentence ( Figure 1 ) . We then split the corpus into two sub-sets , one with 90 per cent of the corpus for the training and with the remaining 10 per cent of the corpus for the testing . 3.3 DEVELOPING BERT CLASSIFICATION MODEL . We set the parameters related to BERT training such as batch size ( 32 ) , epoch ( 50 ) , seed ( 42 ) , epsilon ( 0.00000008 ) , and learning rate ( 0.00002 ) , as advised by McCormick ( 2019 ) . We then employed a pre-trained language model in order to obtain high accuracy of outcomes ; for this purpose , we used a Korean BERT model ( KoBERT ; Jeon et al. , 2019 ) . Before the actual BERT training , we transformed the input data into three embedding types—token embeddings , position embeddings , and segment embeddings ( cf. , Devlin et al. , 2018 ) —in the following ways . First , for the token embedding , we used KoBertTokenizer for the sentence tokenization ( the maximum number of tokens for each sentence was set to 128 ) . Second , we converted each token into numeric values indicating unique indices of the tokens in the vocabulary of KoBERT for the position embedding . Third , for the segment embedding , we converted the number of tokens of each sentence into 128 numeric values using 0 ( i.e. , not existed ) or 1 ( i.e. , existed ) . The labels of the data indicating the intended function of each postposition in the sentence were stored separately . After this transformation step , we proceeded to the model training as follows . We first loaded KoBERT through the function BertForSequenceClassification from transformers ( Wolf et al. , 2019 ) . 2Our corpus is available at : we will propose a URL Next , we fine-tuned the pre-trained model by using the training set , with a view to reducing loss values and updating the learning rate for better classification accuracy of the model . We then loaded the testing set to evaluate whether the fine-tuned model successfully recognized the intended functions of each postposition in each sentence . In this part , the rates of accuracy for each function and the total accuracy rate were calculated by comparing the intended function of each postposition in each test sentence with the classified function of each postposition via the BERT model . Lastly , we employed t-distributed Stochastic Neighbor Embedding ( t-SNE ; Maaten & Hinton , 2008 ) for dimension reduction of classification embeddings from the postposition per epoch . In addition , to statistically confirm the changes of sentence-level embedding outcomes by each epoch , we performed densitybased clustering ( Sander et al. , 1998 ) . These outcomes were fed into the visualization system , which we outline next . | The technical essence of the paper is to use BERT to perform classification and provide basic visualization of the change of contextual word embeddings during training. The classification problem concerns polysemy in Korean: for each adverbial postposition (e.g. -eyse), classify the use of the postposition in a given sentence into one of the available functions (e.g. LOC or SRC), where the number of classes range from two to eight. The contributions of this paper are as follows: the paper studies a relatively understudied language, Korean; the authors made progress on a very specific instance of polysemy classification problem by using contextual word embeddings compared to the previous approach that uses traditional static word embeddings (albeit this claim is not supported by the experiments); and the authors created a visualization tool for the word embeddings and model performance. | SP:623af8b95ab193ee419da1a8d2962faadc82e2d8 |
EAT-C: Environment-Adversarial sub-Task Curriculum for Efficient Reinforcement Learning | Reinforcement learning ( RL ) ’ s efficiency can drastically degrade on long-horizon tasks due to sparse rewards and the learned policy can be fragile to small changes in deployed environments . We address these two challenges by automatically generating a curriculum of tasks with coupled environments . To this end , we train two curriculum policies together with RL : ( 1 ) a co-operative planning policy recursively decomposing a hard task into coarse-to-fine sub-task sequences as a tree ; and ( 2 ) an adversarial policy modifying the environment ( e.g. , position/size of obstacles ) in each sub-task . They are complementary in acquiring more informative feedback for RL : the planning policy provides dense reward of finishing easier sub-tasks while the environment policy modifies these sub-tasks to be adequately challenging and diverse so the RL agent can quickly adapt to different tasks/environments . On the other hand , they are trained using the RL agent ’ s dense feedback on sub-tasks so the sub-task curriculum keeps adaptive to the agent ’ s progress via this “ iterative mutual-boosting ” scheme . Moreover , the sub-task tree naturally enables an easyto-hard curriculum for every policy : its top-down construction gradually increases sub-tasks the planning policy needs to generate , while the adversarial training between the environment policy and the RL policy follows a bottom-up traversal that starts from a dense sequence of easier sub-tasks allowing more frequent modifications to the environment . Therefore , jointly training the three policies leads to efficient RL guided by a curriculum progressively improving the sparse reward and generalization . We compare our method with popular RL/planning approaches targeting similar problems and the ones with environment generators or adversarial agents . Thorough experiments on diverse benchmark tasks demonstrate significant advantages of our method on improving RL ’ s efficiency and generalization . 1 INTRODUCTION . Although RL achieves breakthrough success or even outperform humans on a few challenging tasks ( Lillicrap et al. , 2016 ; Mnih et al. , 2015 ; Florensa et al . ) , it is highly inefficient when targeting long-horizon tasks due to the sparse rewards collected via interactions . In addition , a policy trained in a specific complicated/simulated environment can be sensitive to small changes in the deployed environment and thus generalizes poorly in practice . Hence , selecting or generating more informative tasks and environments interacting with an agent is an essential challenge on the path towards more efficient , robust , and versatile RL . In this paper , we mainly study goal-conditioned RL ( Kaelbling , 1993 ) whose policy is trained to adapt to any given goal/task : it is challenging , but the resulted policy can be applied to multiple goals/tasks . The sparse reward problem has motivated different reward shaping/relabeling methods and curiositydriven exploration for RL , aiming to modify the task reward to be denser than states and actions . Hierarchical RL/planning ( Elbanhawi & Simic , 2014 ; Nasiriany et al. , 2019 ; Jurgenson et al. , 2020 ; Pertsch et al. , 2020 ) decomposes a complicated task/motion by searching a root-to-leaf path of sub-tasks on a tree such that a higher-level task invokes lower-level ones as its actions . However , building the tree requires hierarchical partition of the whole state-action space and prior knowledge to define the sub-tasks , which can be expensive or unavailable . Moreover , the pre-defined sub-tasks can be either too easy or too challenging for the RL-agent in-training . Reward shaping ( Laud , 2004 ) relies on heuristics or external guidance to augment the sparse task-reward with dense rewards for exploring uncertain actions , rarely-visited states , or intermediate goals . Hindsight experience replay ( Andrychowicz et al. , 2017 ) and its variants relabels the achieved states as pseudo-goals with nonzero rewards , but many of them might be redundant to provide informative feedback efficiently improving the policy . Hence , automatically modifying the sparse reward or generating sub-tasks to provide the most informative feedback maximizing the RL agent ’ s learning progress is still an open problem , and resolving it can result in more efficient interactive learning . Another primary challenge in RL is how to improve its robustness and generalization to small changes in the environments . A growing number of recent studies ( Pinto et al. , 2017 ; Vinitsky et al. , 2020 ; Ferguson & Law , 2018 ) show that RL policy can be vulnerable to small perturbations in its inputs so training an adversarial agent to perturb the inputs or compete with the original agent may improve RL ’ s robustness . This paper mainly focuses on improving RL policy ’ s tolerance to the physical differences between the training environment and possible deployed environment , e.g. , changes of the position and size of obstacles/objects . In practice , this is important for deploying an RL policy trained in a simulator successfully to a realistic environment . Moreover , when addressing the sparse reward problem , the engineered sub-tasks or relabeled goals might be redundant or too easy to provide effective feedback to RL . However , adversarially modifying their environments can effectively improve their utility and RL policy ’ s robust generalization against diverse perturbed environments . Although modifying environment models has been recently studied to assist RL ( Co-Reyes et al. , 2020 ; Gur et al. , 2021 ; Wang et al. , 2019 ) , directly applying this strategy to long-horizon tasks might make them even more challenging and reward-sparse , hence detrimental to RL ’ s efficiency . In this paper , we address the above two challenges by automatically generating a curriculum of sub-tasks with adversarially perturbed environments to train the RL agent , where the curriculum is adaptive to the agent ’ s learning progress . To this end , we propose a marriage of subtasking and adversarial environments . The former decomposes a challenging long-horizon task into easier subtasks offering dense rewards , and the latter modifies each sub-task to be sufficiently challenging yet effective for improving RL policy ’ s tolerance to perturbed environments . As illustrated in Fig . 1 , our approach , “ environment-adversarial subTask curriculum ( EAT-C ) ” , generates a tree-structured curriculum by ( 1 ) a path-planning policy that recursively decomposes an assigned task ( e.g. , a state-goal pair ( s , g ) ) to sequences of sub-tasks ( e.g. , navigating between consecutive sub-goals ) with varying granularity ( k = 0 , . . . , 3 ) ; and ( 2 ) an environment policy that adversarially modifies each sub-task ’ s environment . The training objective for the path-planner is finding the most cost-efficient/shortest path in each level k , while the objective for the environment policy is to minimize the expected return of the RL agent on all sub-tasks . In EAT-C , training these two policies does not require external supervision or prior knowledge , we instead collect the time costs and rewards of the RL agent on previous sub-tasks to train them towards generating better curricula adaptive to RL progress , which then guide the training of the next episode . Together with the RL policy , the two curriculum policies can efficiently learn from each other by iterating the above mutual-boosting scheme on the tree-structured curriculum of sub-tasks . A key advantage of the tree is that it naturally enables an easy-to-hard curriculum to train each of the three policies : ( 1 ) the top-down construction of the tree trains the path-planner to merely interpolate a few sub-goals between ( s , g ) at first and then gradually increases their number for more fine-grained and challenging planning ; ( 2 ) the adversarial training between the environment policy and the RL policy follows a bottom-up traversal of the tree , i.e. , they start from learning many easier sub-tasks and more frequent perturbations between ( s , g ) respectively , which are easy for both policies , and then progressively evolve to handle more challenging cases . i.e. , a few long-horizon sub-tasks allow lessfrequent environment changes between ( s , g ) . In experiments for discrete navigation and continuous control tasks , we show that EAT-C is considerably efficient in learning to generalize to perturbed environments during finishing long-horizon tasks . Compared with recent RL methods equipped with hand-crafted curricula , environment generators , hierarchical policies , or path-planning , EAT-C exhibits promising advantages on efficiency and generalization to different tasks and environments . 2 RELATED WORK . Goal-conditioned RL ( Pong et al. , 2018 ; Kaelbling , 1993 ) takes a goal as an additional input to its policy and aims to handle different goals/tasks by the same policy . However , it requires extensive exploration and costly training on various goals but it still easily fails to reach distant goals in practice . Goal relabeling and reward shaping ( Andrychowicz et al. , 2017 ; Nasiriany et al. , 2019 ) have been studied to mitigate these issues by modifying the task rewards to be dense but they introduce extra data bias and can not control the utility of modified goals/rewards to the targeted ones . In order to provide tasks at the appropriate level of difficulty for the agent to learn , Held et al . ( 2018 ) and Racanière et al . ( 2019 ) train a goal/task generator based on the agent ’ s performance on existing tasks . But it is usually non-trivial to determine and tune the appropriate difficulty level for each training stage . In EAT-C , an RL agent achieves dense rewards by finishing a sequence of easier sub-goals towards the final goal , while the adversarial EG modifies each sub-task to be sufficiently challenging and diverse from other sub-tasks . Such sub-task curricula lead to more efficient goal-conditioned RL . Moreover , we do not need to carefully tune the difficulty levels of sub-tasks because the path-planner and EG together automatically generate the most informative tasks for each training stage of RL . Planning algorithms ( Sutton & Barto , 2018 ; LaValle , 2006 ) are helpful to solving long-horizon tasks ( Levine et al. , 2011 ) by interpolating intermediate sub-goals . In navigation , planning usually refers to finding the shortest path between two nodes ( states ) on a graph ( LaValle , 2006 ; Dayan & Hinton , 1993 ) but a distance metric more accurate than Euclidean distance is usually unavailable . Moreover , it suffers from local minimal and performs poorly in narrow regions ( Koren & Borenstein , 1991 ) . Sequentially planning ( Schmidhuber & Wahnsiedler , 1993 ; Zhang et al. , 2020 ) sub-goals from the starting state to the goal state is inefficient in complex tasks , as it needs to search in a large space . In RL , planning requires an environment model or learns a value function to improve the policy ( Levine et al. , 2011 ; Lau & Kuffner , 2005 ; Elbanhawi & Simic , 2014 ) , which can be as challenging as model-free RL . Control-based methods require accurate models for both the robot and the environment ( Howard & Kelly , 2007 ; Werling et al. , 2012 ) , as well as an accurate distance metric ( Eysenbach et al. , 2019 ) , which can be a rather daunting task . In EAT-C , we train a path-planner that “ learns to plan ” and to directly generate the shortest ( in terms of time cost ) path for the RL agent . Hence , it requires neither expensive search procedures nor a predefined distance metric . Moreover , it does not rely on learning a world/robot model or a value function . The easy-to-hard curriculum and dense feedback from the RL agent make it efficient to train . Furthermore , it does not need to plan for every step but only a few crucial waypoints . Several recent works ( Wang et al. , 2019 ; Portelas et al. , 2019 ; Gur et al. , 2021 ) show that the efficiency and generalization of RL can be improved by generating/selecting different environments for training . Wang et al . ( 2019 ) studies a method “ POET ” to pair an adversarial environment generator with two agents to improve the generalization of the RL policy to novel environments . Portelas et al . ( 2019 ) proposes a teacher algorithm to sample environments based on a Gaussian mixture model capturing both the diversity of environments and the learning progress on them . This leads to a curriculum improving the efficiency of RL . However , the generated adversarial environments can be infeasible or too challenging for the RL agent and it is non-trivial to control their hardness . Moreover , modeling the distribution of environments requires collecting many samples from them and evaluating the RL policy on them , which can be costly and inaccurate especially for complicated environments . In EAT-C , we adversarially modify the environment for each easier sub-task , whose hardness is controlled by both the path planner and EG , so the modified sub-task is still feasible for the RL agent to learn . Moreover , a policy network for EG facilitates the modification of complicated environments and training it is efficient in our mutual boosting scheme with an easy-to-hard curriculum . Hierarchical planning ( HP ) methods ( Nasiriany et al. , 2019 ; Jurgenson et al. , 2020 ; Pertsch et al. , 2020 ) search for a sequence of sub-goal on a tree to guide the exploration of an agent but building the hierarchical partition of all possible sub-goals can be expensive . A conventional hierarchical planning algorithm ( Kaelbling & Lozano-Pérez , 2011 ) also learns to predict sub-tasks of a tree structure based on a set of pre-defined motion primitives . Hierarchical RL ( HRL ) for goal-reaching tasks has been recently studied in ( Zhang et al. , 2021 ; Nachum et al. , 2018 ) . Shu et al . ( 2018 ) trains the RL agent on a human-designed curriculum of tree-structured sub-goals . They learn a sequence of primeval policies towards finishing complicated tasks , where the higher-level policies decompose a complex task into easier sub-tasks or motion primitives that can be addressed by the lower-level policies or controllers . However , the hierarchy of sub-tasks need to be either determined by prior knowledge or discovered automatically , which is usually challenging due to the huge space of possible sub-tasks . In contrast , EAT-C trains one planning policy to decompose a hard task into sub-tasks of multiple difficulty levels by only using the RL ’ s time cost data . It directly generates the sub-tasks and thus requires neither hierarchical partition of the task space nor a predefined set of sub-tasks . Compared to HP and HRL , EAT-C ( 1 ) enables a mutual training between the path-planner and the RL agent ; ( 2 ) has an adversarial EG to adjust each sub-task ’ s environment to be more challenging ; and ( 3 ) improves the training efficiency under the guidance of easy-to-hard curricula automatically generated for each stage . | The paper proposes an RL method targeted at long-horizon tasks in environments with perturbed dynamics or configurations. By combining a recently proposed goal-based RL method with an adversarial approach to domain randomization, the authors obtain a method that allows them to learn robust policies in sparse long-horizon settings. The method is compared with different baselines form hierarchical- or curriculum RL, demonstrating promising performance. | SP:ead982cae78d09fddead293d39e1bd7317b03ed5 |
EAT-C: Environment-Adversarial sub-Task Curriculum for Efficient Reinforcement Learning | Reinforcement learning ( RL ) ’ s efficiency can drastically degrade on long-horizon tasks due to sparse rewards and the learned policy can be fragile to small changes in deployed environments . We address these two challenges by automatically generating a curriculum of tasks with coupled environments . To this end , we train two curriculum policies together with RL : ( 1 ) a co-operative planning policy recursively decomposing a hard task into coarse-to-fine sub-task sequences as a tree ; and ( 2 ) an adversarial policy modifying the environment ( e.g. , position/size of obstacles ) in each sub-task . They are complementary in acquiring more informative feedback for RL : the planning policy provides dense reward of finishing easier sub-tasks while the environment policy modifies these sub-tasks to be adequately challenging and diverse so the RL agent can quickly adapt to different tasks/environments . On the other hand , they are trained using the RL agent ’ s dense feedback on sub-tasks so the sub-task curriculum keeps adaptive to the agent ’ s progress via this “ iterative mutual-boosting ” scheme . Moreover , the sub-task tree naturally enables an easyto-hard curriculum for every policy : its top-down construction gradually increases sub-tasks the planning policy needs to generate , while the adversarial training between the environment policy and the RL policy follows a bottom-up traversal that starts from a dense sequence of easier sub-tasks allowing more frequent modifications to the environment . Therefore , jointly training the three policies leads to efficient RL guided by a curriculum progressively improving the sparse reward and generalization . We compare our method with popular RL/planning approaches targeting similar problems and the ones with environment generators or adversarial agents . Thorough experiments on diverse benchmark tasks demonstrate significant advantages of our method on improving RL ’ s efficiency and generalization . 1 INTRODUCTION . Although RL achieves breakthrough success or even outperform humans on a few challenging tasks ( Lillicrap et al. , 2016 ; Mnih et al. , 2015 ; Florensa et al . ) , it is highly inefficient when targeting long-horizon tasks due to the sparse rewards collected via interactions . In addition , a policy trained in a specific complicated/simulated environment can be sensitive to small changes in the deployed environment and thus generalizes poorly in practice . Hence , selecting or generating more informative tasks and environments interacting with an agent is an essential challenge on the path towards more efficient , robust , and versatile RL . In this paper , we mainly study goal-conditioned RL ( Kaelbling , 1993 ) whose policy is trained to adapt to any given goal/task : it is challenging , but the resulted policy can be applied to multiple goals/tasks . The sparse reward problem has motivated different reward shaping/relabeling methods and curiositydriven exploration for RL , aiming to modify the task reward to be denser than states and actions . Hierarchical RL/planning ( Elbanhawi & Simic , 2014 ; Nasiriany et al. , 2019 ; Jurgenson et al. , 2020 ; Pertsch et al. , 2020 ) decomposes a complicated task/motion by searching a root-to-leaf path of sub-tasks on a tree such that a higher-level task invokes lower-level ones as its actions . However , building the tree requires hierarchical partition of the whole state-action space and prior knowledge to define the sub-tasks , which can be expensive or unavailable . Moreover , the pre-defined sub-tasks can be either too easy or too challenging for the RL-agent in-training . Reward shaping ( Laud , 2004 ) relies on heuristics or external guidance to augment the sparse task-reward with dense rewards for exploring uncertain actions , rarely-visited states , or intermediate goals . Hindsight experience replay ( Andrychowicz et al. , 2017 ) and its variants relabels the achieved states as pseudo-goals with nonzero rewards , but many of them might be redundant to provide informative feedback efficiently improving the policy . Hence , automatically modifying the sparse reward or generating sub-tasks to provide the most informative feedback maximizing the RL agent ’ s learning progress is still an open problem , and resolving it can result in more efficient interactive learning . Another primary challenge in RL is how to improve its robustness and generalization to small changes in the environments . A growing number of recent studies ( Pinto et al. , 2017 ; Vinitsky et al. , 2020 ; Ferguson & Law , 2018 ) show that RL policy can be vulnerable to small perturbations in its inputs so training an adversarial agent to perturb the inputs or compete with the original agent may improve RL ’ s robustness . This paper mainly focuses on improving RL policy ’ s tolerance to the physical differences between the training environment and possible deployed environment , e.g. , changes of the position and size of obstacles/objects . In practice , this is important for deploying an RL policy trained in a simulator successfully to a realistic environment . Moreover , when addressing the sparse reward problem , the engineered sub-tasks or relabeled goals might be redundant or too easy to provide effective feedback to RL . However , adversarially modifying their environments can effectively improve their utility and RL policy ’ s robust generalization against diverse perturbed environments . Although modifying environment models has been recently studied to assist RL ( Co-Reyes et al. , 2020 ; Gur et al. , 2021 ; Wang et al. , 2019 ) , directly applying this strategy to long-horizon tasks might make them even more challenging and reward-sparse , hence detrimental to RL ’ s efficiency . In this paper , we address the above two challenges by automatically generating a curriculum of sub-tasks with adversarially perturbed environments to train the RL agent , where the curriculum is adaptive to the agent ’ s learning progress . To this end , we propose a marriage of subtasking and adversarial environments . The former decomposes a challenging long-horizon task into easier subtasks offering dense rewards , and the latter modifies each sub-task to be sufficiently challenging yet effective for improving RL policy ’ s tolerance to perturbed environments . As illustrated in Fig . 1 , our approach , “ environment-adversarial subTask curriculum ( EAT-C ) ” , generates a tree-structured curriculum by ( 1 ) a path-planning policy that recursively decomposes an assigned task ( e.g. , a state-goal pair ( s , g ) ) to sequences of sub-tasks ( e.g. , navigating between consecutive sub-goals ) with varying granularity ( k = 0 , . . . , 3 ) ; and ( 2 ) an environment policy that adversarially modifies each sub-task ’ s environment . The training objective for the path-planner is finding the most cost-efficient/shortest path in each level k , while the objective for the environment policy is to minimize the expected return of the RL agent on all sub-tasks . In EAT-C , training these two policies does not require external supervision or prior knowledge , we instead collect the time costs and rewards of the RL agent on previous sub-tasks to train them towards generating better curricula adaptive to RL progress , which then guide the training of the next episode . Together with the RL policy , the two curriculum policies can efficiently learn from each other by iterating the above mutual-boosting scheme on the tree-structured curriculum of sub-tasks . A key advantage of the tree is that it naturally enables an easy-to-hard curriculum to train each of the three policies : ( 1 ) the top-down construction of the tree trains the path-planner to merely interpolate a few sub-goals between ( s , g ) at first and then gradually increases their number for more fine-grained and challenging planning ; ( 2 ) the adversarial training between the environment policy and the RL policy follows a bottom-up traversal of the tree , i.e. , they start from learning many easier sub-tasks and more frequent perturbations between ( s , g ) respectively , which are easy for both policies , and then progressively evolve to handle more challenging cases . i.e. , a few long-horizon sub-tasks allow lessfrequent environment changes between ( s , g ) . In experiments for discrete navigation and continuous control tasks , we show that EAT-C is considerably efficient in learning to generalize to perturbed environments during finishing long-horizon tasks . Compared with recent RL methods equipped with hand-crafted curricula , environment generators , hierarchical policies , or path-planning , EAT-C exhibits promising advantages on efficiency and generalization to different tasks and environments . 2 RELATED WORK . Goal-conditioned RL ( Pong et al. , 2018 ; Kaelbling , 1993 ) takes a goal as an additional input to its policy and aims to handle different goals/tasks by the same policy . However , it requires extensive exploration and costly training on various goals but it still easily fails to reach distant goals in practice . Goal relabeling and reward shaping ( Andrychowicz et al. , 2017 ; Nasiriany et al. , 2019 ) have been studied to mitigate these issues by modifying the task rewards to be dense but they introduce extra data bias and can not control the utility of modified goals/rewards to the targeted ones . In order to provide tasks at the appropriate level of difficulty for the agent to learn , Held et al . ( 2018 ) and Racanière et al . ( 2019 ) train a goal/task generator based on the agent ’ s performance on existing tasks . But it is usually non-trivial to determine and tune the appropriate difficulty level for each training stage . In EAT-C , an RL agent achieves dense rewards by finishing a sequence of easier sub-goals towards the final goal , while the adversarial EG modifies each sub-task to be sufficiently challenging and diverse from other sub-tasks . Such sub-task curricula lead to more efficient goal-conditioned RL . Moreover , we do not need to carefully tune the difficulty levels of sub-tasks because the path-planner and EG together automatically generate the most informative tasks for each training stage of RL . Planning algorithms ( Sutton & Barto , 2018 ; LaValle , 2006 ) are helpful to solving long-horizon tasks ( Levine et al. , 2011 ) by interpolating intermediate sub-goals . In navigation , planning usually refers to finding the shortest path between two nodes ( states ) on a graph ( LaValle , 2006 ; Dayan & Hinton , 1993 ) but a distance metric more accurate than Euclidean distance is usually unavailable . Moreover , it suffers from local minimal and performs poorly in narrow regions ( Koren & Borenstein , 1991 ) . Sequentially planning ( Schmidhuber & Wahnsiedler , 1993 ; Zhang et al. , 2020 ) sub-goals from the starting state to the goal state is inefficient in complex tasks , as it needs to search in a large space . In RL , planning requires an environment model or learns a value function to improve the policy ( Levine et al. , 2011 ; Lau & Kuffner , 2005 ; Elbanhawi & Simic , 2014 ) , which can be as challenging as model-free RL . Control-based methods require accurate models for both the robot and the environment ( Howard & Kelly , 2007 ; Werling et al. , 2012 ) , as well as an accurate distance metric ( Eysenbach et al. , 2019 ) , which can be a rather daunting task . In EAT-C , we train a path-planner that “ learns to plan ” and to directly generate the shortest ( in terms of time cost ) path for the RL agent . Hence , it requires neither expensive search procedures nor a predefined distance metric . Moreover , it does not rely on learning a world/robot model or a value function . The easy-to-hard curriculum and dense feedback from the RL agent make it efficient to train . Furthermore , it does not need to plan for every step but only a few crucial waypoints . Several recent works ( Wang et al. , 2019 ; Portelas et al. , 2019 ; Gur et al. , 2021 ) show that the efficiency and generalization of RL can be improved by generating/selecting different environments for training . Wang et al . ( 2019 ) studies a method “ POET ” to pair an adversarial environment generator with two agents to improve the generalization of the RL policy to novel environments . Portelas et al . ( 2019 ) proposes a teacher algorithm to sample environments based on a Gaussian mixture model capturing both the diversity of environments and the learning progress on them . This leads to a curriculum improving the efficiency of RL . However , the generated adversarial environments can be infeasible or too challenging for the RL agent and it is non-trivial to control their hardness . Moreover , modeling the distribution of environments requires collecting many samples from them and evaluating the RL policy on them , which can be costly and inaccurate especially for complicated environments . In EAT-C , we adversarially modify the environment for each easier sub-task , whose hardness is controlled by both the path planner and EG , so the modified sub-task is still feasible for the RL agent to learn . Moreover , a policy network for EG facilitates the modification of complicated environments and training it is efficient in our mutual boosting scheme with an easy-to-hard curriculum . Hierarchical planning ( HP ) methods ( Nasiriany et al. , 2019 ; Jurgenson et al. , 2020 ; Pertsch et al. , 2020 ) search for a sequence of sub-goal on a tree to guide the exploration of an agent but building the hierarchical partition of all possible sub-goals can be expensive . A conventional hierarchical planning algorithm ( Kaelbling & Lozano-Pérez , 2011 ) also learns to predict sub-tasks of a tree structure based on a set of pre-defined motion primitives . Hierarchical RL ( HRL ) for goal-reaching tasks has been recently studied in ( Zhang et al. , 2021 ; Nachum et al. , 2018 ) . Shu et al . ( 2018 ) trains the RL agent on a human-designed curriculum of tree-structured sub-goals . They learn a sequence of primeval policies towards finishing complicated tasks , where the higher-level policies decompose a complex task into easier sub-tasks or motion primitives that can be addressed by the lower-level policies or controllers . However , the hierarchy of sub-tasks need to be either determined by prior knowledge or discovered automatically , which is usually challenging due to the huge space of possible sub-tasks . In contrast , EAT-C trains one planning policy to decompose a hard task into sub-tasks of multiple difficulty levels by only using the RL ’ s time cost data . It directly generates the sub-tasks and thus requires neither hierarchical partition of the task space nor a predefined set of sub-tasks . Compared to HP and HRL , EAT-C ( 1 ) enables a mutual training between the path-planner and the RL agent ; ( 2 ) has an adversarial EG to adjust each sub-task ’ s environment to be more challenging ; and ( 3 ) improves the training efficiency under the guidance of easy-to-hard curricula automatically generated for each stage . | This work introduces a method called Environment-Adversarial Sub-Task Curriculum (EAT-C) wjocj automatically generates a curricuulm of task-environment couples for efficient RL. To this end, EAT-C trains two policies, in addition to the main decision-making policy. One policiy recursively decomposes hard task to coarse-to-fine subtasks, while the other performs adversarial modifications of the environment in each of the subtask. The authors show empirically that EAT-C leads to efficient training of the policy and superior generalization capabilities compared to existing baselines. | SP:ead982cae78d09fddead293d39e1bd7317b03ed5 |
