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0000000000000000000000000000000000000000..64396f04122c19c804730d46842584e600b8cbd1 --- /dev/null +++ b/parse/train/B1eB5xSFvr/B1eB5xSFvr.md @@ -0,0 +1,432 @@ +# DIFFTAICHI: DIFFERENTIABLE PROGRAMMING FORPHYSICAL SIMULATION + +Yuanming $\mathbf { H } \mathbf { u } ^ { \dag }$ , Luke Anderson†, Tzu-Mao $\mathbf { L i } ^ { * }$ , Qi $\mathbf { S u n } ^ { \ddagger }$ , Nathan $\mathbf { C a r r } ^ { \dagger }$ , +Jonathan Ragan-Kelley∗, Frédo Durand† +†MIT CSAIL {yuanming,lukea,fredo}@mit.edu +‡Adobe Research {qisu,ncarr}@adobe.com +∗UC Berkeley {tzumao,jrk}@berkeley.edu + +# ABSTRACT + +We present DiffTaichi, a new differentiable programming language tailored for building high-performance differentiable physical simulators. Based on an imperative programming language, DiffTaichi generates gradients of simulation steps using source code transformations that preserve arithmetic intensity and parallelism. A light-weight tape is used to record the whole simulation program structure and replay the gradient kernels in a reversed order, for end-to-end backpropagation. We demonstrate the performance and productivity of our language in gradient-based learning and optimization tasks on 10 different physical simulators. For example, a differentiable elastic object simulator written in our language is $4 . 2 \times$ shorter than the hand-engineered CUDA version yet runs as fast, and is $1 8 8 \times$ faster than the TensorFlow implementation. Using our differentiable programs, neural network controllers are typically optimized within only tens of iterations. + +![](images/31d717138f86fae191183842f0f62dd34da1fdd52913d021ce76254056e3dfcd.jpg) +Figure 1: Left: Our language allows us to seamlessly integrate a neural network (NN) controller and a physical simulation module, and update the weights of the controller or the initial state parameterization (blue). Our simulations typically have $5 1 2 \sim 2 0 4 8$ time steps, and each time step has up to one thousand parallel operations. Right: 10 differentiable simulators built with DiffTaichi. + +Differentiable physical simulators are effective components in machine learning systems. For example, de Avila Belbute-Peres et al. (2018a) and Hu et al. (2019b) have shown that controller optimization with differentiable simulators converges one to four orders of magnitude faster than model-free reinforcement learning algorithms. The presence of differentiable physical simulators in the inner loop of these applications makes their performance vitally important. Unfortunately, using existing tools it is difficult to implement these simulators with high performance. + +We present DiffTaichi, a new differentiable programming language for high performance physical simulations on both CPU and GPU. It is based on the Taichi programming language (Hu et al., + +2019a). The DiffTaichi automatic differentiation system is designed to suit key language features required by physical simulation, yet often missing in existing differentiable programming tools, as detailed below: + +Megakernels Our language uses a “megakernel” approach, allowing the programmer to naturally fuse multiple stages of computation into a single kernel, which is later differentiated using source code transformations and just-in-time compilation. Compared to the linear algebra operators in TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2017), DiffTaichi kernels have higher arithmetic intensity and are therefore more efficient for physical simulation tasks. + +Imperative Parallel Programming In contrast to functional array programming languages that are popular in modern deep learning (Bergstra et al., 2010; Abadi et al., 2016; Li et al., 2018b), most traditional physical simulation programs are written in imperative languages such as Fortran and $\mathrm { C } { + } { + }$ . DiffTaichi likewise adopts an imperative approach. The language provides parallel loops and control flows (such as “if” statements), which are widely used constructs in physical simulations: they simplify common tasks such as handling collisions, evaluating boundary conditions, and building iterative solvers. Using an imperative style makes it easier to port existing physical simulation code to DiffTaichi. + +Flexible Indexing Existing parallel differentiable programming systems provide element-wise operations on arrays of the same shape, e.g. c[i, j] $=$ a[i, j] + b[i, j]. However, many physical simulation operations, such as numerical stencils and particle-grid interactions are not elementwise. Common simulation patterns such as y[p[i] $\star \ 2$ , $\mathbf { j } ] \ = \ \times [ \mathbf { q } [ \mathbf { j } + \mathbf { j } ] ]$ can only be expressed with unintuitive scatter/gather operations in these existing systems, which are not only inefficient but also hard to develop and maintain. On the other hand, in DiffTaichi, the programmer directly manipulates array elements via arbitrary indexing, thus allowing partial updates of global arrays and making these common simulation patterns naturally expressible. The explicit indexing syntax also makes it easy for the compiler to perform access optimizations (Hu et al., 2019a). + +The three requirements motivated us to design a tailored two-scale automatic differentiation system, which makes DiffTaichi especially suitable for developing complex and high-performance differentiable physical simulators, possibly with neural network controllers (Fig. 1, left). Using our language, we are able to quickly implement and automatically differentiate 10 physical simulators1, covering rigid bodies, deformable objects, and fluids (Fig. 1, right). A comprehensive comparison between DiffTaichiand other differentiable programming tools is in Appendix A. + +# 2 BACKGROUND: THE TAICHI PROGRAMMING LANGUAGE