EAT-C: Environment-Adversarial sub-Task Curriculum for Efficient Reinforcement Learning | Reinforcement learning ( RL ) ’ s efficiency can drastically degrade on long-horizon tasks due to sparse rewards and the learned policy can be fragile to small changes in deployed environments . We address these two challenges by automatically generating a curriculum of tasks with coupled environments . To this end , we train two curriculum policies together with RL : ( 1 ) a co-operative planning policy recursively decomposing a hard task into coarse-to-fine sub-task sequences as a tree ; and ( 2 ) an adversarial policy modifying the environment ( e.g. , position/size of obstacles ) in each sub-task . They are complementary in acquiring more informative feedback for RL : the planning policy provides dense reward of finishing easier sub-tasks while the environment policy modifies these sub-tasks to be adequately challenging and diverse so the RL agent can quickly adapt to different tasks/environments . On the other hand , they are trained using the RL agent ’ s dense feedback on sub-tasks so the sub-task curriculum keeps adaptive to the agent ’ s progress via this “ iterative mutual-boosting ” scheme . Moreover , the sub-task tree naturally enables an easyto-hard curriculum for every policy : its top-down construction gradually increases sub-tasks the planning policy needs to generate , while the adversarial training between the environment policy and the RL policy follows a bottom-up traversal that starts from a dense sequence of easier sub-tasks allowing more frequent modifications to the environment . Therefore , jointly training the three policies leads to efficient RL guided by a curriculum progressively improving the sparse reward and generalization . We compare our method with popular RL/planning approaches targeting similar problems and the ones with environment generators or adversarial agents . Thorough experiments on diverse benchmark tasks demonstrate significant advantages of our method on improving RL ’ s efficiency and generalization . 1 INTRODUCTION . Although RL achieves breakthrough success or even outperform humans on a few challenging tasks ( Lillicrap et al. , 2016 ; Mnih et al. , 2015 ; Florensa et al . ) , it is highly inefficient when targeting long-horizon tasks due to the sparse rewards collected via interactions . In addition , a policy trained in a specific complicated/simulated environment can be sensitive to small changes in the deployed environment and thus generalizes poorly in practice . Hence , selecting or generating more informative tasks and environments interacting with an agent is an essential challenge on the path towards more efficient , robust , and versatile RL . In this paper , we mainly study goal-conditioned RL ( Kaelbling , 1993 ) whose policy is trained to adapt to any given goal/task : it is challenging , but the resulted policy can be applied to multiple goals/tasks . The sparse reward problem has motivated different reward shaping/relabeling methods and curiositydriven exploration for RL , aiming to modify the task reward to be denser than states and actions . Hierarchical RL/planning ( Elbanhawi & Simic , 2014 ; Nasiriany et al. , 2019 ; Jurgenson et al. , 2020 ; Pertsch et al. , 2020 ) decomposes a complicated task/motion by searching a root-to-leaf path of sub-tasks on a tree such that a higher-level task invokes lower-level ones as its actions . However , building the tree requires hierarchical partition of the whole state-action space and prior knowledge to define the sub-tasks , which can be expensive or unavailable . Moreover , the pre-defined sub-tasks can be either too easy or too challenging for the RL-agent in-training . Reward shaping ( Laud , 2004 ) relies on heuristics or external guidance to augment the sparse task-reward with dense rewards for exploring uncertain actions , rarely-visited states , or intermediate goals . Hindsight experience replay ( Andrychowicz et al. , 2017 ) and its variants relabels the achieved states as pseudo-goals with nonzero rewards , but many of them might be redundant to provide informative feedback efficiently improving the policy . Hence , automatically modifying the sparse reward or generating sub-tasks to provide the most informative feedback maximizing the RL agent ’ s learning progress is still an open problem , and resolving it can result in more efficient interactive learning . Another primary challenge in RL is how to improve its robustness and generalization to small changes in the environments . A growing number of recent studies ( Pinto et al. , 2017 ; Vinitsky et al. , 2020 ; Ferguson & Law , 2018 ) show that RL policy can be vulnerable to small perturbations in its inputs so training an adversarial agent to perturb the inputs or compete with the original agent may improve RL ’ s robustness . This paper mainly focuses on improving RL policy ’ s tolerance to the physical differences between the training environment and possible deployed environment , e.g. , changes of the position and size of obstacles/objects . In practice , this is important for deploying an RL policy trained in a simulator successfully to a realistic environment . Moreover , when addressing the sparse reward problem , the engineered sub-tasks or relabeled goals might be redundant or too easy to provide effective feedback to RL . However , adversarially modifying their environments can effectively improve their utility and RL policy ’ s robust generalization against diverse perturbed environments . Although modifying environment models has been recently studied to assist RL ( Co-Reyes et al. , 2020 ; Gur et al. , 2021 ; Wang et al. , 2019 ) , directly applying this strategy to long-horizon tasks might make them even more challenging and reward-sparse , hence detrimental to RL ’ s efficiency . In this paper , we address the above two challenges by automatically generating a curriculum of sub-tasks with adversarially perturbed environments to train the RL agent , where the curriculum is adaptive to the agent ’ s learning progress . To this end , we propose a marriage of subtasking and adversarial environments . The former decomposes a challenging long-horizon task into easier subtasks offering dense rewards , and the latter modifies each sub-task to be sufficiently challenging yet effective for improving RL policy ’ s tolerance to perturbed environments . As illustrated in Fig . 1 , our approach , “ environment-adversarial subTask curriculum ( EAT-C ) ” , generates a tree-structured curriculum by ( 1 ) a path-planning policy that recursively decomposes an assigned task ( e.g. , a state-goal pair ( s , g ) ) to sequences of sub-tasks ( e.g. , navigating between consecutive sub-goals ) with varying granularity ( k = 0 , . . . , 3 ) ; and ( 2 ) an environment policy that adversarially modifies each sub-task ’ s environment . The training objective for the path-planner is finding the most cost-efficient/shortest path in each level k , while the objective for the environment policy is to minimize the expected return of the RL agent on all sub-tasks . In EAT-C , training these two policies does not require external supervision or prior knowledge , we instead collect the time costs and rewards of the RL agent on previous sub-tasks to train them towards generating better curricula adaptive to RL progress , which then guide the training of the next episode . Together with the RL policy , the two curriculum policies can efficiently learn from each other by iterating the above mutual-boosting scheme on the tree-structured curriculum of sub-tasks . A key advantage of the tree is that it naturally enables an easy-to-hard curriculum to train each of the three policies : ( 1 ) the top-down construction of the tree trains the path-planner to merely interpolate a few sub-goals between ( s , g ) at first and then gradually increases their number for more fine-grained and challenging planning ; ( 2 ) the adversarial training between the environment policy and the RL policy follows a bottom-up traversal of the tree , i.e. , they start from learning many easier sub-tasks and more frequent perturbations between ( s , g ) respectively , which are easy for both policies , and then progressively evolve to handle more challenging cases . i.e. , a few long-horizon sub-tasks allow lessfrequent environment changes between ( s , g ) . In experiments for discrete navigation and continuous control tasks , we show that EAT-C is considerably efficient in learning to generalize to perturbed environments during finishing long-horizon tasks . Compared with recent RL methods equipped with hand-crafted curricula , environment generators , hierarchical policies , or path-planning , EAT-C exhibits promising advantages on efficiency and generalization to different tasks and environments . 2 RELATED WORK . Goal-conditioned RL ( Pong et al. , 2018 ; Kaelbling , 1993 ) takes a goal as an additional input to its policy and aims to handle different goals/tasks by the same policy . However , it requires extensive exploration and costly training on various goals but it still easily fails to reach distant goals in practice . Goal relabeling and reward shaping ( Andrychowicz et al. , 2017 ; Nasiriany et al. , 2019 ) have been studied to mitigate these issues by modifying the task rewards to be dense but they introduce extra data bias and can not control the utility of modified goals/rewards to the targeted ones . In order to provide tasks at the appropriate level of difficulty for the agent to learn , Held et al . ( 2018 ) and Racanière et al . ( 2019 ) train a goal/task generator based on the agent ’ s performance on existing tasks . But it is usually non-trivial to determine and tune the appropriate difficulty level for each training stage . In EAT-C , an RL agent achieves dense rewards by finishing a sequence of easier sub-goals towards the final goal , while the adversarial EG modifies each sub-task to be sufficiently challenging and diverse from other sub-tasks . Such sub-task curricula lead to more efficient goal-conditioned RL . Moreover , we do not need to carefully tune the difficulty levels of sub-tasks because the path-planner and EG together automatically generate the most informative tasks for each training stage of RL . Planning algorithms ( Sutton & Barto , 2018 ; LaValle , 2006 ) are helpful to solving long-horizon tasks ( Levine et al. , 2011 ) by interpolating intermediate sub-goals . In navigation , planning usually refers to finding the shortest path between two nodes ( states ) on a graph ( LaValle , 2006 ; Dayan & Hinton , 1993 ) but a distance metric more accurate than Euclidean distance is usually unavailable . Moreover , it suffers from local minimal and performs poorly in narrow regions ( Koren & Borenstein , 1991 ) . Sequentially planning ( Schmidhuber & Wahnsiedler , 1993 ; Zhang et al. , 2020 ) sub-goals from the starting state to the goal state is inefficient in complex tasks , as it needs to search in a large space . In RL , planning requires an environment model or learns a value function to improve the policy ( Levine et al. , 2011 ; Lau & Kuffner , 2005 ; Elbanhawi & Simic , 2014 ) , which can be as challenging as model-free RL . Control-based methods require accurate models for both the robot and the environment ( Howard & Kelly , 2007 ; Werling et al. , 2012 ) , as well as an accurate distance metric ( Eysenbach et al. , 2019 ) , which can be a rather daunting task . In EAT-C , we train a path-planner that “ learns to plan ” and to directly generate the shortest ( in terms of time cost ) path for the RL agent . Hence , it requires neither expensive search procedures nor a predefined distance metric . Moreover , it does not rely on learning a world/robot model or a value function . The easy-to-hard curriculum and dense feedback from the RL agent make it efficient to train . Furthermore , it does not need to plan for every step but only a few crucial waypoints . Several recent works ( Wang et al. , 2019 ; Portelas et al. , 2019 ; Gur et al. , 2021 ) show that the efficiency and generalization of RL can be improved by generating/selecting different environments for training . Wang et al . ( 2019 ) studies a method “ POET ” to pair an adversarial environment generator with two agents to improve the generalization of the RL policy to novel environments . Portelas et al . ( 2019 ) proposes a teacher algorithm to sample environments based on a Gaussian mixture model capturing both the diversity of environments and the learning progress on them . This leads to a curriculum improving the efficiency of RL . However , the generated adversarial environments can be infeasible or too challenging for the RL agent and it is non-trivial to control their hardness . Moreover , modeling the distribution of environments requires collecting many samples from them and evaluating the RL policy on them , which can be costly and inaccurate especially for complicated environments . In EAT-C , we adversarially modify the environment for each easier sub-task , whose hardness is controlled by both the path planner and EG , so the modified sub-task is still feasible for the RL agent to learn . Moreover , a policy network for EG facilitates the modification of complicated environments and training it is efficient in our mutual boosting scheme with an easy-to-hard curriculum . Hierarchical planning ( HP ) methods ( Nasiriany et al. , 2019 ; Jurgenson et al. , 2020 ; Pertsch et al. , 2020 ) search for a sequence of sub-goal on a tree to guide the exploration of an agent but building the hierarchical partition of all possible sub-goals can be expensive . A conventional hierarchical planning algorithm ( Kaelbling & Lozano-Pérez , 2011 ) also learns to predict sub-tasks of a tree structure based on a set of pre-defined motion primitives . Hierarchical RL ( HRL ) for goal-reaching tasks has been recently studied in ( Zhang et al. , 2021 ; Nachum et al. , 2018 ) . Shu et al . ( 2018 ) trains the RL agent on a human-designed curriculum of tree-structured sub-goals . They learn a sequence of primeval policies towards finishing complicated tasks , where the higher-level policies decompose a complex task into easier sub-tasks or motion primitives that can be addressed by the lower-level policies or controllers . However , the hierarchy of sub-tasks need to be either determined by prior knowledge or discovered automatically , which is usually challenging due to the huge space of possible sub-tasks . In contrast , EAT-C trains one planning policy to decompose a hard task into sub-tasks of multiple difficulty levels by only using the RL ’ s time cost data . It directly generates the sub-tasks and thus requires neither hierarchical partition of the task space nor a predefined set of sub-tasks . Compared to HP and HRL , EAT-C ( 1 ) enables a mutual training between the path-planner and the RL agent ; ( 2 ) has an adversarial EG to adjust each sub-task ’ s environment to be more challenging ; and ( 3 ) improves the training efficiency under the guidance of easy-to-hard curricula automatically generated for each stage . | This paper introduces an auto-curriculum generation approach to train better low-level policies conditioned on the subgoals generated by a path-planning high-level policy. The core idea is to use a planning policy to decompose a task into subgoals via a tree structure (multi-level subgoal decomposition) and use an environment generation policy to adversarially make the environment harder for the subgoal policy to perform. Thanks to the tree structure, the adversarial training can follow a bottom-up process, i.e., in an easy-to-hard order. The evaluation on a 2D object using task and a 2D grid world environment shows the effectiveness of this approach. | SP:ead982cae78d09fddead293d39e1bd7317b03ed5 |
Depth Without the Magic: Inductive Bias of Natural Gradient Descent | 1 INTRODUCTION There is plenty of empirical evidence that the choice of network architecture is an important determinant of the success of deep learning ( He et al. , 2015 ; Vaswani et al. , 2017 ) . The empirical observations are now supported by theoretical work into the role that parameter-to-hypothesis mapping plays in determining inductive biases of gradient-based learning . Unregularized gradient descent can efficiently find low-rank solutions in matrix completion problems ( Arora et al. , 2019 ) , sparse solutions in separable classification ( Gunasekar et al. , 2018 ) or compressed sensing ( Vaškevičius et al. , 2019 ) . Valle-Pérez et al . ( 2018 ) studied deep neural networks and found evidence that the parameter-hypothesis mapping1 is biased towards simpler functions as measured by Kolmogorov complexity . Taken together , these observations and findings have lead the community to hypothesize that The parameter-to-hypothesis mapping influences the inductive biases of gradient-based learning and may play an important role in generalization . In parallel to improving architectures , considerable research was done to improve optimization algorithms for deep learning , with a focus on faster convergence and robustness to hyperparameters . Among the most advanced optimization methods are natural gradient descent ( NGD ) techniques . An intuitive motivation for NGD is that it improves convergence by implicitly lifting the problem from parameter-space , where the loss is non-convex and poorly behaved to the Riemannian manifold 1The mapping between the parameter space and the set of hypotheses as seen on Figure 1 of hypotheses , where the loss is better behaved . From the perspective of inductive biases , the most interesting aspect of NGD is its approximate invariance to reparametrization . Natural gradient descent eliminates the effect of parameter-to-hypothesis mapping . These two observations invite questions about the nature of inductive biases in NGD as well as the role of parametrization-dependence in generalization . The first , practical , implication is as follows : if the parameter-to-hypothesis mapping really does play an important role in generalization , then eliminating its influence on the optimization path may be undesirable , and consequently the pursuit of implementing exact NGD in deep architectures may be counterproductive . Secondly , studying the behaviour of NGD in various models and tasks may give us new insights about the importance of parametrization , and could perhaps offer a way to experimentally or theoretically test hypotheses . In this paper we study the inductive bias of natural gradient descent in deep linear models . These models are particularly suited for our analysis because ( a ) efficient algorithms exist to calculate exact natural gradients which is otherwise computationally intractable and ( b ) the inductive biases of Euclidean gradient descent ( EGD ) in these models have been thoroughly studied and understood . We make the following contributions : • In linear classification , we show that NGF is invariant under invertible transformations of data ( Theorems 1 & 2 ) and as a consequence it can not recover the ` p large margin solutions that EGD tends to converge to . • We further show that ( in case of separable classification ) when the number of parameters exceeds the number of datapoints , NGF interpolates training labels in a way similar to ordinary least squares or ridgeless regression ( Theorems 3 & 4 ) . • We demonstrate experimentally that there exist learning problems where NGD can not reach good generalization performance , while EGD with the right architecture can succeed . • To perform experiments , we extended the work of Bernacchia et al . ( 2018 ) to derive efficient and numerically stable algorithms for calculating exact natural gradients in diagonal networks ( Gunasekar et al. , 2018 ) and deep matrix factorization ( Arora et al. , 2019 ) . Before stating our main theoretical and experimental results we review some relevant background on parametrization-dependent implicit regularization and natural gradients . 2 BACKGROUND . 2.1 SEPARABLE CLASSIFICATION WITH DEEP LINEAR MODELS . In this article we consider binary classification datasets { ( xn , yn ) , n = 1 , . . . , N } separable by a homogeneous linear classifier with a positive margin ( i. e. ∃β∗ s.t . ynx > nβ ∗ ≥ 1 ∀n ) . ( We use the notation X = ( x1 · · ·xN ) > ) . In such situation β∗ is not unique and there may be many separating hyperplanes which all achieve 0 training loss - it is up to the inductive biases of the learning algorithm to select one . Soudry et al . ( 2017 ) studied the dynamics of unregularized Euclidean gradient descent on logistic loss and found that the iterate β ( t ) converges to the well-known ` 2 large margin classifier in direction , that is limt→∞ β ( t ) |β ( t ) | = β∗ ` 2 |β∗ ` 2 | where β∗ ` 2 = argmin β∈RD ||β||2 s.t . ynx > nβ ≥ 1 ∀n . Importantly , Gunasekar et al . ( 2018 ) later showed that this behaviour changes if the gradient descent is performed on a different parametrization . In this paper we will focus on L-layer linear diagonal networks ( Gunasekar et al. , 2018 ) , where β = w1 w2 . . . wL , using to denote elementwise product . When we adjust parameters w1 , . . . , wL through Euclidean gradient descent , β ( t ) converges to the ` 2 L large margin separator defined as limt→∞ β ( t ) |β ( t ) | = β∗diag |β∗diag| where β∗diag = argmin β∈RD ||β|| 2 L s.t . ynx > nβ ≥ 1 ∀n . A remarkable consequence of this is that unregularized gradient descent can find sparse classifiers , without any form of explicit regularization . In fact , this inductive bias is even more sparsity-seeking than the typically used ` 1 regularization ( see e. g. Koh et al. , 2007 ; Tibshirani , 1996 ) . Figure 2 illustrates this behaviour in a 2D example . 2.2 MATRIX COMPLETION VIA DEEP MATRIX FACTORIZATION . The task of matrix completion involves recovering an unknown matrix β∗ ∈ RD×D from a randomly chosen subset of observed entries2 . The problem is clearly underdefined : there are infinitely many matrices that match the observed entries . It is common to make additional assumptions about β∗ , most commonly that that it has low rank , under which it becomes identifiable . One approach to matrix completion under the low-rank assumption is based on explicit regularization ( e.g . nuclear norm ) which leads to a convex optimization problem . Another common approach is matrix factorization using an underparametrized representation β = UV where the sizes of U ∈ RD×R and V ∈ RR×D are restricted to ensure β ’ s rank is at most R. Learning then proceeds by minimizing the non-convex mean-squared reconstruction error in U , V via gradient descent . Remarkably , Gunasekar et al . ( 2017 ) showed that the gradient-based matrix factorization method tends to converge to low-rank solutions even in the overparametrized setting , i.e . when β = W1W2 where W1 and W2 are full square matrices , without any explicit regularization . This was later extended by Arora et al . ( 2019 ) , who studied the deep matrix product parametrization of the form β =W1W2 · · ·WL . Arora et al . ( 2019 ) ran experiments for different matrix completion tasks varying initialization , depth and number of observations and compared them to minimum nuclear norm solution . When the number of observed entries is large gradient descent in deep matrix factorization models tended to the minimum nuclear norm solution . However , in the interesting case of fewer observed entries , the behaviour was different . Gradient descent pre- ferred solutions with lower effective rank at the expense of higher nuclear norm . From the evolution of the singular values of β they also concluded that the implicit regularization is towards low rank that becomes stronger as depth grows . 2to simplify presentation we assume the matrices are square , but our arguments hold more generally . 2.3 NATURAL GRADIENT DESCENT . In the next section we briefly introduce some notation and key properties of natural gradient descent ( NGD , Amari , 1997 ; Pascanu & Bengio , 2013 ) . Intuitively , one can think of NGD as a gradient descent method , but not in the Euclidean space ( with the Euclidean metric ) of parameters , but instead on the Riemannian manifold of probabilistic models the parameters define ( equipped with a different metric ) . More specifically , let ’ s say that the parameter of interest is θ , where θ defines a probabilistic model p ( y|x , θ ) . We assume that we wish to minimize the log loss under this model , i. e. l ( θ , x , y ) = − log p ( y|x , θ ) and L ( θ ) = ∑N n=1 l ( θ , xn , yn ) . Then , NGD is usually defined as θ ( t+ 1 ) = θ ( t ) − ηF−1 ( θ ) ∇θL ( θ ) , where ( 1 ) F ( θ ) = EX [ EY |X ; θ [ ∇θL ( θ ) ∇ > θ L ( θ ) ] ] ( 2 ) is the average Fisher information matrix and η is the step size . In the above definition , EY |X ; θ is taken over the distribution specified by θ , but distribution with respect to which the expectation EX is calculated can be arbitrarily chosen . In this article we use the empirical distribution of training data , though other choices are possible ( Pascanu & Bengio , 2013 ) . We will also consider natural gradient flow ( NGF ) the continuous limit of NGD , analogously defined as θ̇ = −F−1 ( θ ) ∇θL ( θ ) . ( 3 ) We also note , that F ( θ ) is not generally invertible , and indeed it will not be in some of the cases we will consider . Therefore , it is more correct to define NGF as any trajectory θt which satisfies F ( θ ) θ̇ = −∇θL ( θ ) . ( 4 ) The natural gradient direction is thus only unique within the eigenspace of F ( θ ) . Of all natural gradient directions , one common choice is to use the Moore-Penrose pseudoinverse of F : θ̇ = −F+ ( θ ) ∇θL ( θ ) . ( 5 ) We have seen how in EGD , different parametrization of the same problem leads to drastically different trajectories and optima . However , NGD with infinitesimally small learning rate ( i. e. NGF ) always follows the same trajectory in model-space and this finds the same optimum , irrespective of how it is parametrized , provided that the parametrization is smooth and locally invertible . Below we formally state this property Amari ( 1997 ) , alongside a short proof for illustration . Statement ( Invariance of NGF under reparametrization ) . Let w and θ be two parameter vectors related by the mapping θ = P ( w ) and consider natural gradient flow in w. Assume that ( 1 ) the Jacobian J = ∂θt∂wt and ( 2 ) F ( θt ) are both full rank for all t. If wt follows natural gradient flow starting from w0 then θt = P ( wt ) follows NGF , i. e. it solves θ̇t = −F ( θt ) +∇θtL ( X , θt ) . | The paper studies inductive bias of natural gradient descent (NGD) in deep linear networks. Utilizing _reparameterization invariance_ properties of NGD, the authors attempt to eliminate the role of parameterization and isolate the effect of solution found by gradient descent. The paper studies gradient flow dynamics for 1) separable classification under logistic loss and 2) deep matrix factorization. One of the contributions stated is identifying learning problems where NGD fails to generalize while GD with the right architecture performs well. | SP:a10ef4a336ff1883de5e8801cea611bd9e79883f |
Depth Without the Magic: Inductive Bias of Natural Gradient Descent | 1 INTRODUCTION There is plenty of empirical evidence that the choice of network architecture is an important determinant of the success of deep learning ( He et al. , 2015 ; Vaswani et al. , 2017 ) . The empirical observations are now supported by theoretical work into the role that parameter-to-hypothesis mapping plays in determining inductive biases of gradient-based learning . Unregularized gradient descent can efficiently find low-rank solutions in matrix completion problems ( Arora et al. , 2019 ) , sparse solutions in separable classification ( Gunasekar et al. , 2018 ) or compressed sensing ( Vaškevičius et al. , 2019 ) . Valle-Pérez et al . ( 2018 ) studied deep neural networks and found evidence that the parameter-hypothesis mapping1 is biased towards simpler functions as measured by Kolmogorov complexity . Taken together , these observations and findings have lead the community to hypothesize that The parameter-to-hypothesis mapping influences the inductive biases of gradient-based learning and may play an important role in generalization . In parallel to improving architectures , considerable research was done to improve optimization algorithms for deep learning , with a focus on faster convergence and robustness to hyperparameters . Among the most advanced optimization methods are natural gradient descent ( NGD ) techniques . An intuitive motivation for NGD is that it improves convergence by implicitly lifting the problem from parameter-space , where the loss is non-convex and poorly behaved to the Riemannian manifold 1The mapping between the parameter space and the set of hypotheses as seen on Figure 1 of hypotheses , where the loss is better behaved . From the perspective of inductive biases , the most interesting aspect of NGD is its approximate invariance to reparametrization . Natural gradient descent eliminates the effect of parameter-to-hypothesis mapping . These two observations invite questions about the nature of inductive biases in NGD as well as the role of parametrization-dependence in generalization . The first , practical , implication is as follows : if the parameter-to-hypothesis mapping really does play an important role in generalization , then eliminating its influence on the optimization path may be undesirable , and consequently the pursuit of implementing exact NGD in deep architectures may be counterproductive . Secondly , studying the behaviour of NGD in various models and tasks may give us new insights about the importance of parametrization , and could perhaps offer a way to experimentally or theoretically test hypotheses . In this paper we study the inductive bias of natural gradient descent in deep linear models . These models are particularly suited for our analysis because ( a ) efficient algorithms exist to calculate exact natural gradients which is otherwise computationally intractable and ( b ) the inductive biases of Euclidean gradient descent ( EGD ) in these models have been thoroughly studied and understood . We make the following contributions : • In linear classification , we show that NGF is invariant under invertible transformations of data ( Theorems 1 & 2 ) and as a consequence it can not recover the ` p large margin solutions that EGD tends to converge to . • We further show that ( in case of separable classification ) when the number of parameters exceeds the number of datapoints , NGF interpolates training labels in a way similar to ordinary least squares or ridgeless regression ( Theorems 3 & 4 ) . • We demonstrate experimentally that there exist learning problems where NGD can not reach good generalization performance , while EGD with the right architecture can succeed . • To perform experiments , we extended the work of Bernacchia et al . ( 2018 ) to derive efficient and numerically stable algorithms for calculating exact natural gradients in diagonal networks ( Gunasekar et al. , 2018 ) and deep matrix factorization ( Arora et al. , 2019 ) . Before stating our main theoretical and experimental results we review some relevant background on parametrization-dependent implicit regularization and natural gradients . 2 BACKGROUND . 2.1 SEPARABLE CLASSIFICATION WITH DEEP LINEAR MODELS . In this article we consider binary classification datasets { ( xn , yn ) , n = 1 , . . . , N } separable by a homogeneous linear classifier with a positive margin ( i. e. ∃β∗ s.t . ynx > nβ ∗ ≥ 1 ∀n ) . ( We use the notation X = ( x1 · · ·xN ) > ) . In such situation β∗ is not unique and there may be many separating hyperplanes which all achieve 0 training loss - it is up to the inductive biases of the learning algorithm to select one . Soudry et al . ( 2017 ) studied the dynamics of unregularized Euclidean gradient descent on logistic loss and found that the iterate β ( t ) converges to the well-known ` 2 large margin classifier in direction , that is limt→∞ β ( t ) |β ( t ) | = β∗ ` 2 |β∗ ` 2 | where β∗ ` 2 = argmin β∈RD ||β||2 s.t . ynx > nβ ≥ 1 ∀n . Importantly , Gunasekar et al . ( 2018 ) later showed that this behaviour changes if the gradient descent is performed on a different parametrization . In this paper we will focus on L-layer linear diagonal networks ( Gunasekar et al. , 2018 ) , where β = w1 w2 . . . wL , using to denote elementwise product . When we adjust parameters w1 , . . . , wL through Euclidean gradient descent , β ( t ) converges to the ` 2 L large margin separator defined as limt→∞ β ( t ) |β ( t ) | = β∗diag |β∗diag| where β∗diag = argmin β∈RD ||β|| 2 L s.t . ynx > nβ ≥ 1 ∀n . A remarkable consequence of this is that unregularized gradient descent can find sparse classifiers , without any form of explicit regularization . In fact , this inductive bias is even more sparsity-seeking than the typically used ` 1 regularization ( see e. g. Koh et al. , 2007 ; Tibshirani , 1996 ) . Figure 2 illustrates this behaviour in a 2D example . 2.2 MATRIX COMPLETION VIA DEEP MATRIX FACTORIZATION . The task of matrix completion involves recovering an unknown matrix β∗ ∈ RD×D from a randomly chosen subset of observed entries2 . The problem is clearly underdefined : there are infinitely many matrices that match the observed entries . It is common to make additional assumptions about β∗ , most commonly that that it has low rank , under which it becomes identifiable . One approach to matrix completion under the low-rank assumption is based on explicit regularization ( e.g . nuclear norm ) which leads to a convex optimization problem . Another common approach is matrix factorization using an underparametrized representation β = UV where the sizes of U ∈ RD×R and V ∈ RR×D are restricted to ensure β ’ s rank is at most R. Learning then proceeds by minimizing the non-convex mean-squared reconstruction error in U , V via gradient descent . Remarkably , Gunasekar et al . ( 2017 ) showed that the gradient-based matrix factorization method tends to converge to low-rank solutions even in the overparametrized setting , i.e . when β = W1W2 where W1 and W2 are full square matrices , without any explicit regularization . This was later extended by Arora et al . ( 2019 ) , who studied the deep matrix product parametrization of the form β =W1W2 · · ·WL . Arora et al . ( 2019 ) ran experiments for different matrix completion tasks varying initialization , depth and number of observations and compared them to minimum nuclear norm solution . When the number of observed entries is large gradient descent in deep matrix factorization models tended to the minimum nuclear norm solution . However , in the interesting case of fewer observed entries , the behaviour was different . Gradient descent pre- ferred solutions with lower effective rank at the expense of higher nuclear norm . From the evolution of the singular values of β they also concluded that the implicit regularization is towards low rank that becomes stronger as depth grows . 2to simplify presentation we assume the matrices are square , but our arguments hold more generally . 2.3 NATURAL GRADIENT DESCENT . In the next section we briefly introduce some notation and key properties of natural gradient descent ( NGD , Amari , 1997 ; Pascanu & Bengio , 2013 ) . Intuitively , one can think of NGD as a gradient descent method , but not in the Euclidean space ( with the Euclidean metric ) of parameters , but instead on the Riemannian manifold of probabilistic models the parameters define ( equipped with a different metric ) . More specifically , let ’ s say that the parameter of interest is θ , where θ defines a probabilistic model p ( y|x , θ ) . We assume that we wish to minimize the log loss under this model , i. e. l ( θ , x , y ) = − log p ( y|x , θ ) and L ( θ ) = ∑N n=1 l ( θ , xn , yn ) . Then , NGD is usually defined as θ ( t+ 1 ) = θ ( t ) − ηF−1 ( θ ) ∇θL ( θ ) , where ( 1 ) F ( θ ) = EX [ EY |X ; θ [ ∇θL ( θ ) ∇ > θ L ( θ ) ] ] ( 2 ) is the average Fisher information matrix and η is the step size . In the above definition , EY |X ; θ is taken over the distribution specified by θ , but distribution with respect to which the expectation EX is calculated can be arbitrarily chosen . In this article we use the empirical distribution of training data , though other choices are possible ( Pascanu & Bengio , 2013 ) . We will also consider natural gradient flow ( NGF ) the continuous limit of NGD , analogously defined as θ̇ = −F−1 ( θ ) ∇θL ( θ ) . ( 3 ) We also note , that F ( θ ) is not generally invertible , and indeed it will not be in some of the cases we will consider . Therefore , it is more correct to define NGF as any trajectory θt which satisfies F ( θ ) θ̇ = −∇θL ( θ ) . ( 4 ) The natural gradient direction is thus only unique within the eigenspace of F ( θ ) . Of all natural gradient directions , one common choice is to use the Moore-Penrose pseudoinverse of F : θ̇ = −F+ ( θ ) ∇θL ( θ ) . ( 5 ) We have seen how in EGD , different parametrization of the same problem leads to drastically different trajectories and optima . However , NGD with infinitesimally small learning rate ( i. e. NGF ) always follows the same trajectory in model-space and this finds the same optimum , irrespective of how it is parametrized , provided that the parametrization is smooth and locally invertible . Below we formally state this property Amari ( 1997 ) , alongside a short proof for illustration . Statement ( Invariance of NGF under reparametrization ) . Let w and θ be two parameter vectors related by the mapping θ = P ( w ) and consider natural gradient flow in w. Assume that ( 1 ) the Jacobian J = ∂θt∂wt and ( 2 ) F ( θt ) are both full rank for all t. If wt follows natural gradient flow starting from w0 then θt = P ( wt ) follows NGF , i. e. it solves θ̇t = −F ( θt ) +∇θtL ( X , θt ) . | This paper studies the inductive bias of natural gradient flow in deep linear networks and in matrix factorization. They show that the solution to the empirical risk minimization problem using NGF is invariant to pixel permutation, and conclude that the desirable property of Euclidean gradient descent under logistic loss of finding the minimum l2 norm solution does not hold (theorem 1&2). They also contribute an efficient NGD algorithm for deep diagonal linear networks. | SP:a10ef4a336ff1883de5e8801cea611bd9e79883f |
Depth Without the Magic: Inductive Bias of Natural Gradient Descent | 1 INTRODUCTION There is plenty of empirical evidence that the choice of network architecture is an important determinant of the success of deep learning ( He et al. , 2015 ; Vaswani et al. , 2017 ) . The empirical observations are now supported by theoretical work into the role that parameter-to-hypothesis mapping plays in determining inductive biases of gradient-based learning . Unregularized gradient descent can efficiently find low-rank solutions in matrix completion problems ( Arora et al. , 2019 ) , sparse solutions in separable classification ( Gunasekar et al. , 2018 ) or compressed sensing ( Vaškevičius et al. , 2019 ) . Valle-Pérez et al . ( 2018 ) studied deep neural networks and found evidence that the parameter-hypothesis mapping1 is biased towards simpler functions as measured by Kolmogorov complexity . Taken together , these observations and findings have lead the community to hypothesize that The parameter-to-hypothesis mapping influences the inductive biases of gradient-based learning and may play an important role in generalization . In parallel to improving architectures , considerable research was done to improve optimization algorithms for deep learning , with a focus on faster convergence and robustness to hyperparameters . Among the most advanced optimization methods are natural gradient descent ( NGD ) techniques . An intuitive motivation for NGD is that it improves convergence by implicitly lifting the problem from parameter-space , where the loss is non-convex and poorly behaved to the Riemannian manifold 1The mapping between the parameter space and the set of hypotheses as seen on Figure 1 of hypotheses , where the loss is better behaved . From the perspective of inductive biases , the most interesting aspect of NGD is its approximate invariance to reparametrization . Natural gradient descent eliminates the effect of parameter-to-hypothesis mapping . These two observations invite questions about the nature of inductive biases in NGD as well as the role of parametrization-dependence in generalization . The first , practical , implication is as follows : if the parameter-to-hypothesis mapping really does play an important role in generalization , then eliminating its influence on the optimization path may be undesirable , and consequently the pursuit of implementing exact NGD in deep architectures may be counterproductive . Secondly , studying the behaviour of NGD in various models and tasks may give us new insights about the importance of parametrization , and could perhaps offer a way to experimentally or theoretically test hypotheses . In this paper we study the inductive bias of natural gradient descent in deep linear models . These models are particularly suited for our analysis because ( a ) efficient algorithms exist to calculate exact natural gradients which is otherwise computationally intractable and ( b ) the inductive biases of Euclidean gradient descent ( EGD ) in these models have been thoroughly studied and understood . We make the following contributions : • In linear classification , we show that NGF is invariant under invertible transformations of data ( Theorems 1 & 2 ) and as a consequence it can not recover the ` p large margin solutions that EGD tends to converge to . • We further show that ( in case of separable classification ) when the number of parameters exceeds the number of datapoints , NGF interpolates training labels in a way similar to ordinary least squares or ridgeless regression ( Theorems 3 & 4 ) . • We demonstrate experimentally that there exist learning problems where NGD can not reach good generalization performance , while EGD with the right architecture can succeed . • To perform experiments , we extended the work of Bernacchia et al . ( 2018 ) to derive efficient and numerically stable algorithms for calculating exact natural gradients in diagonal networks ( Gunasekar et al. , 2018 ) and deep matrix factorization ( Arora et al. , 2019 ) . Before stating our main theoretical and experimental results we review some relevant background on parametrization-dependent implicit regularization and natural gradients . 2 BACKGROUND . 2.1 SEPARABLE CLASSIFICATION WITH DEEP LINEAR MODELS . In this article we consider binary classification datasets { ( xn , yn ) , n = 1 , . . . , N } separable by a homogeneous linear classifier with a positive margin ( i. e. ∃β∗ s.t . ynx > nβ ∗ ≥ 1 ∀n ) . ( We use the notation X = ( x1 · · ·xN ) > ) . In such situation β∗ is not unique and there may be many separating hyperplanes which all achieve 0 training loss - it is up to the inductive biases of the learning algorithm to select one . Soudry et al . ( 2017 ) studied the dynamics of unregularized Euclidean gradient descent on logistic loss and found that the iterate β ( t ) converges to the well-known ` 2 large margin classifier in direction , that is limt→∞ β ( t ) |β ( t ) | = β∗ ` 2 |β∗ ` 2 | where β∗ ` 2 = argmin β∈RD ||β||2 s.t . ynx > nβ ≥ 1 ∀n . Importantly , Gunasekar et al . ( 2018 ) later showed that this behaviour changes if the gradient descent is performed on a different parametrization . In this paper we will focus on L-layer linear diagonal networks ( Gunasekar et al. , 2018 ) , where β = w1 w2 . . . wL , using to denote elementwise product . When we adjust parameters w1 , . . . , wL through Euclidean gradient descent , β ( t ) converges to the ` 2 L large margin separator defined as limt→∞ β ( t ) |β ( t ) | = β∗diag |β∗diag| where β∗diag = argmin β∈RD ||β|| 2 L s.t . ynx > nβ ≥ 1 ∀n . A remarkable consequence of this is that unregularized gradient descent can find sparse classifiers , without any form of explicit regularization . In fact , this inductive bias is even more sparsity-seeking than the typically used ` 1 regularization ( see e. g. Koh et al. , 2007 ; Tibshirani , 1996 ) . Figure 2 illustrates this behaviour in a 2D example . 2.2 MATRIX COMPLETION VIA DEEP MATRIX FACTORIZATION . The task of matrix completion involves recovering an unknown matrix β∗ ∈ RD×D from a randomly chosen subset of observed entries2 . The problem is clearly underdefined : there are infinitely many matrices that match the observed entries . It is common to make additional assumptions about β∗ , most commonly that that it has low rank , under which it becomes identifiable . One approach to matrix completion under the low-rank assumption is based on explicit regularization ( e.g . nuclear norm ) which leads to a convex optimization problem . Another common approach is matrix factorization using an underparametrized representation β = UV where the sizes of U ∈ RD×R and V ∈ RR×D are restricted to ensure β ’ s rank is at most R. Learning then proceeds by minimizing the non-convex mean-squared reconstruction error in U , V via gradient descent . Remarkably , Gunasekar et al . ( 2017 ) showed that the gradient-based matrix factorization method tends to converge to low-rank solutions even in the overparametrized setting , i.e . when β = W1W2 where W1 and W2 are full square matrices , without any explicit regularization . This was later extended by Arora et al . ( 2019 ) , who studied the deep matrix product parametrization of the form β =W1W2 · · ·WL . Arora et al . ( 2019 ) ran experiments for different matrix completion tasks varying initialization , depth and number of observations and compared them to minimum nuclear norm solution . When the number of observed entries is large gradient descent in deep matrix factorization models tended to the minimum nuclear norm solution . However , in the interesting case of fewer observed entries , the behaviour was different . Gradient descent pre- ferred solutions with lower effective rank at the expense of higher nuclear norm . From the evolution of the singular values of β they also concluded that the implicit regularization is towards low rank that becomes stronger as depth grows . 2to simplify presentation we assume the matrices are square , but our arguments hold more generally . 2.3 NATURAL GRADIENT DESCENT . In the next section we briefly introduce some notation and key properties of natural gradient descent ( NGD , Amari , 1997 ; Pascanu & Bengio , 2013 ) . Intuitively , one can think of NGD as a gradient descent method , but not in the Euclidean space ( with the Euclidean metric ) of parameters , but instead on the Riemannian manifold of probabilistic models the parameters define ( equipped with a different metric ) . More specifically , let ’ s say that the parameter of interest is θ , where θ defines a probabilistic model p ( y|x , θ ) . We assume that we wish to minimize the log loss under this model , i. e. l ( θ , x , y ) = − log p ( y|x , θ ) and L ( θ ) = ∑N n=1 l ( θ , xn , yn ) . Then , NGD is usually defined as θ ( t+ 1 ) = θ ( t ) − ηF−1 ( θ ) ∇θL ( θ ) , where ( 1 ) F ( θ ) = EX [ EY |X ; θ [ ∇θL ( θ ) ∇ > θ L ( θ ) ] ] ( 2 ) is the average Fisher information matrix and η is the step size . In the above definition , EY |X ; θ is taken over the distribution specified by θ , but distribution with respect to which the expectation EX is calculated can be arbitrarily chosen . In this article we use the empirical distribution of training data , though other choices are possible ( Pascanu & Bengio , 2013 ) . We will also consider natural gradient flow ( NGF ) the continuous limit of NGD , analogously defined as θ̇ = −F−1 ( θ ) ∇θL ( θ ) . ( 3 ) We also note , that F ( θ ) is not generally invertible , and indeed it will not be in some of the cases we will consider . Therefore , it is more correct to define NGF as any trajectory θt which satisfies F ( θ ) θ̇ = −∇θL ( θ ) . ( 4 ) The natural gradient direction is thus only unique within the eigenspace of F ( θ ) . Of all natural gradient directions , one common choice is to use the Moore-Penrose pseudoinverse of F : θ̇ = −F+ ( θ ) ∇θL ( θ ) . ( 5 ) We have seen how in EGD , different parametrization of the same problem leads to drastically different trajectories and optima . However , NGD with infinitesimally small learning rate ( i. e. NGF ) always follows the same trajectory in model-space and this finds the same optimum , irrespective of how it is parametrized , provided that the parametrization is smooth and locally invertible . Below we formally state this property Amari ( 1997 ) , alongside a short proof for illustration . Statement ( Invariance of NGF under reparametrization ) . Let w and θ be two parameter vectors related by the mapping θ = P ( w ) and consider natural gradient flow in w. Assume that ( 1 ) the Jacobian J = ∂θt∂wt and ( 2 ) F ( θt ) are both full rank for all t. If wt follows natural gradient flow starting from w0 then θt = P ( wt ) follows NGF , i. e. it solves θ̇t = −F ( θt ) +∇θtL ( X , θt ) . | This paper studies the role of parameterization in determining inductive biases and influencing generalization in deep learning by studying natural gradient descent. As the authors review, natural gradient descent is invariant to reparameterization. This implies that the trajectory of NGD is in some sense ablating the role of parameterization allowing the authors to (1) determine the importance of parameterization with gradient descent in generalization, and (2) study the inductive biases unique to NGD. The authors consider two settings both theoretically and empirically: separable classification with deep linear models and matrix completion via deep matrix factorization. | SP:a10ef4a336ff1883de5e8801cea611bd9e79883f |
Subspace Regularizers for Few-Shot Class Incremental Learning | 1 INTRODUCTION . Standard approaches to classification in machine learning assume a fixed training dataset and a fixed set of class labels . But for many real-world classification problems , these assumptions are unrealistic . Classifiers must sometimes be updated on-the-fly to recognize new concepts ( e.g . new skills in personal assistants or new road signs in self-driving vehicles ) , while training data is sometimes unavailable for reuse ( e.g . due to privacy regulations , Lesort et al . 2019 ; McClure et al . 2018 ; or storage and retraining costs , Bender et al . 2021 ) . Development of models that support few-shot class-incremental learning ( FSCIL ) , in which classifiers ’ label sets can be easily extended with small numbers of new examples and no retraining , is a key challenge for machine learning systems deployed in the real world ( Masana et al. , 2020 ) . As a concrete example , consider the classification problem depicted in Fig . 1 . A model , initially trained on a large set of examples from several base classes ( snorkel , arctic fox , meerkat ; Fig . 1a ) , must subsequently be updated to additionally recognize two novel classes ( white wolf and poncho ; Fig . 1b ) , and ultimately distinguish among all five classes ( Fig . 1c ) . Training a model to recognize the base classes is straightforward : for example , we can jointly optimize the parameters of a feature extractor ( perhaps a convolutional network parameterized by θ ) and a linear classification layer ( η ) to maximize the regularized likelihood of ( image , label ) pairs from the dataset in Fig . 1a : L ( θ , η ) = 1 n ∑ ( x , y ) log exp ( η > y fθ ( x ) ) ∑ y′ exp ( η > y′fθ ( x ) ) + α ( ‖η‖2 + ‖θ‖2 ) ( 1 ) But how can this model be updated to additionally recognize the classes in Fig . 1b , with only a few examples of each new class and no access to the original training data ? Naı̈vely continuing to optimize Eq . 1 on ( x , y ) pairs drawn from the new dataset will cause several problems . In the absence of any positive examples of those classes , performance on base classes will suffer due to catastrophic forgetting ( Goodfellow et al. , 2013 ) , while performance on novel classes will likely be poor as a result of overfitting ( Anderson & Burnham , 2004 ) . As a consequence , most past work on FSCIL has focused on alternative approaches that use nonstandard prediction architectures ( e.g. , Tao et al. , 2020b ) or optimize non-likelihood objectives ( e.g. , Yoon et al. , 2020 ; Ren et al. , 2019 ) . This divergence between approaches to standard and incremental classification has its own costs—state-of-the-art approaches to FSCIL are complicated , requiring nested optimizers , complex data structures , and numerous hyperparameters . When improved representation learning and optimization techniques are developed for standard classification problems , it is often unclear to how to apply these to the incremental setting . In this paper , we turn the standard approach to classification into a surprisingly effective tool for FSCIL . Specifically , we show that both catastrophic forgetting and overfitting can be reduced by introducing an additional subspace regularizer ( related to one studied by Agarwal et al . 2010 and Kirkpatrick et al . 2017 ) that encourages novel η to lie close to the subspace spanned by the base classes . On its own , the proposed subspace regularizer produces ordinary linear classifiers that achieve state-of-the-art results on FSCIL , improving over existing work in multiple tasks and datasets . Because of its simplicity , this regularization approach can be easily extended to incorporate additional information about relationships between base and novel classes . Using language data as a source of background knowledge about classes , we describe a variation of our approach , which we term semantic subspace regularization , that pulls weight vectors toward particular convex combinations of base classes that capture their semantic similarity to existing classes . This further improves accuracy by up to 2 % over simple subspace regularization across multiple tasks . These results suggest that FSCIL and related problems may not require specialized machinery to solve , and that simple regularization approaches can solve the problems that result from limited access to training data for both base and novel classes.1 1Code will be made publicly available . 2 BACKGROUND . A long line of research has focused on the development of automated decision-making systems that support online expansion of the set of concepts they can recognize and generate . An early example ( closely related to our learning-from-definitions experiment in Section 5 ) appears in the classic SHRDLU language grounding environment ( Winograd , 1972 ) : given the definition a steeple is a small triangle on top of a tall rectangle , SHRDLU acquires the ability to answer questions containing the novel concept steeple . Recent work in machine learning describes several versions of this problem in featuring more complex perception or control : Few-shot and incremental learning Few-shot classification problems test learners ’ ability to distinguish among a fixed set of classes using only a handful of labeled examples per class ( Scheirer et al. , 2012 ) . Most effective approaches to few-shot learning rely on additional data for pre-training ( Tian et al. , 2020 ) or meta-learning ( Vinyals et al. , 2016 ; Finn et al. , 2017 ; Snell et al. , 2017 ; Yoon et al. , 2019 ) . One peculiarity of this evaluation paradigm is that , even when pre-trained , models are evaluated only on new ( few-shot ) classes , and free to update their parameters in ways that cause them to perform poorly on pre-training tasks . As noted by past work , a more realistic evaluation of models ’ ability to rapidly acquire new concepts should consider their ability to discriminate among both new concepts and old ones , a problem usually referred to as few-shot class-incremental learning ( FSCIL ) 2 ( Tao et al. , 2020b ) . FSCIL requires learners to incrementally acquire novel classes with few labeled examples while retaining high accuracy on previously learned classes . It combines the most challenging aspects of class-incremental learning ( Rebuffi et al. , 2017 ) task-incremental learning ( Delange et al. , 2021 ) , and rehearsal-based learning ( Rolnick et al. , 2019 ; Chaudhry et al. , 2019 ) , three related problems with much stronger assumptions about the kind of information available to learners . Existing approaches to this problem either prioritize novel class adaptation ( Ren et al. , 2019 ; Yoon et al. , 2020 ; Chen & Lee , 2021 ; Cheraghian et al. , 2021 ) or reducing forgetting in old classes ( Tao et al. , 2020b ) . Learning class representations Even prior to the widespread use of deep representation learning approaches , the view of classification as problem of learning class representations motivated a number of approaches to multi-class and multi-task learning ( Argyriou et al. , 2007a ; Agarwal et al. , 2010 ) . In few-shot and incremental learning settings , many recent approaches have also focused on the space of class representations ( Tao et al. , 2020a ) . Qi et al . ( 2018 ) initialize novel class representations using the average features from few-shot samples . Others ( Gidaris & Komodakis , 2018 ; Yoon et al. , 2020 ; Zhang et al. , 2021 ) train a class representation predictor via meta-learning , and Tao et al . ( 2020b ) impose topological constraints on the manifold of class representations as new representations are added . Alternatively , Chen & Lee ( 2021 ) models the visual feature space as a Gaussian mixture and use the cluster centers in a similarity-based classification scheme . Lastly , two concurrent works condition both old and new class representations at each session according to an auxiliary scheme ; graph attention network in Zhang et al . ( 2021 ) and relation projection in Zhu et al . ( 2021 ) . Our approach is related to Ren et al . ( 2019 ) , who proposes a nested optimization framework to learn auxiliary parameters for every base and novel class to influence the novel weights via regularization ; we show that these regularization targets can be derived geometrically without the need for an inner optimization step . Also related is the work of Barzilai & Crammer ( 2015 ) , which synthesizes the novel weights as linear combinations of base weights ; we adopt a regularization approach that allows learning of class representations that are not strict linear combinations of base classes . Learning with side information from language The use of background information from other modalities ( especially language ) to bootstrap learning of new classes is widely studied ( Frome et al. , 2013 ; Radford et al. , 2021 ; Reed et al. , 2016 ; Yan et al. , 2021 ) —particularly in the zero-shot learning and generalized zero-shot learning where side information is the only source of information about the novel class ( Chang et al. , 2008 ; Larochelle et al. , 2008 ; Akata et al. , 2013 ; Pourpanah et al. , 2Variants of this problem have gone by numerous names in past work , including generalized few-shot learning ( Schönfeld et al. , 2019 ) , dynamic few-shot learning ( Gidaris & Komodakis , 2018 ) or simply incremental few-shot learning ( Ren et al. , 2019 ; Chen & Lee , 2021 ) . 2020 ) . Specialized approaches exist for integrating side information in few-shot learning settings ( Schwartz et al. , 2019 ; Cheraghian et al. , 2021 ) . 3 PROBLEM FORMULATION . We follow the notation in Tao et al . ( 2020b ) for FSCIL : assume a stream of T learning sessions , each associated with a labeled dataset D ( 0 ) , D ( 1 ) , . . . , D ( T ) . Every D ( t ) consists of a support set S ( t ) ( used for training ) and a query set Q ( t ) ( used for evaluation ) . We will refer to the classes represented in D ( 0 ) as base classes ; as in Fig . 1a , we will assume that it contains a large number of examples for every class . D ( 1 ) ( and subsequent datasets ) introduce novel classes ( Fig . 1b ) . Let C ( S ) = { y : ( x , y ) ∈ S } denote the set of classes expressed in a set of examples S ; we will write C ( t ) = C ( S ( t ) ) and C ( ≤t ) : = ⋃ j≤t C ( S ( j ) ) for convenience . The learning problem we study is incremental in the sense that each support set contains only new classes ( C ( t ) ∩ C ( < t ) = ∅ ) 3 , while each query set evaluates models on both novel classes and previously seen ones ( C ( Q ( t ) ) = C ( ≤t ) ) . It is few-shot in the sense that for t > 0 , |S ( t ) | is small ( containing 1–5 examples for all datasets studied in this paper ) . Given an incremental learning session t > 0 the goal is to fine-tune existing classifier with the limited training data from novel classes such that the classifier performs well in classifying all classes learned thus far . FSCIL with a single session Prior to Tao et al . ( 2020b ) , a simpler version of the multi-session FSCIL was proposed by Qi et al . ( 2018 ) where there is only single incremental learning session after the pre-training stage i.e . T = 1 . This version , which we call single-session FSCIL , has been extensively studied by previous work ( Qi et al. , 2018 ; Gidaris & Komodakis , 2018 ; Ren et al. , 2019 ; Yoon et al. , 2020 ) . This problem formulation is the same as above with T = 1 : a feature extractor is trained on the samples from D ( 0 ) , then D ( 1 ) , then evaluated on samples with classes in C ( 0 ) ∪C ( 1 ) . | The paper proposes subspace regularization technique for incremental few-shot learning. The high-level idea is to find the basis vectors of the subspace spanned by the base classes, and then project the new classes into the subspace. The regularizer encourages the new class weights to be similar to the projected vectors with an L2 loss. Additionally, the paper also proposes to use semantic word vectors to perform regularization. Experimental results confirm that the proposed method is significantly better compared to Tao et al. (2020) and Chen & Lee (2021) either with an additional memory buffer or without, but it is unclear whether it is due to a better few-shot learning algorithm or simply a better pretrained checkpoint on the base classes. Moreover, the benefit of semantic regularization is not significant. | SP:67d187a3fd8e648c620046d8363214cf4563a4a3 |
Subspace Regularizers for Few-Shot Class Incremental Learning | 1 INTRODUCTION . Standard approaches to classification in machine learning assume a fixed training dataset and a fixed set of class labels . But for many real-world classification problems , these assumptions are unrealistic . Classifiers must sometimes be updated on-the-fly to recognize new concepts ( e.g . new skills in personal assistants or new road signs in self-driving vehicles ) , while training data is sometimes unavailable for reuse ( e.g . due to privacy regulations , Lesort et al . 2019 ; McClure et al . 2018 ; or storage and retraining costs , Bender et al . 2021 ) . Development of models that support few-shot class-incremental learning ( FSCIL ) , in which classifiers ’ label sets can be easily extended with small numbers of new examples and no retraining , is a key challenge for machine learning systems deployed in the real world ( Masana et al. , 2020 ) . As a concrete example , consider the classification problem depicted in Fig . 1 . A model , initially trained on a large set of examples from several base classes ( snorkel , arctic fox , meerkat ; Fig . 1a ) , must subsequently be updated to additionally recognize two novel classes ( white wolf and poncho ; Fig . 1b ) , and ultimately distinguish among all five classes ( Fig . 1c ) . Training a model to recognize the base classes is straightforward : for example , we can jointly optimize the parameters of a feature extractor ( perhaps a convolutional network parameterized by θ ) and a linear classification layer ( η ) to maximize the regularized likelihood of ( image , label ) pairs from the dataset in Fig . 1a : L ( θ , η ) = 1 n ∑ ( x , y ) log exp ( η > y fθ ( x ) ) ∑ y′ exp ( η > y′fθ ( x ) ) + α ( ‖η‖2 + ‖θ‖2 ) ( 1 ) But how can this model be updated to additionally recognize the classes in Fig . 1b , with only a few examples of each new class and no access to the original training data ? Naı̈vely continuing to optimize Eq . 1 on ( x , y ) pairs drawn from the new dataset will cause several problems . In the absence of any positive examples of those classes , performance on base classes will suffer due to catastrophic forgetting ( Goodfellow et al. , 2013 ) , while performance on novel classes will likely be poor as a result of overfitting ( Anderson & Burnham , 2004 ) . As a consequence , most past work on FSCIL has focused on alternative approaches that use nonstandard prediction architectures ( e.g. , Tao et al. , 2020b ) or optimize non-likelihood objectives ( e.g. , Yoon et al. , 2020 ; Ren et al. , 2019 ) . This divergence between approaches to standard and incremental classification has its own costs—state-of-the-art approaches to FSCIL are complicated , requiring nested optimizers , complex data structures , and numerous hyperparameters . When improved representation learning and optimization techniques are developed for standard classification problems , it is often unclear to how to apply these to the incremental setting . In this paper , we turn the standard approach to classification into a surprisingly effective tool for FSCIL . Specifically , we show that both catastrophic forgetting and overfitting can be reduced by introducing an additional subspace regularizer ( related to one studied by Agarwal et al . 2010 and Kirkpatrick et al . 2017 ) that encourages novel η to lie close to the subspace spanned by the base classes . On its own , the proposed subspace regularizer produces ordinary linear classifiers that achieve state-of-the-art results on FSCIL , improving over existing work in multiple tasks and datasets . Because of its simplicity , this regularization approach can be easily extended to incorporate additional information about relationships between base and novel classes . Using language data as a source of background knowledge about classes , we describe a variation of our approach , which we term semantic subspace regularization , that pulls weight vectors toward particular convex combinations of base classes that capture their semantic similarity to existing classes . This further improves accuracy by up to 2 % over simple subspace regularization across multiple tasks . These results suggest that FSCIL and related problems may not require specialized machinery to solve , and that simple regularization approaches can solve the problems that result from limited access to training data for both base and novel classes.1 1Code will be made publicly available . 2 BACKGROUND . A long line of research has focused on the development of automated decision-making systems that support online expansion of the set of concepts they can recognize and generate . An early example ( closely related to our learning-from-definitions experiment in Section 5 ) appears in the classic SHRDLU language grounding environment ( Winograd , 1972 ) : given the definition a steeple is a small triangle on top of a tall rectangle , SHRDLU acquires the ability to answer questions containing the novel concept steeple . Recent work in machine learning describes several versions of this problem in featuring more complex perception or control : Few-shot and incremental learning Few-shot classification problems test learners ’ ability to distinguish among a fixed set of classes using only a handful of labeled examples per class ( Scheirer et al. , 2012 ) . Most effective approaches to few-shot learning rely on additional data for pre-training ( Tian et al. , 2020 ) or meta-learning ( Vinyals et al. , 2016 ; Finn et al. , 2017 ; Snell et al. , 2017 ; Yoon et al. , 2019 ) . One peculiarity of this evaluation paradigm is that , even when pre-trained , models are evaluated only on new ( few-shot ) classes , and free to update their parameters in ways that cause them to perform poorly on pre-training tasks . As noted by past work , a more realistic evaluation of models ’ ability to rapidly acquire new concepts should consider their ability to discriminate among both new concepts and old ones , a problem usually referred to as few-shot class-incremental learning ( FSCIL ) 2 ( Tao et al. , 2020b ) . FSCIL requires learners to incrementally acquire novel classes with few labeled examples while retaining high accuracy on previously learned classes . It combines the most challenging aspects of class-incremental learning ( Rebuffi et al. , 2017 ) task-incremental learning ( Delange et al. , 2021 ) , and rehearsal-based learning ( Rolnick et al. , 2019 ; Chaudhry et al. , 2019 ) , three related problems with much stronger assumptions about the kind of information available to learners . Existing approaches to this problem either prioritize novel class adaptation ( Ren et al. , 2019 ; Yoon et al. , 2020 ; Chen & Lee , 2021 ; Cheraghian et al. , 2021 ) or reducing forgetting in old classes ( Tao et al. , 2020b ) . Learning class representations Even prior to the widespread use of deep representation learning approaches , the view of classification as problem of learning class representations motivated a number of approaches to multi-class and multi-task learning ( Argyriou et al. , 2007a ; Agarwal et al. , 2010 ) . In few-shot and incremental learning settings , many recent approaches have also focused on the space of class representations ( Tao et al. , 2020a ) . Qi et al . ( 2018 ) initialize novel class representations using the average features from few-shot samples . Others ( Gidaris & Komodakis , 2018 ; Yoon et al. , 2020 ; Zhang et al. , 2021 ) train a class representation predictor via meta-learning , and Tao et al . ( 2020b ) impose topological constraints on the manifold of class representations as new representations are added . Alternatively , Chen & Lee ( 2021 ) models the visual feature space as a Gaussian mixture and use the cluster centers in a similarity-based classification scheme . Lastly , two concurrent works condition both old and new class representations at each session according to an auxiliary scheme ; graph attention network in Zhang et al . ( 2021 ) and relation projection in Zhu et al . ( 2021 ) . Our approach is related to Ren et al . ( 2019 ) , who proposes a nested optimization framework to learn auxiliary parameters for every base and novel class to influence the novel weights via regularization ; we show that these regularization targets can be derived geometrically without the need for an inner optimization step . Also related is the work of Barzilai & Crammer ( 2015 ) , which synthesizes the novel weights as linear combinations of base weights ; we adopt a regularization approach that allows learning of class representations that are not strict linear combinations of base classes . Learning with side information from language The use of background information from other modalities ( especially language ) to bootstrap learning of new classes is widely studied ( Frome et al. , 2013 ; Radford et al. , 2021 ; Reed et al. , 2016 ; Yan et al. , 2021 ) —particularly in the zero-shot learning and generalized zero-shot learning where side information is the only source of information about the novel class ( Chang et al. , 2008 ; Larochelle et al. , 2008 ; Akata et al. , 2013 ; Pourpanah et al. , 2Variants of this problem have gone by numerous names in past work , including generalized few-shot learning ( Schönfeld et al. , 2019 ) , dynamic few-shot learning ( Gidaris & Komodakis , 2018 ) or simply incremental few-shot learning ( Ren et al. , 2019 ; Chen & Lee , 2021 ) . 2020 ) . Specialized approaches exist for integrating side information in few-shot learning settings ( Schwartz et al. , 2019 ; Cheraghian et al. , 2021 ) . 3 PROBLEM FORMULATION . We follow the notation in Tao et al . ( 2020b ) for FSCIL : assume a stream of T learning sessions , each associated with a labeled dataset D ( 0 ) , D ( 1 ) , . . . , D ( T ) . Every D ( t ) consists of a support set S ( t ) ( used for training ) and a query set Q ( t ) ( used for evaluation ) . We will refer to the classes represented in D ( 0 ) as base classes ; as in Fig . 1a , we will assume that it contains a large number of examples for every class . D ( 1 ) ( and subsequent datasets ) introduce novel classes ( Fig . 1b ) . Let C ( S ) = { y : ( x , y ) ∈ S } denote the set of classes expressed in a set of examples S ; we will write C ( t ) = C ( S ( t ) ) and C ( ≤t ) : = ⋃ j≤t C ( S ( j ) ) for convenience . The learning problem we study is incremental in the sense that each support set contains only new classes ( C ( t ) ∩ C ( < t ) = ∅ ) 3 , while each query set evaluates models on both novel classes and previously seen ones ( C ( Q ( t ) ) = C ( ≤t ) ) . It is few-shot in the sense that for t > 0 , |S ( t ) | is small ( containing 1–5 examples for all datasets studied in this paper ) . Given an incremental learning session t > 0 the goal is to fine-tune existing classifier with the limited training data from novel classes such that the classifier performs well in classifying all classes learned thus far . FSCIL with a single session Prior to Tao et al . ( 2020b ) , a simpler version of the multi-session FSCIL was proposed by Qi et al . ( 2018 ) where there is only single incremental learning session after the pre-training stage i.e . T = 1 . This version , which we call single-session FSCIL , has been extensively studied by previous work ( Qi et al. , 2018 ; Gidaris & Komodakis , 2018 ; Ren et al. , 2019 ; Yoon et al. , 2020 ) . This problem formulation is the same as above with T = 1 : a feature extractor is trained on the samples from D ( 0 ) , then D ( 1 ) , then evaluated on samples with classes in C ( 0 ) ∪C ( 1 ) . | The authors tackle the problem of few-shot class incremental learning (FSCIL). In this setting, a learner is first trained on a set of base classes for which plenty of data is available. Then, it must learn from small sets of novel classes while retaining good performance in the whole sequence. In order to prevent the classifier from overfitting new classes, they authors propose to regularize novel class weights to keep them in the subspace of base classes, thus making the classifier rely on known features rather than new features that are probably spurious. In addition, they also show that it is possible to align new classes with base classes based on their similarity in a learned semantic space such as Glove. Experimental results on miniImagenet and tiered-Imagenet show that their method obtains better performance than previous state of the art. | SP:67d187a3fd8e648c620046d8363214cf4563a4a3 |
Subspace Regularizers for Few-Shot Class Incremental Learning | 1 INTRODUCTION . Standard approaches to classification in machine learning assume a fixed training dataset and a fixed set of class labels . But for many real-world classification problems , these assumptions are unrealistic . Classifiers must sometimes be updated on-the-fly to recognize new concepts ( e.g . new skills in personal assistants or new road signs in self-driving vehicles ) , while training data is sometimes unavailable for reuse ( e.g . due to privacy regulations , Lesort et al . 2019 ; McClure et al . 2018 ; or storage and retraining costs , Bender et al . 2021 ) . Development of models that support few-shot class-incremental learning ( FSCIL ) , in which classifiers ’ label sets can be easily extended with small numbers of new examples and no retraining , is a key challenge for machine learning systems deployed in the real world ( Masana et al. , 2020 ) . As a concrete example , consider the classification problem depicted in Fig . 1 . A model , initially trained on a large set of examples from several base classes ( snorkel , arctic fox , meerkat ; Fig . 1a ) , must subsequently be updated to additionally recognize two novel classes ( white wolf and poncho ; Fig . 1b ) , and ultimately distinguish among all five classes ( Fig . 1c ) . Training a model to recognize the base classes is straightforward : for example , we can jointly optimize the parameters of a feature extractor ( perhaps a convolutional network parameterized by θ ) and a linear classification layer ( η ) to maximize the regularized likelihood of ( image , label ) pairs from the dataset in Fig . 1a : L ( θ , η ) = 1 n ∑ ( x , y ) log exp ( η > y fθ ( x ) ) ∑ y′ exp ( η > y′fθ ( x ) ) + α ( ‖η‖2 + ‖θ‖2 ) ( 1 ) But how can this model be updated to additionally recognize the classes in Fig . 1b , with only a few examples of each new class and no access to the original training data ? Naı̈vely continuing to optimize Eq . 1 on ( x , y ) pairs drawn from the new dataset will cause several problems . In the absence of any positive examples of those classes , performance on base classes will suffer due to catastrophic forgetting ( Goodfellow et al. , 2013 ) , while performance on novel classes will likely be poor as a result of overfitting ( Anderson & Burnham , 2004 ) . As a consequence , most past work on FSCIL has focused on alternative approaches that use nonstandard prediction architectures ( e.g. , Tao et al. , 2020b ) or optimize non-likelihood objectives ( e.g. , Yoon et al. , 2020 ; Ren et al. , 2019 ) . This divergence between approaches to standard and incremental classification has its own costs—state-of-the-art approaches to FSCIL are complicated , requiring nested optimizers , complex data structures , and numerous hyperparameters . When improved representation learning and optimization techniques are developed for standard classification problems , it is often unclear to how to apply these to the incremental setting . In this paper , we turn the standard approach to classification into a surprisingly effective tool for FSCIL . Specifically , we show that both catastrophic forgetting and overfitting can be reduced by introducing an additional subspace regularizer ( related to one studied by Agarwal et al . 2010 and Kirkpatrick et al . 2017 ) that encourages novel η to lie close to the subspace spanned by the base classes . On its own , the proposed subspace regularizer produces ordinary linear classifiers that achieve state-of-the-art results on FSCIL , improving over existing work in multiple tasks and datasets . Because of its simplicity , this regularization approach can be easily extended to incorporate additional information about relationships between base and novel classes . Using language data as a source of background knowledge about classes , we describe a variation of our approach , which we term semantic subspace regularization , that pulls weight vectors toward particular convex combinations of base classes that capture their semantic similarity to existing classes . This further improves accuracy by up to 2 % over simple subspace regularization across multiple tasks . These results suggest that FSCIL and related problems may not require specialized machinery to solve , and that simple regularization approaches can solve the problems that result from limited access to training data for both base and novel classes.1 1Code will be made publicly available . 2 BACKGROUND . A long line of research has focused on the development of automated decision-making systems that support online expansion of the set of concepts they can recognize and generate . An early example ( closely related to our learning-from-definitions experiment in Section 5 ) appears in the classic SHRDLU language grounding environment ( Winograd , 1972 ) : given the definition a steeple is a small triangle on top of a tall rectangle , SHRDLU acquires the ability to answer questions containing the novel concept steeple . Recent work in machine learning describes several versions of this problem in featuring more complex perception or control : Few-shot and incremental learning Few-shot classification problems test learners ’ ability to distinguish among a fixed set of classes using only a handful of labeled examples per class ( Scheirer et al. , 2012 ) . Most effective approaches to few-shot learning rely on additional data for pre-training ( Tian et al. , 2020 ) or meta-learning ( Vinyals et al. , 2016 ; Finn et al. , 2017 ; Snell et al. , 2017 ; Yoon et al. , 2019 ) . One peculiarity of this evaluation paradigm is that , even when pre-trained , models are evaluated only on new ( few-shot ) classes , and free to update their parameters in ways that cause them to perform poorly on pre-training tasks . As noted by past work , a more realistic evaluation of models ’ ability to rapidly acquire new concepts should consider their ability to discriminate among both new concepts and old ones , a problem usually referred to as few-shot class-incremental learning ( FSCIL ) 2 ( Tao et al. , 2020b ) . FSCIL requires learners to incrementally acquire novel classes with few labeled examples while retaining high accuracy on previously learned classes . It combines the most challenging aspects of class-incremental learning ( Rebuffi et al. , 2017 ) task-incremental learning ( Delange et al. , 2021 ) , and rehearsal-based learning ( Rolnick et al. , 2019 ; Chaudhry et al. , 2019 ) , three related problems with much stronger assumptions about the kind of information available to learners . Existing approaches to this problem either prioritize novel class adaptation ( Ren et al. , 2019 ; Yoon et al. , 2020 ; Chen & Lee , 2021 ; Cheraghian et al. , 2021 ) or reducing forgetting in old classes ( Tao et al. , 2020b ) . Learning class representations Even prior to the widespread use of deep representation learning approaches , the view of classification as problem of learning class representations motivated a number of approaches to multi-class and multi-task learning ( Argyriou et al. , 2007a ; Agarwal et al. , 2010 ) . In few-shot and incremental learning settings , many recent approaches have also focused on the space of class representations ( Tao et al. , 2020a ) . Qi et al . ( 2018 ) initialize novel class representations using the average features from few-shot samples . Others ( Gidaris & Komodakis , 2018 ; Yoon et al. , 2020 ; Zhang et al. , 2021 ) train a class representation predictor via meta-learning , and Tao et al . ( 2020b ) impose topological constraints on the manifold of class representations as new representations are added . Alternatively , Chen & Lee ( 2021 ) models the visual feature space as a Gaussian mixture and use the cluster centers in a similarity-based classification scheme . Lastly , two concurrent works condition both old and new class representations at each session according to an auxiliary scheme ; graph attention network in Zhang et al . ( 2021 ) and relation projection in Zhu et al . ( 2021 ) . Our approach is related to Ren et al . ( 2019 ) , who proposes a nested optimization framework to learn auxiliary parameters for every base and novel class to influence the novel weights via regularization ; we show that these regularization targets can be derived geometrically without the need for an inner optimization step . Also related is the work of Barzilai & Crammer ( 2015 ) , which synthesizes the novel weights as linear combinations of base weights ; we adopt a regularization approach that allows learning of class representations that are not strict linear combinations of base classes . Learning with side information from language The use of background information from other modalities ( especially language ) to bootstrap learning of new classes is widely studied ( Frome et al. , 2013 ; Radford et al. , 2021 ; Reed et al. , 2016 ; Yan et al. , 2021 ) —particularly in the zero-shot learning and generalized zero-shot learning where side information is the only source of information about the novel class ( Chang et al. , 2008 ; Larochelle et al. , 2008 ; Akata et al. , 2013 ; Pourpanah et al. , 2Variants of this problem have gone by numerous names in past work , including generalized few-shot learning ( Schönfeld et al. , 2019 ) , dynamic few-shot learning ( Gidaris & Komodakis , 2018 ) or simply incremental few-shot learning ( Ren et al. , 2019 ; Chen & Lee , 2021 ) . 2020 ) . Specialized approaches exist for integrating side information in few-shot learning settings ( Schwartz et al. , 2019 ; Cheraghian et al. , 2021 ) . 3 PROBLEM FORMULATION . We follow the notation in Tao et al . ( 2020b ) for FSCIL : assume a stream of T learning sessions , each associated with a labeled dataset D ( 0 ) , D ( 1 ) , . . . , D ( T ) . Every D ( t ) consists of a support set S ( t ) ( used for training ) and a query set Q ( t ) ( used for evaluation ) . We will refer to the classes represented in D ( 0 ) as base classes ; as in Fig . 1a , we will assume that it contains a large number of examples for every class . D ( 1 ) ( and subsequent datasets ) introduce novel classes ( Fig . 1b ) . Let C ( S ) = { y : ( x , y ) ∈ S } denote the set of classes expressed in a set of examples S ; we will write C ( t ) = C ( S ( t ) ) and C ( ≤t ) : = ⋃ j≤t C ( S ( j ) ) for convenience . The learning problem we study is incremental in the sense that each support set contains only new classes ( C ( t ) ∩ C ( < t ) = ∅ ) 3 , while each query set evaluates models on both novel classes and previously seen ones ( C ( Q ( t ) ) = C ( ≤t ) ) . It is few-shot in the sense that for t > 0 , |S ( t ) | is small ( containing 1–5 examples for all datasets studied in this paper ) . Given an incremental learning session t > 0 the goal is to fine-tune existing classifier with the limited training data from novel classes such that the classifier performs well in classifying all classes learned thus far . FSCIL with a single session Prior to Tao et al . ( 2020b ) , a simpler version of the multi-session FSCIL was proposed by Qi et al . ( 2018 ) where there is only single incremental learning session after the pre-training stage i.e . T = 1 . This version , which we call single-session FSCIL , has been extensively studied by previous work ( Qi et al. , 2018 ; Gidaris & Komodakis , 2018 ; Ren et al. , 2019 ; Yoon et al. , 2020 ) . This problem formulation is the same as above with T = 1 : a feature extractor is trained on the samples from D ( 0 ) , then D ( 1 ) , then evaluated on samples with classes in C ( 0 ) ∪C ( 1 ) . | This paper leverages a simple but effective regularization technique for few-shot incremental class learning. The assumption is that at the first stage we have a lot of data and can train a good feature extractor and a network that can classify all the classes to a good degree. Given a few examples of a new unseen class, the authors enforce that the weights of the final layer of the classifier are close to the space that is spanned by the current classifier's last layer weights. They achieve state-of-the-art results on this very challenging task. | SP:67d187a3fd8e648c620046d8363214cf4563a4a3 |
Distributional Generalization: Structure Beyond Test Error | 1 INTRODUCTION . In learning theory , when we study how well a classifier “ generalizes ” , we usually consider a single metric – its test error ( Shalev-Shwartz & Ben-David , 2014 ) . However , there could be many different classifiers with the same test error that differ substantially in , say , the subgroups of inputs on which they make errors . Reducing classifiers to a single number misses these rich aspects of their behavior . In this work , we propose formally studying the entire joint distribution of classifier inputs and outputs . That is , the distribution ( x , f ( x ) ) for samples from the distribution x ∼ D for a classifier f ( x ) . This distribution reveals many structural properties of the classifier beyond test error ( such as where the errors occur ) . In fact , we discover new behaviors of modern classifiers that can only be understood in this framework . As an example , consider the following experiment ( Figure 1 ) . Experiment 1 . Consider a binary classification version of CIFAR-10 , where CIFAR-10 images x have binary labels Animal/Object . Take 50K samples from this distribution as a train set , but apply the following label noise : flip the label of cats to Object with probability 30 % . Now train a WideResNet f to 0 train error on this train set . How does the trained classifier behave on test samples ? Options below : ( 1 ) The test error is low across all classes , since there is only 3 % overall label noise in the train set . ( 2 ) Test error is “ spread ” across the animal class . After all , the classifier is not explicitly told what a cat or a dog is , just that they are all animals . ( 3 ) The classifier misclassifies roughly 30 % of test cats as “ objects ” , but all other animals are largely unaffected . The reality is closest to option ( 3 ) as shown in Figure 1 . The left panel shows the joint density of train inputs x with train labels Object/Animal . The right panel shows the classifier predictions f ( x ) on test inputs x . There are several notable things about this experiment . First , the error is localized to cats in the test set as it was in the train set , even though no explicit cat labels were provided . The interpolating model is thus sensitive to subgroup-structures in the distribution . Second , the amount of error on the cat class is close to the noise applied on the train set . Thus , the behavior of the classifier on the train set generalizes to the test set in a stronger sense than just average error . Specifically , when conditioned on a subgroup ( cat ) , the distribution of the true labels is close to that of the classifier outputs . Third , this is not the behavior of the Bayes-optimal classifier , which would always output the maximum-likelihood label instead of reproducing the noise in the distribution . The network is thus behaving poorly from the perspective of Bayes-optimality , but behaving well in a certain distributional sense ( which we will formalize soon ) . Now , consider a seemingly unrelated experimental observation . Take an AlexNet trained on ImageNet , a 1000-way classification problem with 116 varieties of dogs . AlexNet only achieves 56.5 % test accuracy on ImageNet . However , it at least classifies most dogs as some variety of dog ( with 98.4 % accuracy ) , though it may mistake the exact breed . In this work , we show that both of these experiments are examples of the same underlying phenomenon . We empirically show that for an interpolating classifier , its classification outputs are close in distribution to the true labels — even when conditioned on many subsets of the domain . For example , in Figure 1 , the distribution of p ( f ( x ) |x = cat ) is close to the true label distribution of p ( y|x = cat ) . We propose a formal conjecture ( Feature Calibration ) , that predicts which subgroups of the domain can be conditioned on for the above distributional closeness to hold . These experimental behaviors could not have been captured solely by looking at average test error , as is done in the classical theory of generalization . In fact , they are special cases of a new kind of generalization , which we call “ Distributional Generalization ” . 1.1 DISTRIBUTIONAL GENERALIZATION . Informally , Distributional Generalization states that the outputs of classifiers f on their train sets and test sets are close as distributions ( as opposed to close in just error ) . That is , the following joint distributions1 are close : ( x , f ( x ) ) x∼TestSet ≈ ( x , f ( x ) ) x∼TrainSet ( 1 ) The remainder of this paper is devoted to making the above statement precise , and empirically checking its validity on real-world tasks . Specifically , we want to formally define the notion of approximation ( ≈ ) , and understand how it depends on the problem parameters ( the type of classifier , number of train samples , etc ) . We focus primarily on interpolating methods , where we formalize Equation ( 1 ) through our Feature Calibration Conjecture . 1.2 OUR CONTRIBUTIONS AND ORGANIZATION . We discover new empirical behaviors of interpolating classifiers , and we propose quantitative conjectures to characterize these behaviors . • In Section 3 , we introduce a “ Feature Calibration ” conjecture , which unifies our experimental observations . Roughly , Feature Calibration says that the outputs of classifiers match the statistics of their training distribution when conditioned on certain subgroups . • In Section 4 , we experimentally stress test our Feature Calibration conjecture across various settings in machine learning , including neural networks , kernel machines , and decision trees . This highlights the universality of our results across machine learning . 1These distributions also include the randomness in sampling the train and test sets , and in training the classifier , as we define more precisely in Section 3 . • In Section 5 , we relate our results to classical generalization , by defining a new notion of Distributional Generalization which extends and subsumes the classical notion . • Finally , in Section 5.2 we informally discuss how Distributional Generalization can be applied even for non-interpolating methods . Our results extend our scientific understanding of of interpolating methods , and introduce a new type of generalization exhibited across many methods in machine learning . 1.3 RELATED WORK AND SIGNIFICANCE . Our work has connections to , and implications for many existing research programs in deep learning . Implicit Bias and Overparameterization . There has been a long line of recent work towards understanding overparameterized and interpolating methods , since these pose challenges for classical theories of generalization ( e.g . Zhang et al . ( 2016 ) ; Belkin et al . ( 2018a ; b ; 2019 ) ; Liang & Rakhlin ( 2018 ) ; Nakkiran et al . ( 2020 ) ; Schapire et al . ( 1998 ) ; Breiman ( 1995 ) ; Soudry et al . ( 2018 ) ; Gunasekar et al . ( 2018 ) ) . The “ implicit bias ” program here aims to answer : Among all models with 0 train error , which model is actually produced by SGD ? Most existing work seeks to characterize the exact implicit bias of models under certain ( sometimes strong ) assumptions on the model , training method or the data distribution . In contrast , our conjecture applies across many different interpolating models ( from neural nets to decision trees ) as they would be used in practice , and thus form a sort of “ universal implicit bias ” of these methods . Moreover , our results place constraints on potential future theories of implicit bias , and guide us towards theories that better capture practice . Benign Overfitting . Most prior works on interpolating classifiers attempt to explain why training to interpolation “ does not harm ” the the model . This has been dubbed “ benign overfitting ” ( Bartlett et al. , 2020 ) and “ harmless interpolation ” ( Muthukumar et al. , 2020 ) , reflecting the widely-held belief that interpolation does not harm the decision boundary of classifiers . In contrast , we find that interpolation actually does “ harm ” classifiers , in predictable ways : fitting the label noise on the train set causes similar noise to be reproduced at test time . Our results thus indicate that interpolation can significantly affect the decision boundary of classifiers , and should not be considered a purely “ benign ” effect . Classical Generalization and Scaling Limits . Our framework of Distributional Generalization is insightful even to study classical generalization , since it reveals much more about models than just their test error . For example , statistical learning theory attempts to understand if and when models will asymptotically converge to Bayes optimal classifiers , in the limit of large data ( “ asymptotic consistency ” Shalev-Shwartz & Ben-David ( 2014 ) ; Wasserman ( 2013 ) ) . In deep learning , there are at least two distinct ways to scale model and data to infinity together : the underparameterized scaling limit , where data-size model-size always , and the overparameterized scaling limit , where data-size model-size always . The underparameterized scaling limit is well-understood : when data is essentially infinite , neural networks will converge to the Bayes-optimal classifier ( provided the model-size is large enough , and the optimization is run for long enough , with enough noise to escape local minima ) . On the other hand , our work suggests that in the overparameterized scaling limit , models will not converge to the Bayes-optimal classifier . Specifically , our Feature Calibration Conjecture implies that in the limit of large data , interpolating models will approach a sampler from the distribution . That is , the limiting model f will be such that the output f ( x ) is a sample from p ( y|x ) , as opposed to the Bayes-optimal f∗ ( x ) = argmaxy p ( y|x ) . This claim— that overparameterized models do not converge to Bayes-optimal classifiers— is unique to our work as far as we know , and highlights the broad implications of our results . Locality and Manifold Learning . Our intuition for the behaviors in this work is that they arise due to some form of “ locality ” of the trained classifiers , in an appropriate embedding space . For example , the behavior observed in Experiment 1 would be consistent with that of a 1-Nearest-Neighbor classifier in a embedding that separates the CIFAR-10 classes well . This intuition that classifiers learn good embeddings is present in various forms in the literature , for example : the so-called called “ manifold hypothesis , ” that natural data lie on a low-dimensional manifold ( Narayanan & Mitter , 2010 ; Sharma & Kaplan , 2020 ) , as well as works on local stiffness of the loss landscape ( Fort et al. , 2019 ) , and works showing that overparameterized neural networks can learn hidden low-dimensional structure in high-dimensional settings ( Gerace et al. , 2020 ; Bach , 2017 ; Chizat & Bach , 2020 ) . It is open to more formally understand connections between our work and the above . Uncertainty Estimation . Since the appearance of the current work on arXiv , it has been directly built on by other authors . The work of Jiang et al . ( 2021 ) investigates our conjectures further , and extends them to develop a method for out-of-distribution uncertainty estimation . This highlights the fundamental nature and importance of our results , since they have already been used in a practical application . A full discussion of related works is in Appendix A . | The paper proposes a new notion of ``distributional generalization". It formalizes this through a conjecture (feature calibration conjecture) which says that the output distribution of an interpolating classifier matches the distributions of the labels on a certain class of sub-groups of the data. The paper evaluates this conjecture empirically, and also proves it for certain nearest neighbor models. | SP:0cca38c3b824ab3e789e47aff6e5c9fc67b1fb1c |
Distributional Generalization: Structure Beyond Test Error | 1 INTRODUCTION . In learning theory , when we study how well a classifier “ generalizes ” , we usually consider a single metric – its test error ( Shalev-Shwartz & Ben-David , 2014 ) . However , there could be many different classifiers with the same test error that differ substantially in , say , the subgroups of inputs on which they make errors . Reducing classifiers to a single number misses these rich aspects of their behavior . In this work , we propose formally studying the entire joint distribution of classifier inputs and outputs . That is , the distribution ( x , f ( x ) ) for samples from the distribution x ∼ D for a classifier f ( x ) . This distribution reveals many structural properties of the classifier beyond test error ( such as where the errors occur ) . In fact , we discover new behaviors of modern classifiers that can only be understood in this framework . As an example , consider the following experiment ( Figure 1 ) . Experiment 1 . Consider a binary classification version of CIFAR-10 , where CIFAR-10 images x have binary labels Animal/Object . Take 50K samples from this distribution as a train set , but apply the following label noise : flip the label of cats to Object with probability 30 % . Now train a WideResNet f to 0 train error on this train set . How does the trained classifier behave on test samples ? Options below : ( 1 ) The test error is low across all classes , since there is only 3 % overall label noise in the train set . ( 2 ) Test error is “ spread ” across the animal class . After all , the classifier is not explicitly told what a cat or a dog is , just that they are all animals . ( 3 ) The classifier misclassifies roughly 30 % of test cats as “ objects ” , but all other animals are largely unaffected . The reality is closest to option ( 3 ) as shown in Figure 1 . The left panel shows the joint density of train inputs x with train labels Object/Animal . The right panel shows the classifier predictions f ( x ) on test inputs x . There are several notable things about this experiment . First , the error is localized to cats in the test set as it was in the train set , even though no explicit cat labels were provided . The interpolating model is thus sensitive to subgroup-structures in the distribution . Second , the amount of error on the cat class is close to the noise applied on the train set . Thus , the behavior of the classifier on the train set generalizes to the test set in a stronger sense than just average error . Specifically , when conditioned on a subgroup ( cat ) , the distribution of the true labels is close to that of the classifier outputs . Third , this is not the behavior of the Bayes-optimal classifier , which would always output the maximum-likelihood label instead of reproducing the noise in the distribution . The network is thus behaving poorly from the perspective of Bayes-optimality , but behaving well in a certain distributional sense ( which we will formalize soon ) . Now , consider a seemingly unrelated experimental observation . Take an AlexNet trained on ImageNet , a 1000-way classification problem with 116 varieties of dogs . AlexNet only achieves 56.5 % test accuracy on ImageNet . However , it at least classifies most dogs as some variety of dog ( with 98.4 % accuracy ) , though it may mistake the exact breed . In this work , we show that both of these experiments are examples of the same underlying phenomenon . We empirically show that for an interpolating classifier , its classification outputs are close in distribution to the true labels — even when conditioned on many subsets of the domain . For example , in Figure 1 , the distribution of p ( f ( x ) |x = cat ) is close to the true label distribution of p ( y|x = cat ) . We propose a formal conjecture ( Feature Calibration ) , that predicts which subgroups of the domain can be conditioned on for the above distributional closeness to hold . These experimental behaviors could not have been captured solely by looking at average test error , as is done in the classical theory of generalization . In fact , they are special cases of a new kind of generalization , which we call “ Distributional Generalization ” . 1.1 DISTRIBUTIONAL GENERALIZATION . Informally , Distributional Generalization states that the outputs of classifiers f on their train sets and test sets are close as distributions ( as opposed to close in just error ) . That is , the following joint distributions1 are close : ( x , f ( x ) ) x∼TestSet ≈ ( x , f ( x ) ) x∼TrainSet ( 1 ) The remainder of this paper is devoted to making the above statement precise , and empirically checking its validity on real-world tasks . Specifically , we want to formally define the notion of approximation ( ≈ ) , and understand how it depends on the problem parameters ( the type of classifier , number of train samples , etc ) . We focus primarily on interpolating methods , where we formalize Equation ( 1 ) through our Feature Calibration Conjecture . 1.2 OUR CONTRIBUTIONS AND ORGANIZATION . We discover new empirical behaviors of interpolating classifiers , and we propose quantitative conjectures to characterize these behaviors . • In Section 3 , we introduce a “ Feature Calibration ” conjecture , which unifies our experimental observations . Roughly , Feature Calibration says that the outputs of classifiers match the statistics of their training distribution when conditioned on certain subgroups . • In Section 4 , we experimentally stress test our Feature Calibration conjecture across various settings in machine learning , including neural networks , kernel machines , and decision trees . This highlights the universality of our results across machine learning . 1These distributions also include the randomness in sampling the train and test sets , and in training the classifier , as we define more precisely in Section 3 . • In Section 5 , we relate our results to classical generalization , by defining a new notion of Distributional Generalization which extends and subsumes the classical notion . • Finally , in Section 5.2 we informally discuss how Distributional Generalization can be applied even for non-interpolating methods . Our results extend our scientific understanding of of interpolating methods , and introduce a new type of generalization exhibited across many methods in machine learning . 1.3 RELATED WORK AND SIGNIFICANCE . Our work has connections to , and implications for many existing research programs in deep learning . Implicit Bias and Overparameterization . There has been a long line of recent work towards understanding overparameterized and interpolating methods , since these pose challenges for classical theories of generalization ( e.g . Zhang et al . ( 2016 ) ; Belkin et al . ( 2018a ; b ; 2019 ) ; Liang & Rakhlin ( 2018 ) ; Nakkiran et al . ( 2020 ) ; Schapire et al . ( 1998 ) ; Breiman ( 1995 ) ; Soudry et al . ( 2018 ) ; Gunasekar et al . ( 2018 ) ) . The “ implicit bias ” program here aims to answer : Among all models with 0 train error , which model is actually produced by SGD ? Most existing work seeks to characterize the exact implicit bias of models under certain ( sometimes strong ) assumptions on the model , training method or the data distribution . In contrast , our conjecture applies across many different interpolating models ( from neural nets to decision trees ) as they would be used in practice , and thus form a sort of “ universal implicit bias ” of these methods . Moreover , our results place constraints on potential future theories of implicit bias , and guide us towards theories that better capture practice . Benign Overfitting . Most prior works on interpolating classifiers attempt to explain why training to interpolation “ does not harm ” the the model . This has been dubbed “ benign overfitting ” ( Bartlett et al. , 2020 ) and “ harmless interpolation ” ( Muthukumar et al. , 2020 ) , reflecting the widely-held belief that interpolation does not harm the decision boundary of classifiers . In contrast , we find that interpolation actually does “ harm ” classifiers , in predictable ways : fitting the label noise on the train set causes similar noise to be reproduced at test time . Our results thus indicate that interpolation can significantly affect the decision boundary of classifiers , and should not be considered a purely “ benign ” effect . Classical Generalization and Scaling Limits . Our framework of Distributional Generalization is insightful even to study classical generalization , since it reveals much more about models than just their test error . For example , statistical learning theory attempts to understand if and when models will asymptotically converge to Bayes optimal classifiers , in the limit of large data ( “ asymptotic consistency ” Shalev-Shwartz & Ben-David ( 2014 ) ; Wasserman ( 2013 ) ) . In deep learning , there are at least two distinct ways to scale model and data to infinity together : the underparameterized scaling limit , where data-size model-size always , and the overparameterized scaling limit , where data-size model-size always . The underparameterized scaling limit is well-understood : when data is essentially infinite , neural networks will converge to the Bayes-optimal classifier ( provided the model-size is large enough , and the optimization is run for long enough , with enough noise to escape local minima ) . On the other hand , our work suggests that in the overparameterized scaling limit , models will not converge to the Bayes-optimal classifier . Specifically , our Feature Calibration Conjecture implies that in the limit of large data , interpolating models will approach a sampler from the distribution . That is , the limiting model f will be such that the output f ( x ) is a sample from p ( y|x ) , as opposed to the Bayes-optimal f∗ ( x ) = argmaxy p ( y|x ) . This claim— that overparameterized models do not converge to Bayes-optimal classifiers— is unique to our work as far as we know , and highlights the broad implications of our results . Locality and Manifold Learning . Our intuition for the behaviors in this work is that they arise due to some form of “ locality ” of the trained classifiers , in an appropriate embedding space . For example , the behavior observed in Experiment 1 would be consistent with that of a 1-Nearest-Neighbor classifier in a embedding that separates the CIFAR-10 classes well . This intuition that classifiers learn good embeddings is present in various forms in the literature , for example : the so-called called “ manifold hypothesis , ” that natural data lie on a low-dimensional manifold ( Narayanan & Mitter , 2010 ; Sharma & Kaplan , 2020 ) , as well as works on local stiffness of the loss landscape ( Fort et al. , 2019 ) , and works showing that overparameterized neural networks can learn hidden low-dimensional structure in high-dimensional settings ( Gerace et al. , 2020 ; Bach , 2017 ; Chizat & Bach , 2020 ) . It is open to more formally understand connections between our work and the above . Uncertainty Estimation . Since the appearance of the current work on arXiv , it has been directly built on by other authors . The work of Jiang et al . ( 2021 ) investigates our conjectures further , and extends them to develop a method for out-of-distribution uncertainty estimation . This highlights the fundamental nature and importance of our results , since they have already been used in a practical application . A full discussion of related works is in Appendix A . | The paper argues about considering entire distribution of classifier behavior rather than a traditional single metric view (test error). The distribution can highlight the areas where the errors actually occur. The paper analyzes datasets with perturbed labels and shows how the classifiers will perform on these perturbed training data. It presents a conjecture that when conditioned on "distinguishable features" the distribution on output is similar to that of distribution of true labels. The authors present some examples where "distinguishable features" depends on classifier. They prove the conjecture for nearest neighbors classifier under some regularity conditions. | SP:0cca38c3b824ab3e789e47aff6e5c9fc67b1fb1c |
Distributional Generalization: Structure Beyond Test Error | 1 INTRODUCTION . In learning theory , when we study how well a classifier “ generalizes ” , we usually consider a single metric – its test error ( Shalev-Shwartz & Ben-David , 2014 ) . However , there could be many different classifiers with the same test error that differ substantially in , say , the subgroups of inputs on which they make errors . Reducing classifiers to a single number misses these rich aspects of their behavior . In this work , we propose formally studying the entire joint distribution of classifier inputs and outputs . That is , the distribution ( x , f ( x ) ) for samples from the distribution x ∼ D for a classifier f ( x ) . This distribution reveals many structural properties of the classifier beyond test error ( such as where the errors occur ) . In fact , we discover new behaviors of modern classifiers that can only be understood in this framework . As an example , consider the following experiment ( Figure 1 ) . Experiment 1 . Consider a binary classification version of CIFAR-10 , where CIFAR-10 images x have binary labels Animal/Object . Take 50K samples from this distribution as a train set , but apply the following label noise : flip the label of cats to Object with probability 30 % . Now train a WideResNet f to 0 train error on this train set . How does the trained classifier behave on test samples ? Options below : ( 1 ) The test error is low across all classes , since there is only 3 % overall label noise in the train set . ( 2 ) Test error is “ spread ” across the animal class . After all , the classifier is not explicitly told what a cat or a dog is , just that they are all animals . ( 3 ) The classifier misclassifies roughly 30 % of test cats as “ objects ” , but all other animals are largely unaffected . The reality is closest to option ( 3 ) as shown in Figure 1 . The left panel shows the joint density of train inputs x with train labels Object/Animal . The right panel shows the classifier predictions f ( x ) on test inputs x . There are several notable things about this experiment . First , the error is localized to cats in the test set as it was in the train set , even though no explicit cat labels were provided . The interpolating model is thus sensitive to subgroup-structures in the distribution . Second , the amount of error on the cat class is close to the noise applied on the train set . Thus , the behavior of the classifier on the train set generalizes to the test set in a stronger sense than just average error . Specifically , when conditioned on a subgroup ( cat ) , the distribution of the true labels is close to that of the classifier outputs . Third , this is not the behavior of the Bayes-optimal classifier , which would always output the maximum-likelihood label instead of reproducing the noise in the distribution . The network is thus behaving poorly from the perspective of Bayes-optimality , but behaving well in a certain distributional sense ( which we will formalize soon ) . Now , consider a seemingly unrelated experimental observation . Take an AlexNet trained on ImageNet , a 1000-way classification problem with 116 varieties of dogs . AlexNet only achieves 56.5 % test accuracy on ImageNet . However , it at least classifies most dogs as some variety of dog ( with 98.4 % accuracy ) , though it may mistake the exact breed . In this work , we show that both of these experiments are examples of the same underlying phenomenon . We empirically show that for an interpolating classifier , its classification outputs are close in distribution to the true labels — even when conditioned on many subsets of the domain . For example , in Figure 1 , the distribution of p ( f ( x ) |x = cat ) is close to the true label distribution of p ( y|x = cat ) . We propose a formal conjecture ( Feature Calibration ) , that predicts which subgroups of the domain can be conditioned on for the above distributional closeness to hold . These experimental behaviors could not have been captured solely by looking at average test error , as is done in the classical theory of generalization . In fact , they are special cases of a new kind of generalization , which we call “ Distributional Generalization ” . 1.1 DISTRIBUTIONAL GENERALIZATION . Informally , Distributional Generalization states that the outputs of classifiers f on their train sets and test sets are close as distributions ( as opposed to close in just error ) . That is , the following joint distributions1 are close : ( x , f ( x ) ) x∼TestSet ≈ ( x , f ( x ) ) x∼TrainSet ( 1 ) The remainder of this paper is devoted to making the above statement precise , and empirically checking its validity on real-world tasks . Specifically , we want to formally define the notion of approximation ( ≈ ) , and understand how it depends on the problem parameters ( the type of classifier , number of train samples , etc ) . We focus primarily on interpolating methods , where we formalize Equation ( 1 ) through our Feature Calibration Conjecture . 1.2 OUR CONTRIBUTIONS AND ORGANIZATION . We discover new empirical behaviors of interpolating classifiers , and we propose quantitative conjectures to characterize these behaviors . • In Section 3 , we introduce a “ Feature Calibration ” conjecture , which unifies our experimental observations . Roughly , Feature Calibration says that the outputs of classifiers match the statistics of their training distribution when conditioned on certain subgroups . • In Section 4 , we experimentally stress test our Feature Calibration conjecture across various settings in machine learning , including neural networks , kernel machines , and decision trees . This highlights the universality of our results across machine learning . 1These distributions also include the randomness in sampling the train and test sets , and in training the classifier , as we define more precisely in Section 3 . • In Section 5 , we relate our results to classical generalization , by defining a new notion of Distributional Generalization which extends and subsumes the classical notion . • Finally , in Section 5.2 we informally discuss how Distributional Generalization can be applied even for non-interpolating methods . Our results extend our scientific understanding of of interpolating methods , and introduce a new type of generalization exhibited across many methods in machine learning . 1.3 RELATED WORK AND SIGNIFICANCE . Our work has connections to , and implications for many existing research programs in deep learning . Implicit Bias and Overparameterization . There has been a long line of recent work towards understanding overparameterized and interpolating methods , since these pose challenges for classical theories of generalization ( e.g . Zhang et al . ( 2016 ) ; Belkin et al . ( 2018a ; b ; 2019 ) ; Liang & Rakhlin ( 2018 ) ; Nakkiran et al . ( 2020 ) ; Schapire et al . ( 1998 ) ; Breiman ( 1995 ) ; Soudry et al . ( 2018 ) ; Gunasekar et al . ( 2018 ) ) . The “ implicit bias ” program here aims to answer : Among all models with 0 train error , which model is actually produced by SGD ? Most existing work seeks to characterize the exact implicit bias of models under certain ( sometimes strong ) assumptions on the model , training method or the data distribution . In contrast , our conjecture applies across many different interpolating models ( from neural nets to decision trees ) as they would be used in practice , and thus form a sort of “ universal implicit bias ” of these methods . Moreover , our results place constraints on potential future theories of implicit bias , and guide us towards theories that better capture practice . Benign Overfitting . Most prior works on interpolating classifiers attempt to explain why training to interpolation “ does not harm ” the the model . This has been dubbed “ benign overfitting ” ( Bartlett et al. , 2020 ) and “ harmless interpolation ” ( Muthukumar et al. , 2020 ) , reflecting the widely-held belief that interpolation does not harm the decision boundary of classifiers . In contrast , we find that interpolation actually does “ harm ” classifiers , in predictable ways : fitting the label noise on the train set causes similar noise to be reproduced at test time . Our results thus indicate that interpolation can significantly affect the decision boundary of classifiers , and should not be considered a purely “ benign ” effect . Classical Generalization and Scaling Limits . Our framework of Distributional Generalization is insightful even to study classical generalization , since it reveals much more about models than just their test error . For example , statistical learning theory attempts to understand if and when models will asymptotically converge to Bayes optimal classifiers , in the limit of large data ( “ asymptotic consistency ” Shalev-Shwartz & Ben-David ( 2014 ) ; Wasserman ( 2013 ) ) . In deep learning , there are at least two distinct ways to scale model and data to infinity together : the underparameterized scaling limit , where data-size model-size always , and the overparameterized scaling limit , where data-size model-size always . The underparameterized scaling limit is well-understood : when data is essentially infinite , neural networks will converge to the Bayes-optimal classifier ( provided the model-size is large enough , and the optimization is run for long enough , with enough noise to escape local minima ) . On the other hand , our work suggests that in the overparameterized scaling limit , models will not converge to the Bayes-optimal classifier . Specifically , our Feature Calibration Conjecture implies that in the limit of large data , interpolating models will approach a sampler from the distribution . That is , the limiting model f will be such that the output f ( x ) is a sample from p ( y|x ) , as opposed to the Bayes-optimal f∗ ( x ) = argmaxy p ( y|x ) . This claim— that overparameterized models do not converge to Bayes-optimal classifiers— is unique to our work as far as we know , and highlights the broad implications of our results . Locality and Manifold Learning . Our intuition for the behaviors in this work is that they arise due to some form of “ locality ” of the trained classifiers , in an appropriate embedding space . For example , the behavior observed in Experiment 1 would be consistent with that of a 1-Nearest-Neighbor classifier in a embedding that separates the CIFAR-10 classes well . This intuition that classifiers learn good embeddings is present in various forms in the literature , for example : the so-called called “ manifold hypothesis , ” that natural data lie on a low-dimensional manifold ( Narayanan & Mitter , 2010 ; Sharma & Kaplan , 2020 ) , as well as works on local stiffness of the loss landscape ( Fort et al. , 2019 ) , and works showing that overparameterized neural networks can learn hidden low-dimensional structure in high-dimensional settings ( Gerace et al. , 2020 ; Bach , 2017 ; Chizat & Bach , 2020 ) . It is open to more formally understand connections between our work and the above . Uncertainty Estimation . Since the appearance of the current work on arXiv , it has been directly built on by other authors . The work of Jiang et al . ( 2021 ) investigates our conjectures further , and extends them to develop a method for out-of-distribution uncertainty estimation . This highlights the fundamental nature and importance of our results , since they have already been used in a practical application . A full discussion of related works is in Appendix A . | The paper reports an interesting phenomenon in deep over-parameterized networks which train to zero train error (known as interpolating networks). Not only does the accuracy on test mimic the accuracy on train, but the accuracy measured on certain subsets of the train set also matches that on the corresponding subsets in test set. The paper coins the term distributional generalization for this phenomenon. The paper then proceeds to perform a number of experiments across a bunch of image datasets to exhibit the generality of this phenomenon. | SP:0cca38c3b824ab3e789e47aff6e5c9fc67b1fb1c |
VUT: Versatile UI Transformer for Multimodal Multi-Task User Interface Modeling | 1 INTRODUCTION . Modern graphical user interfaces specifically touchscreen mobile UIs enable a rich problem space for modeling where the input is inherently multimodal , which consists of several distinct types of data . A user interface screen exists in both a visual form , i.e. , a screenshot , and a structural representation , i.e. , a tree-like view hierarchy . Based on graphical user interfaces , there is a wide spectrum of modeling tasks that either directly enhance user experiences or advance the development of intelligent user interfaces . For example , previous work developed models and datasets for grounding a language command into an executable UI action ( Li et al. , 2020a ) , generating language description for accessibility on mobile devices ( Li et al. , 2020b ; Wang et al. , 2021 ) , and understanding the usability of user interfaces ( Swearngin & Li , 2019 ) or identifying the objects on the screen ( Zhang et al. , 2021 ) . Previous work has also started learning effective representation of user interface screens ( He et al. , 2020 ; Li et al. , 2021a ) , which can potentially benefit downstream tasks . Although these previous works have made progress in addressing individual problems , it is important to investigate the feasibility of learning all these tasks with a single model . In addition to achieving a scientific understanding of how these UI tasks are related , it is extremely valuable to obtain such a multi-task model , which can potentially reduce the number of models that need to be developed and deployed . This is crucial for mobile devices that have limited computing resources . In this work , we propose VUT—Versatile UI Transformer , which handles three types of data : images , structures ( view hierarchies ) and language , and simultaneously performs five unique tasks that are representative in the UI modeling literature , including UI object detection , natural language command grounding , widget captioning , screen summarization and UI tappability prediction . A major challenge we need to address is how to unify these distinct tasks as well as their heterogeneous datasets such that they can be learned by a single model . To this end , we devise a general formulation for UI modeling tasks based on five inherent types of information that define a task . We also aim to design a compact model architecture such that it remains stable for addressing a diverse and potentially growing set of tasks , for which we make each model component multi-purpose . Specifically , VUT comprises two Transformer architectures ( Figure 1 ) : the Image-Structure model , and the Question-Answer model . The Image-Structure model encodes the entire screenshot of a UI along its view hierarchy tree , with early fusion of the two modalities , which is guided by a focus map when a given object is inquired . In addition to being the UI encoder , the Image-Structure model predicts UI objects when the view hierarchy is absent on the input , which achieves the UI object detection task . The Question-Answer model encodes a question while attending to the UI encodings from the Image-Structure model . It decodes a text answer when the task response is language , e.g. , widget captioning ( Li et al. , 2020c ) or screen summarization ( Wang et al. , 2021 ) . For the grounding task whose output is an object reference , the Question-Answer model serves as the question encoder and its hidden state is used to locate UI elements to be acted upon . We highlight the relation of VUT with previous works in Table 1 , and discuss their differences further in the following section . We experiment with our model on 5 datasets , and compare the accuracy of VUT when it is trained alone for each task and jointly with multiple tasks . Our experiments show that VUT is able to perform all the five tasks simultaneously and achieve the performance on par with or surpass that when each task is learned alone . The main contributions of our work are as follows . • We formulate multi-modal multi-task learning for a new domain—graphical user interfaces— with one model to accomplish a wide range of tasks for enhancing mobile user experiences . • We design VUT based on a two-tower Transformer architecture , one for handling imagestructure and the other for language data , where each Transformer is multi-purpose by both encoding and decoding its own modality , with cross-tower attention . • We experiment with VUT on 5 distinct UI tasks , and thoroughly investigated the effect of these tasks when learned alone or jointly with ablations and analysis , which show the feasibility for achieving diverse UI tasks using a single model , which offers the value for reducing the number of models and storage footprints needed for deployment ( Appendix E ) . 2 RELATED WORK . Extensive work has been conducted in multi-modal modeling with vision and languages ( Li et al. , 2019 ; Lu et al. , 2019 ; Hu & Singh , 2021 ; Lu et al. , 2020 ; Tan & Bansal , 2019 ; Kim et al. , 2021 ; Zhou et al. , 2020 ; Gupta et al. , 2021 ) . Existing works differ in the form of input they consume and the objectives of modeling . One category of work focuses on pretraining to learn an effective cross-modality representation for downstream tasks and the other directly learns multiple tasks end-to-end ( Table 1 ) . VUT belongs to the latter . In terms of the forms of multimodal data these models consume , most models handle image and text input . However , a unique form of data in UI modeling is the structure input of view hierarchies , which only VUT , UIBert and ActionBert use . Many existing works feed object regions , instead of the entire image to the model , which requires a pretrained object detection model ( Lu et al. , 2019 ; Bai et al. , 2021 ; He et al. , 2020 ; Zhou et al. , 2020 ; Tan & Bansal , 2019 ; Lu et al. , 2020 ) or address tasks only regarding the entire image ( Kim et al. , 2021 ; Hu & Singh , 2021 ) . Although ActionBert ( He et al. , 2020 ) and UIBert ( Bai et al. , 2021 ) also address the UI domain , they are targeted for representation learning , and do not support multiple tasks simultaneously . As a result , they do not deal with language input of task descriptions . Their text input is those scraped from the UI screen , e.g. , using OCR . In addition , these models require predetermined object regions similar to many BERT-based multi-modal models . In contrast , object detection is one of the tasks that VUT addresses . In terms of modeling techniques , we designed a novel Transformer architecture for multi-task modeling of the UI domain , based on building blocks previously proposed for natural images and language , e.g. , ( Hu et al. , 2020 ; Lu et al. , 2020 ) . the work that is closely related to ours is GPVI ( Gupta et al. , 2021 ) , which uses DETR ( Carion et al. , 2020 ) for object detection , and ViBERT ( Lu et al. , 2019 ) for multimodal modeling . In addition to the obvious deviation our work , e.g. , VUT uses structure input but GPV-I does not , there are several important architecture differences . While GPV-I directly embeds DETR , an encoder-decoder model , into its architecture , VUT uses a single tower design where both the image and object queries are fed to the same Transformer encoder . This design choice is motivated by our goal to achieve a compact architecture , which the Image-Structure model serves both image-structure encoding and object detection ( when the structure input is absent in the input ) . As shown in our experiment , the single tower architecture of VUT ’ s Image-Structure model showed clear advantage over the encoder-decoder architecture in DETR for the UI object detection task . To address the unique domin of UI tasks , we also introduce focus map to guide the model towards the object being inquired . VUT ’ s question-answer Transformer is designed based on existing auto-regressive multi-task language models ( Raffel et al. , 2019 ; Brown et al. , 2020 ) where a question or a command is fed to the model as a prefix , and the responses are decoded token by token . One difference is that for the language command grounding task , instead of generating a language response , the last hidden state of the language model is used , as a question encoding , to retrieve a UI object on the screen . 3 PROBLEM FORMULATION . A graphical user interface contains a collection of UI elements for fulfilling a coherent set of tasks . There are often five types of data involved to formulate a UI task : < S , V , T , Q , A > ( Figure 1 ) . S is the screenshot image that captures the visual appearance of the UI screen . V is the view hierarchy tree that represents the underlying structure of the screen . T is the target object on the screen to be inquired . Q is the natural language description of the task , which can be an open-ended question such as `` What is the caption of the element ? `` , a yes-or-no question such as `` Does the object look clickable ? '' or a command such as `` Click on the Next button. '' . See the full list of Q used in our experiments in Appendix A . Finally , A is the natural language answer to the question Q when the form of the response for the task is supposed to be natural language . Depending on each task setup , these data types appear as either input or output . We elaborate on the formation of each task here , and use F to denote the function for achieving each task . 3.1 UI OBJECT DETECTION . Given the screenshot image , S , the task is to detect each UI element on the screen , such as Text Field , Toggle Button , or Image View . This task is similar to the typical object detection task in natural images ( Carion et al. , 2020 ) or recent UI object detection work ( Zhang et al. , 2021 ) . However , our task is more challenging in that it needs to detect different types of container objects , which determine how UI objects are visually structured of the screen , such as Linear Layout , Frame Layout or List . In total there are 21 types of leaf or non-leaf objects in a view hierarchy . See Appendix D for the full list of objects we detect . UI object detection is important for improving accessibility and enabling other intelligent features such as UI adaptation when view hierarchy is not available . As a screen understanding task , UI object detection is beneficial to other UI modeling tasks as we will show in our experiments . The task is formulated as the follow ( Equation 1 ) . V = F ( S , V∅ , T∅ , Q∅ ) ( 1 ) Note that this task is achieved solely based on the single-tower Image-Structure Transformer ( Figure 1 ) and does not rely on the question-answer model . V∅ , T∅ and Q∅ represent each type of data masked out or missing in the input . | The paper proposes an architecture for graphical user interfaces which involve multi-modal inputs (UI screenshots, Hierarchy structures, Natural Language) and multi-task learning (UI Object Detection, Widget Captioning, Screen summarization, Language grounding, and Tappability).The proposed architecture consists of seperate transformer blocks to encode image and text modalities. The two transformer blocks attends to each other to produce multi-modal outputs, which is then used for downstream tasks. The authors that with the proposed architecture, training on all tasks simultaneously is better than training on individual task alone. | SP:90c1c51f85a7f24748017abb66dcaf2d0c2f6c3e |
VUT: Versatile UI Transformer for Multimodal Multi-Task User Interface Modeling | 1 INTRODUCTION . Modern graphical user interfaces specifically touchscreen mobile UIs enable a rich problem space for modeling where the input is inherently multimodal , which consists of several distinct types of data . A user interface screen exists in both a visual form , i.e. , a screenshot , and a structural representation , i.e. , a tree-like view hierarchy . Based on graphical user interfaces , there is a wide spectrum of modeling tasks that either directly enhance user experiences or advance the development of intelligent user interfaces . For example , previous work developed models and datasets for grounding a language command into an executable UI action ( Li et al. , 2020a ) , generating language description for accessibility on mobile devices ( Li et al. , 2020b ; Wang et al. , 2021 ) , and understanding the usability of user interfaces ( Swearngin & Li , 2019 ) or identifying the objects on the screen ( Zhang et al. , 2021 ) . Previous work has also started learning effective representation of user interface screens ( He et al. , 2020 ; Li et al. , 2021a ) , which can potentially benefit downstream tasks . Although these previous works have made progress in addressing individual problems , it is important to investigate the feasibility of learning all these tasks with a single model . In addition to achieving a scientific understanding of how these UI tasks are related , it is extremely valuable to obtain such a multi-task model , which can potentially reduce the number of models that need to be developed and deployed . This is crucial for mobile devices that have limited computing resources . In this work , we propose VUT—Versatile UI Transformer , which handles three types of data : images , structures ( view hierarchies ) and language , and simultaneously performs five unique tasks that are representative in the UI modeling literature , including UI object detection , natural language command grounding , widget captioning , screen summarization and UI tappability prediction . A major challenge we need to address is how to unify these distinct tasks as well as their heterogeneous datasets such that they can be learned by a single model . To this end , we devise a general formulation for UI modeling tasks based on five inherent types of information that define a task . We also aim to design a compact model architecture such that it remains stable for addressing a diverse and potentially growing set of tasks , for which we make each model component multi-purpose . Specifically , VUT comprises two Transformer architectures ( Figure 1 ) : the Image-Structure model , and the Question-Answer model . The Image-Structure model encodes the entire screenshot of a UI along its view hierarchy tree , with early fusion of the two modalities , which is guided by a focus map when a given object is inquired . In addition to being the UI encoder , the Image-Structure model predicts UI objects when the view hierarchy is absent on the input , which achieves the UI object detection task . The Question-Answer model encodes a question while attending to the UI encodings from the Image-Structure model . It decodes a text answer when the task response is language , e.g. , widget captioning ( Li et al. , 2020c ) or screen summarization ( Wang et al. , 2021 ) . For the grounding task whose output is an object reference , the Question-Answer model serves as the question encoder and its hidden state is used to locate UI elements to be acted upon . We highlight the relation of VUT with previous works in Table 1 , and discuss their differences further in the following section . We experiment with our model on 5 datasets , and compare the accuracy of VUT when it is trained alone for each task and jointly with multiple tasks . Our experiments show that VUT is able to perform all the five tasks simultaneously and achieve the performance on par with or surpass that when each task is learned alone . The main contributions of our work are as follows . • We formulate multi-modal multi-task learning for a new domain—graphical user interfaces— with one model to accomplish a wide range of tasks for enhancing mobile user experiences . • We design VUT based on a two-tower Transformer architecture , one for handling imagestructure and the other for language data , where each Transformer is multi-purpose by both encoding and decoding its own modality , with cross-tower attention . • We experiment with VUT on 5 distinct UI tasks , and thoroughly investigated the effect of these tasks when learned alone or jointly with ablations and analysis , which show the feasibility for achieving diverse UI tasks using a single model , which offers the value for reducing the number of models and storage footprints needed for deployment ( Appendix E ) . 2 RELATED WORK . Extensive work has been conducted in multi-modal modeling with vision and languages ( Li et al. , 2019 ; Lu et al. , 2019 ; Hu & Singh , 2021 ; Lu et al. , 2020 ; Tan & Bansal , 2019 ; Kim et al. , 2021 ; Zhou et al. , 2020 ; Gupta et al. , 2021 ) . Existing works differ in the form of input they consume and the objectives of modeling . One category of work focuses on pretraining to learn an effective cross-modality representation for downstream tasks and the other directly learns multiple tasks end-to-end ( Table 1 ) . VUT belongs to the latter . In terms of the forms of multimodal data these models consume , most models handle image and text input . However , a unique form of data in UI modeling is the structure input of view hierarchies , which only VUT , UIBert and ActionBert use . Many existing works feed object regions , instead of the entire image to the model , which requires a pretrained object detection model ( Lu et al. , 2019 ; Bai et al. , 2021 ; He et al. , 2020 ; Zhou et al. , 2020 ; Tan & Bansal , 2019 ; Lu et al. , 2020 ) or address tasks only regarding the entire image ( Kim et al. , 2021 ; Hu & Singh , 2021 ) . Although ActionBert ( He et al. , 2020 ) and UIBert ( Bai et al. , 2021 ) also address the UI domain , they are targeted for representation learning , and do not support multiple tasks simultaneously . As a result , they do not deal with language input of task descriptions . Their text input is those scraped from the UI screen , e.g. , using OCR . In addition , these models require predetermined object regions similar to many BERT-based multi-modal models . In contrast , object detection is one of the tasks that VUT addresses . In terms of modeling techniques , we designed a novel Transformer architecture for multi-task modeling of the UI domain , based on building blocks previously proposed for natural images and language , e.g. , ( Hu et al. , 2020 ; Lu et al. , 2020 ) . the work that is closely related to ours is GPVI ( Gupta et al. , 2021 ) , which uses DETR ( Carion et al. , 2020 ) for object detection , and ViBERT ( Lu et al. , 2019 ) for multimodal modeling . In addition to the obvious deviation our work , e.g. , VUT uses structure input but GPV-I does not , there are several important architecture differences . While GPV-I directly embeds DETR , an encoder-decoder model , into its architecture , VUT uses a single tower design where both the image and object queries are fed to the same Transformer encoder . This design choice is motivated by our goal to achieve a compact architecture , which the Image-Structure model serves both image-structure encoding and object detection ( when the structure input is absent in the input ) . As shown in our experiment , the single tower architecture of VUT ’ s Image-Structure model showed clear advantage over the encoder-decoder architecture in DETR for the UI object detection task . To address the unique domin of UI tasks , we also introduce focus map to guide the model towards the object being inquired . VUT ’ s question-answer Transformer is designed based on existing auto-regressive multi-task language models ( Raffel et al. , 2019 ; Brown et al. , 2020 ) where a question or a command is fed to the model as a prefix , and the responses are decoded token by token . One difference is that for the language command grounding task , instead of generating a language response , the last hidden state of the language model is used , as a question encoding , to retrieve a UI object on the screen . 3 PROBLEM FORMULATION . A graphical user interface contains a collection of UI elements for fulfilling a coherent set of tasks . There are often five types of data involved to formulate a UI task : < S , V , T , Q , A > ( Figure 1 ) . S is the screenshot image that captures the visual appearance of the UI screen . V is the view hierarchy tree that represents the underlying structure of the screen . T is the target object on the screen to be inquired . Q is the natural language description of the task , which can be an open-ended question such as `` What is the caption of the element ? `` , a yes-or-no question such as `` Does the object look clickable ? '' or a command such as `` Click on the Next button. '' . See the full list of Q used in our experiments in Appendix A . Finally , A is the natural language answer to the question Q when the form of the response for the task is supposed to be natural language . Depending on each task setup , these data types appear as either input or output . We elaborate on the formation of each task here , and use F to denote the function for achieving each task . 3.1 UI OBJECT DETECTION . Given the screenshot image , S , the task is to detect each UI element on the screen , such as Text Field , Toggle Button , or Image View . This task is similar to the typical object detection task in natural images ( Carion et al. , 2020 ) or recent UI object detection work ( Zhang et al. , 2021 ) . However , our task is more challenging in that it needs to detect different types of container objects , which determine how UI objects are visually structured of the screen , such as Linear Layout , Frame Layout or List . In total there are 21 types of leaf or non-leaf objects in a view hierarchy . See Appendix D for the full list of objects we detect . UI object detection is important for improving accessibility and enabling other intelligent features such as UI adaptation when view hierarchy is not available . As a screen understanding task , UI object detection is beneficial to other UI modeling tasks as we will show in our experiments . The task is formulated as the follow ( Equation 1 ) . V = F ( S , V∅ , T∅ , Q∅ ) ( 1 ) Note that this task is achieved solely based on the single-tower Image-Structure Transformer ( Figure 1 ) and does not rely on the question-answer model . V∅ , T∅ and Q∅ represent each type of data masked out or missing in the input . | A multi-modal multi-task model is proposed for UI understanding. It has an Image-Structure Transformer and Question-Answer Transformer to encode/decode image, structure and language information. The method is evaluated thoroughly on 5 different data sets and achieves good results. | SP:90c1c51f85a7f24748017abb66dcaf2d0c2f6c3e |
VUT: Versatile UI Transformer for Multimodal Multi-Task User Interface Modeling | 1 INTRODUCTION . Modern graphical user interfaces specifically touchscreen mobile UIs enable a rich problem space for modeling where the input is inherently multimodal , which consists of several distinct types of data . A user interface screen exists in both a visual form , i.e. , a screenshot , and a structural representation , i.e. , a tree-like view hierarchy . Based on graphical user interfaces , there is a wide spectrum of modeling tasks that either directly enhance user experiences or advance the development of intelligent user interfaces . For example , previous work developed models and datasets for grounding a language command into an executable UI action ( Li et al. , 2020a ) , generating language description for accessibility on mobile devices ( Li et al. , 2020b ; Wang et al. , 2021 ) , and understanding the usability of user interfaces ( Swearngin & Li , 2019 ) or identifying the objects on the screen ( Zhang et al. , 2021 ) . Previous work has also started learning effective representation of user interface screens ( He et al. , 2020 ; Li et al. , 2021a ) , which can potentially benefit downstream tasks . Although these previous works have made progress in addressing individual problems , it is important to investigate the feasibility of learning all these tasks with a single model . In addition to achieving a scientific understanding of how these UI tasks are related , it is extremely valuable to obtain such a multi-task model , which can potentially reduce the number of models that need to be developed and deployed . This is crucial for mobile devices that have limited computing resources . In this work , we propose VUT—Versatile UI Transformer , which handles three types of data : images , structures ( view hierarchies ) and language , and simultaneously performs five unique tasks that are representative in the UI modeling literature , including UI object detection , natural language command grounding , widget captioning , screen summarization and UI tappability prediction . A major challenge we need to address is how to unify these distinct tasks as well as their heterogeneous datasets such that they can be learned by a single model . To this end , we devise a general formulation for UI modeling tasks based on five inherent types of information that define a task . We also aim to design a compact model architecture such that it remains stable for addressing a diverse and potentially growing set of tasks , for which we make each model component multi-purpose . Specifically , VUT comprises two Transformer architectures ( Figure 1 ) : the Image-Structure model , and the Question-Answer model . The Image-Structure model encodes the entire screenshot of a UI along its view hierarchy tree , with early fusion of the two modalities , which is guided by a focus map when a given object is inquired . In addition to being the UI encoder , the Image-Structure model predicts UI objects when the view hierarchy is absent on the input , which achieves the UI object detection task . The Question-Answer model encodes a question while attending to the UI encodings from the Image-Structure model . It decodes a text answer when the task response is language , e.g. , widget captioning ( Li et al. , 2020c ) or screen summarization ( Wang et al. , 2021 ) . For the grounding task whose output is an object reference , the Question-Answer model serves as the question encoder and its hidden state is used to locate UI elements to be acted upon . We highlight the relation of VUT with previous works in Table 1 , and discuss their differences further in the following section . We experiment with our model on 5 datasets , and compare the accuracy of VUT when it is trained alone for each task and jointly with multiple tasks . Our experiments show that VUT is able to perform all the five tasks simultaneously and achieve the performance on par with or surpass that when each task is learned alone . The main contributions of our work are as follows . • We formulate multi-modal multi-task learning for a new domain—graphical user interfaces— with one model to accomplish a wide range of tasks for enhancing mobile user experiences . • We design VUT based on a two-tower Transformer architecture , one for handling imagestructure and the other for language data , where each Transformer is multi-purpose by both encoding and decoding its own modality , with cross-tower attention . • We experiment with VUT on 5 distinct UI tasks , and thoroughly investigated the effect of these tasks when learned alone or jointly with ablations and analysis , which show the feasibility for achieving diverse UI tasks using a single model , which offers the value for reducing the number of models and storage footprints needed for deployment ( Appendix E ) . 2 RELATED WORK . Extensive work has been conducted in multi-modal modeling with vision and languages ( Li et al. , 2019 ; Lu et al. , 2019 ; Hu & Singh , 2021 ; Lu et al. , 2020 ; Tan & Bansal , 2019 ; Kim et al. , 2021 ; Zhou et al. , 2020 ; Gupta et al. , 2021 ) . Existing works differ in the form of input they consume and the objectives of modeling . One category of work focuses on pretraining to learn an effective cross-modality representation for downstream tasks and the other directly learns multiple tasks end-to-end ( Table 1 ) . VUT belongs to the latter . In terms of the forms of multimodal data these models consume , most models handle image and text input . However , a unique form of data in UI modeling is the structure input of view hierarchies , which only VUT , UIBert and ActionBert use . Many existing works feed object regions , instead of the entire image to the model , which requires a pretrained object detection model ( Lu et al. , 2019 ; Bai et al. , 2021 ; He et al. , 2020 ; Zhou et al. , 2020 ; Tan & Bansal , 2019 ; Lu et al. , 2020 ) or address tasks only regarding the entire image ( Kim et al. , 2021 ; Hu & Singh , 2021 ) . Although ActionBert ( He et al. , 2020 ) and UIBert ( Bai et al. , 2021 ) also address the UI domain , they are targeted for representation learning , and do not support multiple tasks simultaneously . As a result , they do not deal with language input of task descriptions . Their text input is those scraped from the UI screen , e.g. , using OCR . In addition , these models require predetermined object regions similar to many BERT-based multi-modal models . In contrast , object detection is one of the tasks that VUT addresses . In terms of modeling techniques , we designed a novel Transformer architecture for multi-task modeling of the UI domain , based on building blocks previously proposed for natural images and language , e.g. , ( Hu et al. , 2020 ; Lu et al. , 2020 ) . the work that is closely related to ours is GPVI ( Gupta et al. , 2021 ) , which uses DETR ( Carion et al. , 2020 ) for object detection , and ViBERT ( Lu et al. , 2019 ) for multimodal modeling . In addition to the obvious deviation our work , e.g. , VUT uses structure input but GPV-I does not , there are several important architecture differences . While GPV-I directly embeds DETR , an encoder-decoder model , into its architecture , VUT uses a single tower design where both the image and object queries are fed to the same Transformer encoder . This design choice is motivated by our goal to achieve a compact architecture , which the Image-Structure model serves both image-structure encoding and object detection ( when the structure input is absent in the input ) . As shown in our experiment , the single tower architecture of VUT ’ s Image-Structure model showed clear advantage over the encoder-decoder architecture in DETR for the UI object detection task . To address the unique domin of UI tasks , we also introduce focus map to guide the model towards the object being inquired . VUT ’ s question-answer Transformer is designed based on existing auto-regressive multi-task language models ( Raffel et al. , 2019 ; Brown et al. , 2020 ) where a question or a command is fed to the model as a prefix , and the responses are decoded token by token . One difference is that for the language command grounding task , instead of generating a language response , the last hidden state of the language model is used , as a question encoding , to retrieve a UI object on the screen . 3 PROBLEM FORMULATION . A graphical user interface contains a collection of UI elements for fulfilling a coherent set of tasks . There are often five types of data involved to formulate a UI task : < S , V , T , Q , A > ( Figure 1 ) . S is the screenshot image that captures the visual appearance of the UI screen . V is the view hierarchy tree that represents the underlying structure of the screen . T is the target object on the screen to be inquired . Q is the natural language description of the task , which can be an open-ended question such as `` What is the caption of the element ? `` , a yes-or-no question such as `` Does the object look clickable ? '' or a command such as `` Click on the Next button. '' . See the full list of Q used in our experiments in Appendix A . Finally , A is the natural language answer to the question Q when the form of the response for the task is supposed to be natural language . Depending on each task setup , these data types appear as either input or output . We elaborate on the formation of each task here , and use F to denote the function for achieving each task . 3.1 UI OBJECT DETECTION . Given the screenshot image , S , the task is to detect each UI element on the screen , such as Text Field , Toggle Button , or Image View . This task is similar to the typical object detection task in natural images ( Carion et al. , 2020 ) or recent UI object detection work ( Zhang et al. , 2021 ) . However , our task is more challenging in that it needs to detect different types of container objects , which determine how UI objects are visually structured of the screen , such as Linear Layout , Frame Layout or List . In total there are 21 types of leaf or non-leaf objects in a view hierarchy . See Appendix D for the full list of objects we detect . UI object detection is important for improving accessibility and enabling other intelligent features such as UI adaptation when view hierarchy is not available . As a screen understanding task , UI object detection is beneficial to other UI modeling tasks as we will show in our experiments . The task is formulated as the follow ( Equation 1 ) . V = F ( S , V∅ , T∅ , Q∅ ) ( 1 ) Note that this task is achieved solely based on the single-tower Image-Structure Transformer ( Figure 1 ) and does not rely on the question-answer model . V∅ , T∅ and Q∅ represent each type of data masked out or missing in the input . | This paper presents a multi-modal Transformer for multi-task modeling of user interfaces. It is potentially worthy to investigate how to incorporate multiple UI related tasks into a model. The authors hence design a so-called Versatile UI Transformer model which involves three modals inputs to handle five unique tasks. | SP:90c1c51f85a7f24748017abb66dcaf2d0c2f6c3e |
How Faithful is your Synthetic Data? Sample-level Metrics for Evaluating and Auditing Generative Models | Devising domain- and model-agnostic evaluation metrics for generative models is an important and as yet unresolved problem . Most existing metrics , which were tailored solely to the image synthesis setup , exhibit a limited capacity for diagnosing the modes of failure of generative models across broader application domains . In this paper , we introduce a 3-dimensional metric , ( α-Precision , β-Recall , Authenticity ) , that characterizes the fidelity , diversity and generalization performance of any generative model in a domain-agnostic fashion . Our metric unifies statistical divergence measures with precision-recall analysis , enabling sample-level and distribution-level diagnoses of model fidelity and diversity . We introduce generalization as an additional dimension for model performance that quantifies the extent to which a model copies training data—a crucial performance indicator when modeling sensitive data with requirements on privacy . The three metric components correspond to ( interpretable ) probabilistic quantities , and are estimated via sample-level binary classification . The sample-level nature of our metric inspires a novel use case which we call model auditing , wherein we judge the quality of individual samples generated by a ( black-box ) model , discarding low-quality samples and hence improving the overall model performance in a post-hoc manner . 1 INTRODUCTION . Intuitively , it would seem that evaluating the likelihood function of a generative model is all it takes to assess its performance . As it turns out , the problem of evaluating generative models is far more complicated . This is not only because state-of-the-art models , such as Variational Autoencoders ( VAE ) ( Kingma & Welling ( 2013 ) ) and Generative Adversarial Networks ( GANs ) ( Goodfellow et al . ( 2014 ) ) , do not possess tractable likelihood functions , but also because the likelihood score itself is a flawed measure of performance—it scales badly in high dimensions , and it obscures distinct modes of model failure into a single uninterpretable score ( Theis et al . ( 2015 ) ) . Absent domain-agnostic metrics , earlier work focused on crafting domain-specific scores , e.g. , Inception score ( Salimans et al . ( 2016 ) ) , with an exclusive emphasis on image data ( Lucic et al . ( 2018 ) ) . In this paper , we introduce an alternative approach to evaluating generative models , where instead of assessing the generative distribution by looking at all synthetic samples collectively to compute likelihood or divergence , we classify each sample individually as being of high or low quality . In this way , our metric comprises interpretable probabilistic quantities—resembling those used to evaluate discriminative models ( e.g. , AUC-ROC ) —which describe the rate by which a model makes errors . When averaged over all samples , our sample-level scores reflect discrepancy between real and generative distributions in a way similar to statistical divergence measures ( e.g. , KL divergence , Fréchet distance ( Heusel et al . ( 2017 ) ) , or maximum mean discrepancy ( Sutherland et al . ( 2016 ) ) . In this sense , our metric enables diagnosing model performance on both the sample and distribution levels . Our metric represents the performance of a generative model as a point in a 3-dimensional space— each dimension corresponds to a distinct quality of the model . These qualities are : Fidelity , Diversity and Generalization . Fidelity measures the quality of a model ’ s synthetic samples , and Diversity is Figure 1 : Pictorial depiction for the proposed evaluation metrics . The blue and red spheres correspond to the α- and β-supports of real and generative distributions , respectively . Blue and red points correspond to real and synthetic data . ( a ) Synthetic data falling outside the blue sphere will look unrealistic or noisy . ( b ) Overfitted models can generate ostensibly high-quality data samples that are “ unauthentic ” because they are copied from the training data . ( c ) High-quality data samples should reside inside the blue sphere . ( d ) Outliers do not count in the β-Recall metric . ( Here , α=β=0.9 , α-Precision=8/9 , βRecall = 4/9 , and Authenticity = 9/10 . ) Support of real distribution Support of synthetic distribution Real Synthetic Noisy synthetic sample Unauthentic high-quality sample Authentic high-quality sample Outlier ( a ) ( b ) ( c ) ( d ) the extent to which these samples cover the full variability of real samples , whereas Generalization quantifies the extent to which a model overfits ( copies ) training data . How do we quantify the 3 dimensions of performance ? We build on the precision-recall analysis framework proposed in ( Sajjadi et al . ( 2018 ) ) , and introduce the α-Precision and β-Recall metrics to quantify model Fidelity and Diversity , respectively . Both metrics assume that a fraction 1 − α ( or 1 − β ) of the real ( and synthetic ) data are “ outliers ” , and α ( or β ) are “ typical ” . α-Precision is the fraction of synthetic samples that resemble the “ most typical ” α real samples , whereas β-Recall is the fraction of real samples covered by the most typical β synthetic samples . α-Precision and β-Recall are evaluated for all α , β ∈ [ 0 , 1 ] , providing entire precision and recall curves instead of single numbers . To compute both metrics , we embed the ( real and synthetic ) data into hyperspheres with most samples concentrated around the centers , i.e. , the real and generative distributions ( Pr and Pg ) has spherical-shaped supports . In this transformed feature space , typical samples would be located near the centers of the spheres , whereas outliers would be closer to the boundaries . To quantify Generalization , we introduce the Authenticity metric , which reflects the likelihood of a synthetic sample being copied from training data . We derive the Authenticity metric from a hypothesis test for data copying based on the observed proximity of synthetic samples to real ones in the embedded feature space . A pictorial illustration for all metrics is shown in Figure 1 . How is our metric different ? If one think of standard precision and recall metrics as “ hard ” binary classifiers of real and synthetic samples , our α-Precision and β-Recall can be thought of as softboundary classifiers that do not only compare the supports of Pr and Pg , but also assesses whether both distributions are calibrated . Precision and recall metrics are special cases of α-Precision and β-Recall for α = β = 1 . As we show later , our new metric definitions solve many of the drawbacks of standard precision-recall analysis , such as lack of robustness to outliers and failure to detect distributional mismatches ( Naeem et al . ( 2020 ) ) . They also enable detailed diagnostics of different types of model failure , such as mode collapse and mode invention . Moreover , optimal values of our metrics are achieved only when Pr and Pg are identical , thereby eliminating the need to augment the evaluation procedure with additional measures of statistical divergence ( e.g. , KL divergence ) . Previous works relied on pre-trained embeddings ( using ImageNet feature extractors ( Deng et al . ( 2009 ) ) ) . In this work , we propose feature embeddings that are model- and domain-agnostic , and are tailored to our metric definitions and data set at hand . Our proposed feature embedding step can be completely bespoke to raw data , or augmented with pre-trained embeddings . This enables our metric to remain operable in application domains where no pre-trained representations exist . Overfitting is a crucial mode of failure of generative models , especially when modeling sensitive data ( e.g. , clinical data ) for which data copying may violate privacy requirements ( Yoon et al . ( 2020 ) ) , but it has been overlooked in previous works which focused exclusively on quantifying the FidelityDiversity characterization ( Brock et al . ( 2018 ) ) . As we show in our experiments ( Section 5 ) , because our metric accounts for Generalization , it can provide a fuller picture of a generative model ’ s performance . Precisely , we show that some of the celebrated generative models score highly for Fidelity and Diversity simply because they memorize real samples , rendering them inappropriate for privacysensitive applications . A comprehensive survey of prior work is provided in the Appendix . Model auditing as a novel use case . The sample-level nature of our metrics inspires the new use case of model auditing , wherein we judge individual synthetic samples by their quality , and reject samples that have low Fidelity or are unauthentic . In Section 5 , we show that model audits can improve the outputs of a black-box model in a post-hoc fashion without any modifications to the model itself , and demonstrate this use case in synthesizing clinical data for COVID-19 patients . 2 EVALUATING AND AUDITING GENERATIVE MODELS . 2.1 PROBLEM SETUP . We denote real and generated data asXr ∼ Pr andXg ∼ Pg , respectively , whereXr , Xg ∈ X , with Pr and Pg being the real and generative distributions , and X being the input space . The generative distribution , Pg , is estimated using a generative model ( e.g. , GAN ) . The real and synthetic data sets are Dreal = { Xr , i } ni=1 and Dsynth = { Xg , j } mj=1 , where Xr , i ∼ Pr and Xg , j ∼ Pg . 2.2 WHAT MAKES A GOOD SYNTHETIC DATA SET ? . Our goal is to construct a metric E ( Dreal , Dsynth ) for the quality of Dsynth in order to ( i ) evaluate the performance of the underlying generative model Pg , and ( ii ) audit the model outputs by discarding ( individual ) “ low-quality ” samples , thereby improving the overall quality ofDsynth . In order for E to fulfill the evaluation and auditing tasks , it must satisfy the following desiderata : ( 1 ) it should be able to disentangle the different modes of failure of Pg through interpretable measures of performance , and ( 2 ) it should be sample-wise computable , i.e. , we should be able to tell if a given ( individual ) synthetic sample Xg ∼ Pg is of a low quality . Having outlined the desiderata for our sought-after evaluation metric , we now propose three qualities of synthetic data that the metric E should be able to quantify . Failure to fulfill any of these three qualities correspond to independent modes of failure of the model Pg . These qualities are : 1 . Fidelity—the generated samples resemble real samples from Pr . A high-fidelity synthetic data set should contain “ realistic ” samples , e.g . visually-realistic images . 2 . Diversity—the generated samples are diverse enough to cover the variability of real data , i.e. , a model should be able to generate a wide variety of good samples . 3 . Generalization—the generated samples should not be mere copies of the ( real ) samples in training data , i.e. , models that overfit to Dreal are not truly “ generative ” . In Section 3 , we propose a three-dimensional evaluation metric E that captures all of the qualities above . Our proposed metric can be succinctly described as follows : E , ( α-Precision︸ ︷︷ ︸ Fidelity , β-Recall︸ ︷︷ ︸ Diversity , Authenticity︸ ︷︷ ︸ Generalization ) . ( 1 ) The α-Precision and β-Recall metrics are generalizations of the conventional notions of precision and recall used in binary classification analysis ( Flach & Kull ( 2015 ) ) . Precision measures the rate by which the model synthesizes “ realistic-looking ” samples , whereas the recall measures the fraction of real samples that are covered by Pg . The authenticity score measures the fraction of synthetic samples that are invented by the model and not copied from the training data . | This paper targets to a 3-dim metric: alpha-Precision, beta-Recall, and Authenticity, that quantifies the fidelity, diversity, and generalization performance of a generative model. The proposed metric serves as soft-boundary classifiers between real and generated spherical-shaped supports, and improves the robustness against outliers and generation failure cases. This sample-wise metric can audit models by judging individual synthetic samples by their quality. | SP:e48d1c232568b92d53a89e035830e800b9970b8e |
How Faithful is your Synthetic Data? Sample-level Metrics for Evaluating and Auditing Generative Models | Devising domain- and model-agnostic evaluation metrics for generative models is an important and as yet unresolved problem . Most existing metrics , which were tailored solely to the image synthesis setup , exhibit a limited capacity for diagnosing the modes of failure of generative models across broader application domains . In this paper , we introduce a 3-dimensional metric , ( α-Precision , β-Recall , Authenticity ) , that characterizes the fidelity , diversity and generalization performance of any generative model in a domain-agnostic fashion . Our metric unifies statistical divergence measures with precision-recall analysis , enabling sample-level and distribution-level diagnoses of model fidelity and diversity . We introduce generalization as an additional dimension for model performance that quantifies the extent to which a model copies training data—a crucial performance indicator when modeling sensitive data with requirements on privacy . The three metric components correspond to ( interpretable ) probabilistic quantities , and are estimated via sample-level binary classification . The sample-level nature of our metric inspires a novel use case which we call model auditing , wherein we judge the quality of individual samples generated by a ( black-box ) model , discarding low-quality samples and hence improving the overall model performance in a post-hoc manner . 1 INTRODUCTION . Intuitively , it would seem that evaluating the likelihood function of a generative model is all it takes to assess its performance . As it turns out , the problem of evaluating generative models is far more complicated . This is not only because state-of-the-art models , such as Variational Autoencoders ( VAE ) ( Kingma & Welling ( 2013 ) ) and Generative Adversarial Networks ( GANs ) ( Goodfellow et al . ( 2014 ) ) , do not possess tractable likelihood functions , but also because the likelihood score itself is a flawed measure of performance—it scales badly in high dimensions , and it obscures distinct modes of model failure into a single uninterpretable score ( Theis et al . ( 2015 ) ) . Absent domain-agnostic metrics , earlier work focused on crafting domain-specific scores , e.g. , Inception score ( Salimans et al . ( 2016 ) ) , with an exclusive emphasis on image data ( Lucic et al . ( 2018 ) ) . In this paper , we introduce an alternative approach to evaluating generative models , where instead of assessing the generative distribution by looking at all synthetic samples collectively to compute likelihood or divergence , we classify each sample individually as being of high or low quality . In this way , our metric comprises interpretable probabilistic quantities—resembling those used to evaluate discriminative models ( e.g. , AUC-ROC ) —which describe the rate by which a model makes errors . When averaged over all samples , our sample-level scores reflect discrepancy between real and generative distributions in a way similar to statistical divergence measures ( e.g. , KL divergence , Fréchet distance ( Heusel et al . ( 2017 ) ) , or maximum mean discrepancy ( Sutherland et al . ( 2016 ) ) . In this sense , our metric enables diagnosing model performance on both the sample and distribution levels . Our metric represents the performance of a generative model as a point in a 3-dimensional space— each dimension corresponds to a distinct quality of the model . These qualities are : Fidelity , Diversity and Generalization . Fidelity measures the quality of a model ’ s synthetic samples , and Diversity is Figure 1 : Pictorial depiction for the proposed evaluation metrics . The blue and red spheres correspond to the α- and β-supports of real and generative distributions , respectively . Blue and red points correspond to real and synthetic data . ( a ) Synthetic data falling outside the blue sphere will look unrealistic or noisy . ( b ) Overfitted models can generate ostensibly high-quality data samples that are “ unauthentic ” because they are copied from the training data . ( c ) High-quality data samples should reside inside the blue sphere . ( d ) Outliers do not count in the β-Recall metric . ( Here , α=β=0.9 , α-Precision=8/9 , βRecall = 4/9 , and Authenticity = 9/10 . ) Support of real distribution Support of synthetic distribution Real Synthetic Noisy synthetic sample Unauthentic high-quality sample Authentic high-quality sample Outlier ( a ) ( b ) ( c ) ( d ) the extent to which these samples cover the full variability of real samples , whereas Generalization quantifies the extent to which a model overfits ( copies ) training data . How do we quantify the 3 dimensions of performance ? We build on the precision-recall analysis framework proposed in ( Sajjadi et al . ( 2018 ) ) , and introduce the α-Precision and β-Recall metrics to quantify model Fidelity and Diversity , respectively . Both metrics assume that a fraction 1 − α ( or 1 − β ) of the real ( and synthetic ) data are “ outliers ” , and α ( or β ) are “ typical ” . α-Precision is the fraction of synthetic samples that resemble the “ most typical ” α real samples , whereas β-Recall is the fraction of real samples covered by the most typical β synthetic samples . α-Precision and β-Recall are evaluated for all α , β ∈ [ 0 , 1 ] , providing entire precision and recall curves instead of single numbers . To compute both metrics , we embed the ( real and synthetic ) data into hyperspheres with most samples concentrated around the centers , i.e. , the real and generative distributions ( Pr and Pg ) has spherical-shaped supports . In this transformed feature space , typical samples would be located near the centers of the spheres , whereas outliers would be closer to the boundaries . To quantify Generalization , we introduce the Authenticity metric , which reflects the likelihood of a synthetic sample being copied from training data . We derive the Authenticity metric from a hypothesis test for data copying based on the observed proximity of synthetic samples to real ones in the embedded feature space . A pictorial illustration for all metrics is shown in Figure 1 . How is our metric different ? If one think of standard precision and recall metrics as “ hard ” binary classifiers of real and synthetic samples , our α-Precision and β-Recall can be thought of as softboundary classifiers that do not only compare the supports of Pr and Pg , but also assesses whether both distributions are calibrated . Precision and recall metrics are special cases of α-Precision and β-Recall for α = β = 1 . As we show later , our new metric definitions solve many of the drawbacks of standard precision-recall analysis , such as lack of robustness to outliers and failure to detect distributional mismatches ( Naeem et al . ( 2020 ) ) . They also enable detailed diagnostics of different types of model failure , such as mode collapse and mode invention . Moreover , optimal values of our metrics are achieved only when Pr and Pg are identical , thereby eliminating the need to augment the evaluation procedure with additional measures of statistical divergence ( e.g. , KL divergence ) . Previous works relied on pre-trained embeddings ( using ImageNet feature extractors ( Deng et al . ( 2009 ) ) ) . In this work , we propose feature embeddings that are model- and domain-agnostic , and are tailored to our metric definitions and data set at hand . Our proposed feature embedding step can be completely bespoke to raw data , or augmented with pre-trained embeddings . This enables our metric to remain operable in application domains where no pre-trained representations exist . Overfitting is a crucial mode of failure of generative models , especially when modeling sensitive data ( e.g. , clinical data ) for which data copying may violate privacy requirements ( Yoon et al . ( 2020 ) ) , but it has been overlooked in previous works which focused exclusively on quantifying the FidelityDiversity characterization ( Brock et al . ( 2018 ) ) . As we show in our experiments ( Section 5 ) , because our metric accounts for Generalization , it can provide a fuller picture of a generative model ’ s performance . Precisely , we show that some of the celebrated generative models score highly for Fidelity and Diversity simply because they memorize real samples , rendering them inappropriate for privacysensitive applications . A comprehensive survey of prior work is provided in the Appendix . Model auditing as a novel use case . The sample-level nature of our metrics inspires the new use case of model auditing , wherein we judge individual synthetic samples by their quality , and reject samples that have low Fidelity or are unauthentic . In Section 5 , we show that model audits can improve the outputs of a black-box model in a post-hoc fashion without any modifications to the model itself , and demonstrate this use case in synthesizing clinical data for COVID-19 patients . 2 EVALUATING AND AUDITING GENERATIVE MODELS . 2.1 PROBLEM SETUP . We denote real and generated data asXr ∼ Pr andXg ∼ Pg , respectively , whereXr , Xg ∈ X , with Pr and Pg being the real and generative distributions , and X being the input space . The generative distribution , Pg , is estimated using a generative model ( e.g. , GAN ) . The real and synthetic data sets are Dreal = { Xr , i } ni=1 and Dsynth = { Xg , j } mj=1 , where Xr , i ∼ Pr and Xg , j ∼ Pg . 2.2 WHAT MAKES A GOOD SYNTHETIC DATA SET ? . Our goal is to construct a metric E ( Dreal , Dsynth ) for the quality of Dsynth in order to ( i ) evaluate the performance of the underlying generative model Pg , and ( ii ) audit the model outputs by discarding ( individual ) “ low-quality ” samples , thereby improving the overall quality ofDsynth . In order for E to fulfill the evaluation and auditing tasks , it must satisfy the following desiderata : ( 1 ) it should be able to disentangle the different modes of failure of Pg through interpretable measures of performance , and ( 2 ) it should be sample-wise computable , i.e. , we should be able to tell if a given ( individual ) synthetic sample Xg ∼ Pg is of a low quality . Having outlined the desiderata for our sought-after evaluation metric , we now propose three qualities of synthetic data that the metric E should be able to quantify . Failure to fulfill any of these three qualities correspond to independent modes of failure of the model Pg . These qualities are : 1 . Fidelity—the generated samples resemble real samples from Pr . A high-fidelity synthetic data set should contain “ realistic ” samples , e.g . visually-realistic images . 2 . Diversity—the generated samples are diverse enough to cover the variability of real data , i.e. , a model should be able to generate a wide variety of good samples . 3 . Generalization—the generated samples should not be mere copies of the ( real ) samples in training data , i.e. , models that overfit to Dreal are not truly “ generative ” . In Section 3 , we propose a three-dimensional evaluation metric E that captures all of the qualities above . Our proposed metric can be succinctly described as follows : E , ( α-Precision︸ ︷︷ ︸ Fidelity , β-Recall︸ ︷︷ ︸ Diversity , Authenticity︸ ︷︷ ︸ Generalization ) . ( 1 ) The α-Precision and β-Recall metrics are generalizations of the conventional notions of precision and recall used in binary classification analysis ( Flach & Kull ( 2015 ) ) . Precision measures the rate by which the model synthesizes “ realistic-looking ” samples , whereas the recall measures the fraction of real samples that are covered by Pg . The authenticity score measures the fraction of synthetic samples that are invented by the model and not copied from the training data . | This work addressed an interesting problem and a very relevant one - how to audit generative models? The most popular metric to evaluate generative models is FID but it is extremely opaque and population based. This paper proposes new fidelity metrics that overcome both these challenges. While the definition of these metrics are well grounded, their measurement process itself - which is done via trained NN classifiers - is a bit unconvincing. The authors demonstrate that the proposed metric can recover the real ranking of generative models on a synthetic data generation task and can discover issues like mode collapse. | SP:e48d1c232568b92d53a89e035830e800b9970b8e |
How Faithful is your Synthetic Data? Sample-level Metrics for Evaluating and Auditing Generative Models | Devising domain- and model-agnostic evaluation metrics for generative models is an important and as yet unresolved problem . Most existing metrics , which were tailored solely to the image synthesis setup , exhibit a limited capacity for diagnosing the modes of failure of generative models across broader application domains . In this paper , we introduce a 3-dimensional metric , ( α-Precision , β-Recall , Authenticity ) , that characterizes the fidelity , diversity and generalization performance of any generative model in a domain-agnostic fashion . Our metric unifies statistical divergence measures with precision-recall analysis , enabling sample-level and distribution-level diagnoses of model fidelity and diversity . We introduce generalization as an additional dimension for model performance that quantifies the extent to which a model copies training data—a crucial performance indicator when modeling sensitive data with requirements on privacy . The three metric components correspond to ( interpretable ) probabilistic quantities , and are estimated via sample-level binary classification . The sample-level nature of our metric inspires a novel use case which we call model auditing , wherein we judge the quality of individual samples generated by a ( black-box ) model , discarding low-quality samples and hence improving the overall model performance in a post-hoc manner . 1 INTRODUCTION . Intuitively , it would seem that evaluating the likelihood function of a generative model is all it takes to assess its performance . As it turns out , the problem of evaluating generative models is far more complicated . This is not only because state-of-the-art models , such as Variational Autoencoders ( VAE ) ( Kingma & Welling ( 2013 ) ) and Generative Adversarial Networks ( GANs ) ( Goodfellow et al . ( 2014 ) ) , do not possess tractable likelihood functions , but also because the likelihood score itself is a flawed measure of performance—it scales badly in high dimensions , and it obscures distinct modes of model failure into a single uninterpretable score ( Theis et al . ( 2015 ) ) . Absent domain-agnostic metrics , earlier work focused on crafting domain-specific scores , e.g. , Inception score ( Salimans et al . ( 2016 ) ) , with an exclusive emphasis on image data ( Lucic et al . ( 2018 ) ) . In this paper , we introduce an alternative approach to evaluating generative models , where instead of assessing the generative distribution by looking at all synthetic samples collectively to compute likelihood or divergence , we classify each sample individually as being of high or low quality . In this way , our metric comprises interpretable probabilistic quantities—resembling those used to evaluate discriminative models ( e.g. , AUC-ROC ) —which describe the rate by which a model makes errors . When averaged over all samples , our sample-level scores reflect discrepancy between real and generative distributions in a way similar to statistical divergence measures ( e.g. , KL divergence , Fréchet distance ( Heusel et al . ( 2017 ) ) , or maximum mean discrepancy ( Sutherland et al . ( 2016 ) ) . In this sense , our metric enables diagnosing model performance on both the sample and distribution levels . Our metric represents the performance of a generative model as a point in a 3-dimensional space— each dimension corresponds to a distinct quality of the model . These qualities are : Fidelity , Diversity and Generalization . Fidelity measures the quality of a model ’ s synthetic samples , and Diversity is Figure 1 : Pictorial depiction for the proposed evaluation metrics . The blue and red spheres correspond to the α- and β-supports of real and generative distributions , respectively . Blue and red points correspond to real and synthetic data . ( a ) Synthetic data falling outside the blue sphere will look unrealistic or noisy . ( b ) Overfitted models can generate ostensibly high-quality data samples that are “ unauthentic ” because they are copied from the training data . ( c ) High-quality data samples should reside inside the blue sphere . ( d ) Outliers do not count in the β-Recall metric . ( Here , α=β=0.9 , α-Precision=8/9 , βRecall = 4/9 , and Authenticity = 9/10 . ) Support of real distribution Support of synthetic distribution Real Synthetic Noisy synthetic sample Unauthentic high-quality sample Authentic high-quality sample Outlier ( a ) ( b ) ( c ) ( d ) the extent to which these samples cover the full variability of real samples , whereas Generalization quantifies the extent to which a model overfits ( copies ) training data . How do we quantify the 3 dimensions of performance ? We build on the precision-recall analysis framework proposed in ( Sajjadi et al . ( 2018 ) ) , and introduce the α-Precision and β-Recall metrics to quantify model Fidelity and Diversity , respectively . Both metrics assume that a fraction 1 − α ( or 1 − β ) of the real ( and synthetic ) data are “ outliers ” , and α ( or β ) are “ typical ” . α-Precision is the fraction of synthetic samples that resemble the “ most typical ” α real samples , whereas β-Recall is the fraction of real samples covered by the most typical β synthetic samples . α-Precision and β-Recall are evaluated for all α , β ∈ [ 0 , 1 ] , providing entire precision and recall curves instead of single numbers . To compute both metrics , we embed the ( real and synthetic ) data into hyperspheres with most samples concentrated around the centers , i.e. , the real and generative distributions ( Pr and Pg ) has spherical-shaped supports . In this transformed feature space , typical samples would be located near the centers of the spheres , whereas outliers would be closer to the boundaries . To quantify Generalization , we introduce the Authenticity metric , which reflects the likelihood of a synthetic sample being copied from training data . We derive the Authenticity metric from a hypothesis test for data copying based on the observed proximity of synthetic samples to real ones in the embedded feature space . A pictorial illustration for all metrics is shown in Figure 1 . How is our metric different ? If one think of standard precision and recall metrics as “ hard ” binary classifiers of real and synthetic samples , our α-Precision and β-Recall can be thought of as softboundary classifiers that do not only compare the supports of Pr and Pg , but also assesses whether both distributions are calibrated . Precision and recall metrics are special cases of α-Precision and β-Recall for α = β = 1 . As we show later , our new metric definitions solve many of the drawbacks of standard precision-recall analysis , such as lack of robustness to outliers and failure to detect distributional mismatches ( Naeem et al . ( 2020 ) ) . They also enable detailed diagnostics of different types of model failure , such as mode collapse and mode invention . Moreover , optimal values of our metrics are achieved only when Pr and Pg are identical , thereby eliminating the need to augment the evaluation procedure with additional measures of statistical divergence ( e.g. , KL divergence ) . Previous works relied on pre-trained embeddings ( using ImageNet feature extractors ( Deng et al . ( 2009 ) ) ) . In this work , we propose feature embeddings that are model- and domain-agnostic , and are tailored to our metric definitions and data set at hand . Our proposed feature embedding step can be completely bespoke to raw data , or augmented with pre-trained embeddings . This enables our metric to remain operable in application domains where no pre-trained representations exist . Overfitting is a crucial mode of failure of generative models , especially when modeling sensitive data ( e.g. , clinical data ) for which data copying may violate privacy requirements ( Yoon et al . ( 2020 ) ) , but it has been overlooked in previous works which focused exclusively on quantifying the FidelityDiversity characterization ( Brock et al . ( 2018 ) ) . As we show in our experiments ( Section 5 ) , because our metric accounts for Generalization , it can provide a fuller picture of a generative model ’ s performance . Precisely , we show that some of the celebrated generative models score highly for Fidelity and Diversity simply because they memorize real samples , rendering them inappropriate for privacysensitive applications . A comprehensive survey of prior work is provided in the Appendix . Model auditing as a novel use case . The sample-level nature of our metrics inspires the new use case of model auditing , wherein we judge individual synthetic samples by their quality , and reject samples that have low Fidelity or are unauthentic . In Section 5 , we show that model audits can improve the outputs of a black-box model in a post-hoc fashion without any modifications to the model itself , and demonstrate this use case in synthesizing clinical data for COVID-19 patients . 2 EVALUATING AND AUDITING GENERATIVE MODELS . 2.1 PROBLEM SETUP . We denote real and generated data asXr ∼ Pr andXg ∼ Pg , respectively , whereXr , Xg ∈ X , with Pr and Pg being the real and generative distributions , and X being the input space . The generative distribution , Pg , is estimated using a generative model ( e.g. , GAN ) . The real and synthetic data sets are Dreal = { Xr , i } ni=1 and Dsynth = { Xg , j } mj=1 , where Xr , i ∼ Pr and Xg , j ∼ Pg . 2.2 WHAT MAKES A GOOD SYNTHETIC DATA SET ? . Our goal is to construct a metric E ( Dreal , Dsynth ) for the quality of Dsynth in order to ( i ) evaluate the performance of the underlying generative model Pg , and ( ii ) audit the model outputs by discarding ( individual ) “ low-quality ” samples , thereby improving the overall quality ofDsynth . In order for E to fulfill the evaluation and auditing tasks , it must satisfy the following desiderata : ( 1 ) it should be able to disentangle the different modes of failure of Pg through interpretable measures of performance , and ( 2 ) it should be sample-wise computable , i.e. , we should be able to tell if a given ( individual ) synthetic sample Xg ∼ Pg is of a low quality . Having outlined the desiderata for our sought-after evaluation metric , we now propose three qualities of synthetic data that the metric E should be able to quantify . Failure to fulfill any of these three qualities correspond to independent modes of failure of the model Pg . These qualities are : 1 . Fidelity—the generated samples resemble real samples from Pr . A high-fidelity synthetic data set should contain “ realistic ” samples , e.g . visually-realistic images . 2 . Diversity—the generated samples are diverse enough to cover the variability of real data , i.e. , a model should be able to generate a wide variety of good samples . 3 . Generalization—the generated samples should not be mere copies of the ( real ) samples in training data , i.e. , models that overfit to Dreal are not truly “ generative ” . In Section 3 , we propose a three-dimensional evaluation metric E that captures all of the qualities above . Our proposed metric can be succinctly described as follows : E , ( α-Precision︸ ︷︷ ︸ Fidelity , β-Recall︸ ︷︷ ︸ Diversity , Authenticity︸ ︷︷ ︸ Generalization ) . ( 1 ) The α-Precision and β-Recall metrics are generalizations of the conventional notions of precision and recall used in binary classification analysis ( Flach & Kull ( 2015 ) ) . Precision measures the rate by which the model synthesizes “ realistic-looking ” samples , whereas the recall measures the fraction of real samples that are covered by Pg . The authenticity score measures the fraction of synthetic samples that are invented by the model and not copied from the training data . | The paper presents a methodology assessing the performance of a generative model in a domain agnostic fashion. A 3-dimensional metric space is proposed: fidelity (output quality), diversity (coverage of expected variability of output) and generalization (to what degree model avoids memorizing training data i.e. is truly generative) is proposed. Particularly the latter element is novel and the manner in which it is may be evaluated by developing an `authenticity’ measure for the task. Three illustrative use cases in image (mnist) and medical patient data (COVID-19, `Hide and Seek’ seq2seq data) domains are provided. | SP:e48d1c232568b92d53a89e035830e800b9970b8e |
Continual Learning of Neural Networks for Realtime Wireline Cable Position Inference | 1 INTRODUCTION . With the emergence of AI-powered computer vision techniques in recent years , more and more vision-based applications are being introduced into the oil fields Wan ( 2019 ) Ant ( 2019 ) . As an example , to automate the Wireline spooling process , one can mount a surveillance camera inside the cabin behind the window facing the drum , feed the live spooling video into a trained neural network and infer the so-called cable position in real time ( Fig.1 ) . This inferred cable position will then be passed to a spooling controller , which moves the spooling arm accordingly so that the cable can be laid onto the drum without gaps or crossovers Su et al . ( 2021b ) . To train the neural network , example video frames with ground truth cable position were first collected offline . The weights of the network were then learnt from these training frames to minimize the prediction error . After testing and optimization , the network was then deployed to the Wireline units . Training did not stop after the deployment , however , because the example frames in the first training process may not cover all the operational conditions ( different Wireline units , drum sizes , cable types , lighting and weather conditions , operations in daytime vs. nighttime , etc ) . To continuously improve the network performance on the out-of-distribution frames , i.e. , frames that look significantly different from those in the training set , we kept collecting new training videos to help generalize the performance of the network to various operational environments . This process is called continual , lifelong or incremental neural network training . The most straightforward setup for continual learning is to combine the new videos with all the old ones and re-train the network using the ever-increasing number of training frames Su et al . ( 2021a ) . This , however , is not a scalable solution . Disk space required to hold all the past videos increases without bound over time . Also , the time to re-train a new network increases rapidly with more frames to extract , process and learn . A naive scalable solution could be to re-train the network only on the new videos and remove all the old ones . This solution , however , does not work well . During re-training , a neural network tends to forget what it learned previously , causing large prediction error when applying the updated network on the old videos . This phenomenon is called catastrophic forgetting in the literature Kirkpatrick et al . ( 2017 ) . To help the network learn the characteristics of the new videos without forgetting the information of the old ones , continual learning techniques are usually applied . There exist three types of continual learning techniques Liu et al . ( 2020 ) : the regularization-based method that penalizes large weight changes during re-training Kirkpatrick et al . ( 2017 ) , the parameter isolation method that assigns different weights for different datasets Mallya & Lazebnik ( 2018 ) , and the replay-based method that replays some of the old information to the network together with the new data Hayes et al . ( 2020 ) Wang et al . ( 2021 ) Prabhu et al . ( 2020 ) . More descriptions of these methods will be provided in next section . One realization of the replay-based method is the REMIND network Hayes et al . ( 2020 ) . Instead of saving all the raw videos , the REMIND network compresses and stores the features of the old videos and then replays them to the network for re-training . Feature compression is done by Product Quantization ( PQ ) Jegou et al . ( 2010 ) , which clusters the features into K-means centroids and compactly represents them using the centroid indices . Although efficient in disk space usage , the original REMIND network replaces the previously stored centroids with those generated by the new data at each re-train iteration , which is not suitable for multiple re-training process . Besides , we found that the original REMIND network has memory overflow issue when the size of dataset is large . In this paper , we proposed MREMIND , which modifies the dictionary update policy of the REMIND network so the previously stored centroids won ’ t be replaced and deleted . To optimize memory allocation , we also partitioned the dataset into subsets , which enables the method to work on devices with limited RAM . Additionally , we investigated other state-of-the-art continual learning methods and compared their performance on the cable position inference problem . In summary , our contributions in this work are : 1 . We applied continual learning techniques to train neural networks for real-time vision-based cable position inference to enable Wireline auto-spooling . 2 . We modified the REMIND network with an extendable dictionary update policy and an improved memory management . 3 . We compared the accuracy and the disk space usage of multiple state-of-the-art continual learning techniques on the cable position inference problem . 2 RELATED WORKS . Continual learning solutions aim to achieve the following goals : ( 1 ) Continuously train and re-train a network to learn from new data ; ( 2 ) Keep the performance of the network on the previously seen old data ; and ( 3 ) Avoid saving all the old data in its raw form to make the solution scalable for disk space usage . Related works in this area include : • The regularization-based method . One example is the Elastic Weight Consolidation Method ( EWC ) Kirkpatrick et al . ( 2017 ) . This method adds a regularization term to the loss function to penalize the change of the weights that are sensitive for the old datasets . Here ” sensitivity ” is quantified using the Fisher information . This method is simple to implement . However , setting and tuning the penalty stiffness can be highly problem-dependent . Also , the method was reported to be vulnerable to domain shift Wang et al . ( 2021 ) Aljundi et al . ( 2017 ) . • The the parameter isolation method . This method isolates the parameters and train different datasets on different parameters in the network Mallya & Lazebnik ( 2018 ) . This method indeed can solve the catastrophic forgetting problem because the previously learnt weights won ’ t be affected by subsequent training . However , the amount of weights needed increases without bounds as more data becomes available . Therefore , it is not a scalable solution . • The replay-based method . This method tackles the catastrophic forgetting problem by replaying the old data during retraining . The old data may not in its raw form . It could be compressed , transformed , or reduced to a subset of the original data . For example , the REMIND network Hayes et al . ( 2020 ) extracted and stored features from the old data and replayed these features for network re-training . The ACAE-REMIND work Wang et al . ( 2021 ) increased the number of trainable parameters in plastic layer and introduced an auto-encoder to further enhance the inference accuracy . Wu et al . ( 2019 ) used generative adversarial network ( GAN ) to generate images of the old datasets for future replay . Interestingly , a recent paper , GDumb Prabhu et al . ( 2020 ) , shows that by greedily storing the old images and then retraining the network from scratch at each iteration , one can get similar performance compared to other existing methods . For these replay-based methods , it is important to balance the distribution of replay samples for each dataset to avoid data bias . 3 METHODS . In this section we propose a new continual learning method for wireline cable position inference task . In section 3.1 , we will define the objective of the paper . In 3.2 we first give an introduction to current replay-based continual learning method and point out the drawbacks in actual deployment . Then we introduce our proposed MREMIND method . 3.1 CONTINUAL LEARNING FOR WIRELINE AUTOMATION . Upon actual deployment , wireline cable position inference might occur huge performance drop under different environments including lightness , weather and camera position setting . In this paper , our objective is to train a neural network that has high prediction accuracy in different configurations . Data under different settings will arrive sequentially . In order to avoid catastrophic forgetting issue as well as balance inference accuracy with training time and disk space usage , we apply continual learning technique to tackle this problem . 3.2 MREMIND : MODIFED REMIND NETWORK . 3.2.1 CURRENT REMIND NETWORK . Current REMIND method divides a neural network into two parts : ( 1 ) a feature extractor , also called the encoder or the frozen layer , and ( 2 ) a decoder , also called the plastic layer Hayes et al . ( 2020 ) Wang et al . ( 2021 ) . The weights in the extractor are frozen after the first training , while those in the decoder are continuously trained on the new data iteration after iteration . In this paper , one ” iteration ” refers to one re-training of the network when a new dataset arrives . It could contain multiple training epochs . To avoid saving all the old raw videos , the REMIND network uses the frozen layer to extract the mid-level features from the old frames , then compresses and stores them into dictionaries . When a new dataset arrives , the old features are decompressed from the dictionaries and replayed to the network along with the new frames for network re-training . In this way , the network learns from the new data and keeps its performance on the old ones . Training Procedure To explain the training procedure of the REMIND network , denote Enc and Dec as the encoder and decoder of the network , respectively , Feeding an input image x to the network produces a mid-level feature ( z = Enc ( x ) ) and a predicted cable position ( ŷ = Dec ( Enc ( x ) ) ) . Additionally , we use y to denote the ground truth cable position , so that the prediction error is ŷ− y . The optimal weights of the network are trained by minimizing the squared prediction error . The training procedure is shown in Fig . 2 and is summarized in the steps below : 1 . With all the weights in the encoder and decoder set to be tunable , train the network on all images in the first dataset 1 . The trained network is the backbone pre-trained model . 2 . Apply the PQ method to compress all the mid-level features from dataset 1 . In this step , we obtain two base dictionaries representing dataset 1 . 3 . When a new dataset arrives , feed the new images into the encoder to extract features . 4 . Uniformly and randomly select the compressed old features from the previous dataset , decompress and feed them together with the new features to the plastic layer . Re-train the decoder using these combined features by minimizing the prediction error . 5 . Compress the new features , and replace the stored old features . In this step , we obtain two updated dictionaries representing the new features . Product Quantization The extracted features from the old frames are compressed and stored using the Product Quantization ( PQ ) method Jegou et al . ( 2010 ) . The PQ method clusters the features into K-means centroids , and compactly represents the original features with the centroids indices . Each batch of features , with batch size b , number of channels d , and m× n pixels on each channel , can be flatten into a 2D matrix of size ( b×m× n , d ) . The PQ method divides the d channels into s sub-sections . K-means clustering is then applied on each sub-section , resulting in K numbers of d/s-dimension centroids for each sub-section . The features can then be represented using the IDs of the closest centroids ( of integer type ) . After compression , the original features are represented by the clustered centroids and the centroid IDs that tell us which centroid each sub-feature corresponds to . For example , image a batch of original features with the size of b×m× n× d × 32 bits ( 32 bits type ) . After PQ compression , we will get ( 1 ) an integer ID dictionary of the size b×m×n× log2 K × s bits , and ( 2 ) a centroid dictionary of the size K × s× ( d/s ) × 32 ) bits ( Fig.3 ) . These two dictionaries are used to recover the features during decompression . It has been reported that the PQ method can store 50 times more features compared to raw images with the same disk space Wang et al . ( 2021 ) . We used the library developed by Faiss for implementing the PQ method Johnson et al . ( 2017 ) . | In this paper, the authors proposed a method called the modified-REMIND (MREMIND) network. It is a replay-based continual learning method with a longer memory to historical data and no memory overflow issues. Information of old data is kept for multiple iterations using a new dictionary update rule. The modified-REMIND is applied to the cable position inference problem. experiment section shows that the modified-REMIND network gave the best performance among all other tested solutions. | SP:bb0ea700c272794b1005c1e245805b84bc64e89a |
Continual Learning of Neural Networks for Realtime Wireline Cable Position Inference | 1 INTRODUCTION . With the emergence of AI-powered computer vision techniques in recent years , more and more vision-based applications are being introduced into the oil fields Wan ( 2019 ) Ant ( 2019 ) . As an example , to automate the Wireline spooling process , one can mount a surveillance camera inside the cabin behind the window facing the drum , feed the live spooling video into a trained neural network and infer the so-called cable position in real time ( Fig.1 ) . This inferred cable position will then be passed to a spooling controller , which moves the spooling arm accordingly so that the cable can be laid onto the drum without gaps or crossovers Su et al . ( 2021b ) . To train the neural network , example video frames with ground truth cable position were first collected offline . The weights of the network were then learnt from these training frames to minimize the prediction error . After testing and optimization , the network was then deployed to the Wireline units . Training did not stop after the deployment , however , because the example frames in the first training process may not cover all the operational conditions ( different Wireline units , drum sizes , cable types , lighting and weather conditions , operations in daytime vs. nighttime , etc ) . To continuously improve the network performance on the out-of-distribution frames , i.e. , frames that look significantly different from those in the training set , we kept collecting new training videos to help generalize the performance of the network to various operational environments . This process is called continual , lifelong or incremental neural network training . The most straightforward setup for continual learning is to combine the new videos with all the old ones and re-train the network using the ever-increasing number of training frames Su et al . ( 2021a ) . This , however , is not a scalable solution . Disk space required to hold all the past videos increases without bound over time . Also , the time to re-train a new network increases rapidly with more frames to extract , process and learn . A naive scalable solution could be to re-train the network only on the new videos and remove all the old ones . This solution , however , does not work well . During re-training , a neural network tends to forget what it learned previously , causing large prediction error when applying the updated network on the old videos . This phenomenon is called catastrophic forgetting in the literature Kirkpatrick et al . ( 2017 ) . To help the network learn the characteristics of the new videos without forgetting the information of the old ones , continual learning techniques are usually applied . There exist three types of continual learning techniques Liu et al . ( 2020 ) : the regularization-based method that penalizes large weight changes during re-training Kirkpatrick et al . ( 2017 ) , the parameter isolation method that assigns different weights for different datasets Mallya & Lazebnik ( 2018 ) , and the replay-based method that replays some of the old information to the network together with the new data Hayes et al . ( 2020 ) Wang et al . ( 2021 ) Prabhu et al . ( 2020 ) . More descriptions of these methods will be provided in next section . One realization of the replay-based method is the REMIND network Hayes et al . ( 2020 ) . Instead of saving all the raw videos , the REMIND network compresses and stores the features of the old videos and then replays them to the network for re-training . Feature compression is done by Product Quantization ( PQ ) Jegou et al . ( 2010 ) , which clusters the features into K-means centroids and compactly represents them using the centroid indices . Although efficient in disk space usage , the original REMIND network replaces the previously stored centroids with those generated by the new data at each re-train iteration , which is not suitable for multiple re-training process . Besides , we found that the original REMIND network has memory overflow issue when the size of dataset is large . In this paper , we proposed MREMIND , which modifies the dictionary update policy of the REMIND network so the previously stored centroids won ’ t be replaced and deleted . To optimize memory allocation , we also partitioned the dataset into subsets , which enables the method to work on devices with limited RAM . Additionally , we investigated other state-of-the-art continual learning methods and compared their performance on the cable position inference problem . In summary , our contributions in this work are : 1 . We applied continual learning techniques to train neural networks for real-time vision-based cable position inference to enable Wireline auto-spooling . 2 . We modified the REMIND network with an extendable dictionary update policy and an improved memory management . 3 . We compared the accuracy and the disk space usage of multiple state-of-the-art continual learning techniques on the cable position inference problem . 2 RELATED WORKS . Continual learning solutions aim to achieve the following goals : ( 1 ) Continuously train and re-train a network to learn from new data ; ( 2 ) Keep the performance of the network on the previously seen old data ; and ( 3 ) Avoid saving all the old data in its raw form to make the solution scalable for disk space usage . Related works in this area include : • The regularization-based method . One example is the Elastic Weight Consolidation Method ( EWC ) Kirkpatrick et al . ( 2017 ) . This method adds a regularization term to the loss function to penalize the change of the weights that are sensitive for the old datasets . Here ” sensitivity ” is quantified using the Fisher information . This method is simple to implement . However , setting and tuning the penalty stiffness can be highly problem-dependent . Also , the method was reported to be vulnerable to domain shift Wang et al . ( 2021 ) Aljundi et al . ( 2017 ) . • The the parameter isolation method . This method isolates the parameters and train different datasets on different parameters in the network Mallya & Lazebnik ( 2018 ) . This method indeed can solve the catastrophic forgetting problem because the previously learnt weights won ’ t be affected by subsequent training . However , the amount of weights needed increases without bounds as more data becomes available . Therefore , it is not a scalable solution . • The replay-based method . This method tackles the catastrophic forgetting problem by replaying the old data during retraining . The old data may not in its raw form . It could be compressed , transformed , or reduced to a subset of the original data . For example , the REMIND network Hayes et al . ( 2020 ) extracted and stored features from the old data and replayed these features for network re-training . The ACAE-REMIND work Wang et al . ( 2021 ) increased the number of trainable parameters in plastic layer and introduced an auto-encoder to further enhance the inference accuracy . Wu et al . ( 2019 ) used generative adversarial network ( GAN ) to generate images of the old datasets for future replay . Interestingly , a recent paper , GDumb Prabhu et al . ( 2020 ) , shows that by greedily storing the old images and then retraining the network from scratch at each iteration , one can get similar performance compared to other existing methods . For these replay-based methods , it is important to balance the distribution of replay samples for each dataset to avoid data bias . 3 METHODS . In this section we propose a new continual learning method for wireline cable position inference task . In section 3.1 , we will define the objective of the paper . In 3.2 we first give an introduction to current replay-based continual learning method and point out the drawbacks in actual deployment . Then we introduce our proposed MREMIND method . 3.1 CONTINUAL LEARNING FOR WIRELINE AUTOMATION . Upon actual deployment , wireline cable position inference might occur huge performance drop under different environments including lightness , weather and camera position setting . In this paper , our objective is to train a neural network that has high prediction accuracy in different configurations . Data under different settings will arrive sequentially . In order to avoid catastrophic forgetting issue as well as balance inference accuracy with training time and disk space usage , we apply continual learning technique to tackle this problem . 3.2 MREMIND : MODIFED REMIND NETWORK . 3.2.1 CURRENT REMIND NETWORK . Current REMIND method divides a neural network into two parts : ( 1 ) a feature extractor , also called the encoder or the frozen layer , and ( 2 ) a decoder , also called the plastic layer Hayes et al . ( 2020 ) Wang et al . ( 2021 ) . The weights in the extractor are frozen after the first training , while those in the decoder are continuously trained on the new data iteration after iteration . In this paper , one ” iteration ” refers to one re-training of the network when a new dataset arrives . It could contain multiple training epochs . To avoid saving all the old raw videos , the REMIND network uses the frozen layer to extract the mid-level features from the old frames , then compresses and stores them into dictionaries . When a new dataset arrives , the old features are decompressed from the dictionaries and replayed to the network along with the new frames for network re-training . In this way , the network learns from the new data and keeps its performance on the old ones . Training Procedure To explain the training procedure of the REMIND network , denote Enc and Dec as the encoder and decoder of the network , respectively , Feeding an input image x to the network produces a mid-level feature ( z = Enc ( x ) ) and a predicted cable position ( ŷ = Dec ( Enc ( x ) ) ) . Additionally , we use y to denote the ground truth cable position , so that the prediction error is ŷ− y . The optimal weights of the network are trained by minimizing the squared prediction error . The training procedure is shown in Fig . 2 and is summarized in the steps below : 1 . With all the weights in the encoder and decoder set to be tunable , train the network on all images in the first dataset 1 . The trained network is the backbone pre-trained model . 2 . Apply the PQ method to compress all the mid-level features from dataset 1 . In this step , we obtain two base dictionaries representing dataset 1 . 3 . When a new dataset arrives , feed the new images into the encoder to extract features . 4 . Uniformly and randomly select the compressed old features from the previous dataset , decompress and feed them together with the new features to the plastic layer . Re-train the decoder using these combined features by minimizing the prediction error . 5 . Compress the new features , and replace the stored old features . In this step , we obtain two updated dictionaries representing the new features . Product Quantization The extracted features from the old frames are compressed and stored using the Product Quantization ( PQ ) method Jegou et al . ( 2010 ) . The PQ method clusters the features into K-means centroids , and compactly represents the original features with the centroids indices . Each batch of features , with batch size b , number of channels d , and m× n pixels on each channel , can be flatten into a 2D matrix of size ( b×m× n , d ) . The PQ method divides the d channels into s sub-sections . K-means clustering is then applied on each sub-section , resulting in K numbers of d/s-dimension centroids for each sub-section . The features can then be represented using the IDs of the closest centroids ( of integer type ) . After compression , the original features are represented by the clustered centroids and the centroid IDs that tell us which centroid each sub-feature corresponds to . For example , image a batch of original features with the size of b×m× n× d × 32 bits ( 32 bits type ) . After PQ compression , we will get ( 1 ) an integer ID dictionary of the size b×m×n× log2 K × s bits , and ( 2 ) a centroid dictionary of the size K × s× ( d/s ) × 32 ) bits ( Fig.3 ) . These two dictionaries are used to recover the features during decompression . It has been reported that the PQ method can store 50 times more features compared to raw images with the same disk space Wang et al . ( 2021 ) . We used the library developed by Faiss for implementing the PQ method Johnson et al . ( 2017 ) . | Original REMIND network compresses and stores features from old videos and replays them on the network at the time of retraining. Previously stored centroids are replaced with new ones and this is not suitable for a multiple re-training process. The authors proposed a new memory update rule to overcome catastrophic forgetting more efficiently and a strategy of resolving limited resource issues | SP:bb0ea700c272794b1005c1e245805b84bc64e89a |
Continual Learning of Neural Networks for Realtime Wireline Cable Position Inference | 1 INTRODUCTION . With the emergence of AI-powered computer vision techniques in recent years , more and more vision-based applications are being introduced into the oil fields Wan ( 2019 ) Ant ( 2019 ) . As an example , to automate the Wireline spooling process , one can mount a surveillance camera inside the cabin behind the window facing the drum , feed the live spooling video into a trained neural network and infer the so-called cable position in real time ( Fig.1 ) . This inferred cable position will then be passed to a spooling controller , which moves the spooling arm accordingly so that the cable can be laid onto the drum without gaps or crossovers Su et al . ( 2021b ) . To train the neural network , example video frames with ground truth cable position were first collected offline . The weights of the network were then learnt from these training frames to minimize the prediction error . After testing and optimization , the network was then deployed to the Wireline units . Training did not stop after the deployment , however , because the example frames in the first training process may not cover all the operational conditions ( different Wireline units , drum sizes , cable types , lighting and weather conditions , operations in daytime vs. nighttime , etc ) . To continuously improve the network performance on the out-of-distribution frames , i.e. , frames that look significantly different from those in the training set , we kept collecting new training videos to help generalize the performance of the network to various operational environments . This process is called continual , lifelong or incremental neural network training . The most straightforward setup for continual learning is to combine the new videos with all the old ones and re-train the network using the ever-increasing number of training frames Su et al . ( 2021a ) . This , however , is not a scalable solution . Disk space required to hold all the past videos increases without bound over time . Also , the time to re-train a new network increases rapidly with more frames to extract , process and learn . A naive scalable solution could be to re-train the network only on the new videos and remove all the old ones . This solution , however , does not work well . During re-training , a neural network tends to forget what it learned previously , causing large prediction error when applying the updated network on the old videos . This phenomenon is called catastrophic forgetting in the literature Kirkpatrick et al . ( 2017 ) . To help the network learn the characteristics of the new videos without forgetting the information of the old ones , continual learning techniques are usually applied . There exist three types of continual learning techniques Liu et al . ( 2020 ) : the regularization-based method that penalizes large weight changes during re-training Kirkpatrick et al . ( 2017 ) , the parameter isolation method that assigns different weights for different datasets Mallya & Lazebnik ( 2018 ) , and the replay-based method that replays some of the old information to the network together with the new data Hayes et al . ( 2020 ) Wang et al . ( 2021 ) Prabhu et al . ( 2020 ) . More descriptions of these methods will be provided in next section . One realization of the replay-based method is the REMIND network Hayes et al . ( 2020 ) . Instead of saving all the raw videos , the REMIND network compresses and stores the features of the old videos and then replays them to the network for re-training . Feature compression is done by Product Quantization ( PQ ) Jegou et al . ( 2010 ) , which clusters the features into K-means centroids and compactly represents them using the centroid indices . Although efficient in disk space usage , the original REMIND network replaces the previously stored centroids with those generated by the new data at each re-train iteration , which is not suitable for multiple re-training process . Besides , we found that the original REMIND network has memory overflow issue when the size of dataset is large . In this paper , we proposed MREMIND , which modifies the dictionary update policy of the REMIND network so the previously stored centroids won ’ t be replaced and deleted . To optimize memory allocation , we also partitioned the dataset into subsets , which enables the method to work on devices with limited RAM . Additionally , we investigated other state-of-the-art continual learning methods and compared their performance on the cable position inference problem . In summary , our contributions in this work are : 1 . We applied continual learning techniques to train neural networks for real-time vision-based cable position inference to enable Wireline auto-spooling . 2 . We modified the REMIND network with an extendable dictionary update policy and an improved memory management . 3 . We compared the accuracy and the disk space usage of multiple state-of-the-art continual learning techniques on the cable position inference problem . 2 RELATED WORKS . Continual learning solutions aim to achieve the following goals : ( 1 ) Continuously train and re-train a network to learn from new data ; ( 2 ) Keep the performance of the network on the previously seen old data ; and ( 3 ) Avoid saving all the old data in its raw form to make the solution scalable for disk space usage . Related works in this area include : • The regularization-based method . One example is the Elastic Weight Consolidation Method ( EWC ) Kirkpatrick et al . ( 2017 ) . This method adds a regularization term to the loss function to penalize the change of the weights that are sensitive for the old datasets . Here ” sensitivity ” is quantified using the Fisher information . This method is simple to implement . However , setting and tuning the penalty stiffness can be highly problem-dependent . Also , the method was reported to be vulnerable to domain shift Wang et al . ( 2021 ) Aljundi et al . ( 2017 ) . • The the parameter isolation method . This method isolates the parameters and train different datasets on different parameters in the network Mallya & Lazebnik ( 2018 ) . This method indeed can solve the catastrophic forgetting problem because the previously learnt weights won ’ t be affected by subsequent training . However , the amount of weights needed increases without bounds as more data becomes available . Therefore , it is not a scalable solution . • The replay-based method . This method tackles the catastrophic forgetting problem by replaying the old data during retraining . The old data may not in its raw form . It could be compressed , transformed , or reduced to a subset of the original data . For example , the REMIND network Hayes et al . ( 2020 ) extracted and stored features from the old data and replayed these features for network re-training . The ACAE-REMIND work Wang et al . ( 2021 ) increased the number of trainable parameters in plastic layer and introduced an auto-encoder to further enhance the inference accuracy . Wu et al . ( 2019 ) used generative adversarial network ( GAN ) to generate images of the old datasets for future replay . Interestingly , a recent paper , GDumb Prabhu et al . ( 2020 ) , shows that by greedily storing the old images and then retraining the network from scratch at each iteration , one can get similar performance compared to other existing methods . For these replay-based methods , it is important to balance the distribution of replay samples for each dataset to avoid data bias . 3 METHODS . In this section we propose a new continual learning method for wireline cable position inference task . In section 3.1 , we will define the objective of the paper . In 3.2 we first give an introduction to current replay-based continual learning method and point out the drawbacks in actual deployment . Then we introduce our proposed MREMIND method . 3.1 CONTINUAL LEARNING FOR WIRELINE AUTOMATION . Upon actual deployment , wireline cable position inference might occur huge performance drop under different environments including lightness , weather and camera position setting . In this paper , our objective is to train a neural network that has high prediction accuracy in different configurations . Data under different settings will arrive sequentially . In order to avoid catastrophic forgetting issue as well as balance inference accuracy with training time and disk space usage , we apply continual learning technique to tackle this problem . 3.2 MREMIND : MODIFED REMIND NETWORK . 3.2.1 CURRENT REMIND NETWORK . Current REMIND method divides a neural network into two parts : ( 1 ) a feature extractor , also called the encoder or the frozen layer , and ( 2 ) a decoder , also called the plastic layer Hayes et al . ( 2020 ) Wang et al . ( 2021 ) . The weights in the extractor are frozen after the first training , while those in the decoder are continuously trained on the new data iteration after iteration . In this paper , one ” iteration ” refers to one re-training of the network when a new dataset arrives . It could contain multiple training epochs . To avoid saving all the old raw videos , the REMIND network uses the frozen layer to extract the mid-level features from the old frames , then compresses and stores them into dictionaries . When a new dataset arrives , the old features are decompressed from the dictionaries and replayed to the network along with the new frames for network re-training . In this way , the network learns from the new data and keeps its performance on the old ones . Training Procedure To explain the training procedure of the REMIND network , denote Enc and Dec as the encoder and decoder of the network , respectively , Feeding an input image x to the network produces a mid-level feature ( z = Enc ( x ) ) and a predicted cable position ( ŷ = Dec ( Enc ( x ) ) ) . Additionally , we use y to denote the ground truth cable position , so that the prediction error is ŷ− y . The optimal weights of the network are trained by minimizing the squared prediction error . The training procedure is shown in Fig . 2 and is summarized in the steps below : 1 . With all the weights in the encoder and decoder set to be tunable , train the network on all images in the first dataset 1 . The trained network is the backbone pre-trained model . 2 . Apply the PQ method to compress all the mid-level features from dataset 1 . In this step , we obtain two base dictionaries representing dataset 1 . 3 . When a new dataset arrives , feed the new images into the encoder to extract features . 4 . Uniformly and randomly select the compressed old features from the previous dataset , decompress and feed them together with the new features to the plastic layer . Re-train the decoder using these combined features by minimizing the prediction error . 5 . Compress the new features , and replace the stored old features . In this step , we obtain two updated dictionaries representing the new features . Product Quantization The extracted features from the old frames are compressed and stored using the Product Quantization ( PQ ) method Jegou et al . ( 2010 ) . The PQ method clusters the features into K-means centroids , and compactly represents the original features with the centroids indices . Each batch of features , with batch size b , number of channels d , and m× n pixels on each channel , can be flatten into a 2D matrix of size ( b×m× n , d ) . The PQ method divides the d channels into s sub-sections . K-means clustering is then applied on each sub-section , resulting in K numbers of d/s-dimension centroids for each sub-section . The features can then be represented using the IDs of the closest centroids ( of integer type ) . After compression , the original features are represented by the clustered centroids and the centroid IDs that tell us which centroid each sub-feature corresponds to . For example , image a batch of original features with the size of b×m× n× d × 32 bits ( 32 bits type ) . After PQ compression , we will get ( 1 ) an integer ID dictionary of the size b×m×n× log2 K × s bits , and ( 2 ) a centroid dictionary of the size K × s× ( d/s ) × 32 ) bits ( Fig.3 ) . These two dictionaries are used to recover the features during decompression . It has been reported that the PQ method can store 50 times more features compared to raw images with the same disk space Wang et al . ( 2021 ) . We used the library developed by Faiss for implementing the PQ method Johnson et al . ( 2017 ) . | This submission modified REMIND with different dictionary update policy to keep track of previously stored centroids and partition the datasets to subsets to optimize memory usage. The proposed method has been verified on wireline auto-spooling (predict cable position). Several other state-of-the-art continual learning techniques have been tested on this task too. | SP:bb0ea700c272794b1005c1e245805b84bc64e89a |
Boundary-aware Pre-training for Video Scene Segmentation | 1 INTRODUCTION . The video scene segmentation is a task of identifying scene boundaries from a video where a scene is defined as a semantic unit for making a story and is composed of a series of semantically cohesive shots—a set of frames captured by the same camera during an uninterrupted period of time—in the same context . Localizing scene boundaries is a significant step towards the high-level video understanding because dividing a long video into a set of meaningful scenes enables models to inspect the individual incidents from complex story . One of the biggest challenges with temporal semantic segmentation is that it is not achieved simply by detecting changes in visual cues . As shown in Figure 1 ( a ) , we present an example of nine shots , all of which belong to the same scene , where two characters are talking on the phone . We can see that the overall visual cues within the scene do not stay the same but rather change repeatedly when each character appears . On the other hand , as presented in Figure 1 ( b ) , the other example shows two different scenes which contain visually similar shots ( highlighted in blue ) where the same character appears in the same place . Therefore , it is expected that two adjacent scenes which share shots with similar visual cues need to be contextually discriminated . From this observation , it is important for the video scene segmentation task to model contextual relation between shots by maximizing 1 ) intra-scene similarity ( i.e. , the shots belonging to the same scene should be close to each other ) , and 2 ) inter-scene discrimination across two adjacent scenes ( i.e. , two neighbor shots across the scene boundary should be distinguishable ) . Supervised learning approaches ( e.g. , Rao et al . ( 2020 ) ) are clearly limited due to the lack of largescale datasets with reliable ground-truth annotations . Recently , self-supervision ( Chen et al. , 2020a ; Caron et al. , 2020 ; He et al. , 2020 ; Roh et al. , 2021 ) is spotlighted through its effectiveness in learning in-domain representation without relying on costly ground truth annotations . The selfsupervised learning methods ( Chen et al. , 2021 ; Feichtenhofer et al. , 2021 ; Dave et al. , 2021 ; Qian et al. , 2021 ) in the video domain are often designed to learn spatio-temporal patterns in short clips ( e.g. , shots in movies ) . This kind of learned representation is generic and can be applied to many video understanding tasks ( e.g. , action classification ) . However , such representation is not sufficient for video scene segmentation because this task requires not only a good representation for individual shots but also contextual representation considering neighboring shots at a higher level as illustrated in Figure 1 . Motivated by this , we set our main goal to design effective self-supervised objectives ( i.e. , pretext tasks ) that maximize intra-scene similarity as well as discriminate shots from different scenes . For the purpose , this raises a penetrating question : how can we design boundary-relevant pretext tasks without access to ground truth boundary annotations ? We introduce a novel Boundary-aware Self-Supervised Learning ( BaSSL ) framework . Our main idea of BaSSL is to localize a pseudo-boundary , which is obtained by dividing the input sequence of shots into two semantically disjoint sub-sequences , and use it to define pretext tasks that are beneficial to the video scene segmentation task . On top of the discovered two sub-sequences and a pseudo-boundary , three boundary-aware pretext tasks are proposed : 1 ) Shot-Scene Matching ( SSM ) ; 2 ) Contextual Group Matching ( CGM ) ; and 3 ) Pseudo-boundary Prediction ( PP ) . Note that SSM and CGM encourage the model to maximize intra-scene similarity and inter-scene discrimination while PP enables the model to learn the capability of identifying transitional moments . In addition , we perform Masked Shot Modeling ( MSM ) task inspired by Sun et al . ( 2019a ) to further learn temporal relationship between shots . The comprehensive analysis demonstrates the effectiveness of the proposed framework ( i.e. , pre-training of contextual relationship between shots ) as well as the contribution of the proposed individual components ( i.e. , the algorithm for pseudo-boundary discovery and boundary-aware pretext tasks ) . Our main contributions are summarized as follows : ( i ) We introduce a novel boundary-aware pre-training framework which adopts dynamic time warping ( DTW ) algorithm to identify pseudoboundaries and use them as self-supervision to facilitate the pre-training ; ( ii ) we propose three boundary-aware pretext tasks , which are carefully designed to learn essential capabilities required for the video scene segmentation task ; ( iii ) we perform extensive ablations to demonstrate the effectiveness of the proposed framework , including the observation that our framework is complementary to the existing framework ; ( iv ) we achieve the new state-of-the-art on the MovieNet-SSeg benchmark while outperforming existing self-supervised learning-based methods by large margins . 2 RELATED WORK . Video scene segmentation approaches formulate the task as a problem of temporal grouping of shots . In this formulation , the optimal grouping can be achieved by clustering-based ( Rui et al. , 1998 ; Rasheed & Shah , 2003 ; 2005 ; Chasanis et al. , 2008 ) , dynamic programming-based ( Han & Wu , 2011 ; Tapaswi et al. , 2014 ; Rotman et al. , 2017 ) or multi-modal input-based ( Liang et al. , 2009 ; Sidiropoulos et al. , 2011 ) methods . However , the aforementioned methods have been trained and evaluated on small-scale datasets such as OVSD ( Rotman et al. , 2016 ) and BBC ( Baraldi et al. , 2015 ) which can produce a poorly generalized model . Recently , Huang et al . ( 2020 ) introduce a largescale video scene segmentation dataset ( i.e. , MovieNet-SSeg ) that contains hundreds of movies . Training with large-scale data , Rao et al . ( 2020 ) proposes a strong supervised baseline model that performs a shot-level binary classification followed by grouping using the prediction scores . In addition , Chen et al . ( 2021 ) proposes a shot contrastive pre-training method that learns shot-level representation . We found ShotCoL ( Chen et al. , 2021 ) to be the most similar work to our method . However , our method is different from ShotCoL in that we specifically focus on learning contextual representations by considering the relationship between shots . We refer interested readers to the supplementary material for a more detailed analysis of this . Action segmentation in videos is one of the related works for video scene segmentation , which identifies action labels of individual frames , thus divides a video into a series of action segments . Supervised methods ( Lea et al. , 2016 ; Farha & Gall , 2019 ) proposed CNN-based architectures to effectively capture temporal relationship between frames in order to address an over-segmentation issue . As frame-level annotations are prohibitively costly to acquire , weakly supervised methods ( Chang et al. , 2019 ; Li et al. , 2019 ; Li & Todorovic , 2020 ; Souri et al. , 2021 ; Shen et al. , 2021 ; Zhukov et al. , 2019 ; Fried et al. , 2020 ) have been suggested to use an ordered list of actions occurring in a video as supervision . Most of the methods are trained to find ( temporal ) semantic alignment between frames and a given action list using an HMM-based architecture ( Kuehne et al. , 2018 ) , a dynamic programming-based assignment algorithm ( Fried et al. , 2020 ) or a DTW-based temporal alignment method ( Chang et al. , 2019 ) . Recently , unsupervised methods ( Kumar et al. , 2021 ; Wang et al. , 2021 ; Kukleva et al. , 2019 ; Li & Todorovic , 2021 ; VidalMata et al. , 2021 ) have been further proposed ; in a nutshell , clustering-based prototypes are discovered from unlabeled videos , then the methods segment the videos by assigning prototypes ( corresponding to one of the actions ) into frames . In contrast to the action segmentation task that is limited to localizing segments each of which represents a single action within an activity , video scene segmentation requires localizing more complex segments each of which may be composed of more than two actions ( or activities ) . Self-supervised learning in videos has been actively studied for the recent years with approaches proposing various pretext tasks such as future frame prediction ( Srivastava et al. , 2015 ; Vondrick et al. , 2016 ; Ahsan et al. , 2018 ) , temporal ordering of frames ( Misra et al. , 2016 ; Lee et al. , 2017 ; Xu et al. , 2019 ) , geometric transformations prediction ( Jing & Tian , 2018 ) , colorization of videos ( Vondrick et al. , 2018 ) and contrastive prediction ( Feichtenhofer et al. , 2021 ; Qian et al. , 2021 ; Dave et al. , 2021 ) . In addition , CBT ( Sun et al. , 2019a ; b ) proposes a pretext task of masked frame modeling to learn temporal dependency between frames ( or clips ) . Note that since most of those methods are proposed for the classification task , they would be sub-optimal to the video scene segmentation task . On the other hand , BSP ( Xu et al. , 2020 ) proposes boundary-sensitive pre-text tasks based on synthesized pseudo-boundaries that are obtained by concatenating two clips sampled from different videos . However , strictly speaking , BSP is not a self-supervised learning algorithm since it requires video-level class labels to synthesize pseudo-boundaries ; the proposed pretext tasks are not applicable to videos such as movies that are hard to define semantic labels . Also , note that we empirically show that pseudo-boundaries identified by our method are more effective for pre-training than synthesized pseudo-boundaries . 3 BOUNDARY-AWARE SELF-SUPERVISED LEARNING ( BASSL ) . In this section , we introduce our proposed approach , Boundary-aware Self-Supervised Learning ( BaSSL ) . We start with the problem formulation followed by the model overview . Then , we describe our novel boundary-aware pretext tasks for pre-training . 3.1 PROBLEM FORMULATION . Terminologies A video ( e.g. , documentaries , TV episodes and movies ) is a sequence of scenes , defined as a semantic unit for making a story . A scene is a series of shots , which is a set of frames physically captured by the same camera during an uninterrupted period of time . Problem Definition Given a video , which contains a series of N shots { s1 , ... , sN } with class labels { y1 , ... , yN } where yi ∈ { 0 , 1 } indicating if it is at the scene boundary ( more precisely , if it is the last shot of a scene ) , the video scene segmentation task is formulated as a simple binary classification problem at individual shot level . By definition , a scene boundary is where the semantic of a shot is considerably different from its ( one-way ) neighbors . Thus , it is in nature important to capture and leverage contextual transition across the scenes . Consequently , it is a common practice that the information of the neighbor shots are leveraged together when determining scene boundaries . With this formulation , existing supervised learning approaches typically train a parameterized ( θ ) model by maximizing the expected log-likelihood : Pre-training Stage Fine-tuning Stage Pseudo-boundary Shot θ∗ = arg max θ E [ log pθ ( yn|Sn ) ] , ( 1 ) where Sn = { sn−K , ... , sn , ... , sn+K } is a set of 2K + 1 shots centered at nth shot sn , and K is the number of neighbor shots before and after sn . Note that each shot s is given by a set of Nk keyframes , resulting in a tensor with size of ( Nk , C , H , W ) where C , H and W are the RGB channels , the frame height and the frame width , respectively . | The paper introduces a boundary-aware self-supervised learning framework for video scene segmentation. Based on the contrastive learning protocol, the authors employ DTW to generate dimidiate video segmentation from unlabeled video. Using these segments, the authors employ four loss functions to pre-train a backbone network. | SP:6254bbdb33e5ff8cee7a06f1c44f3653408de7cb |
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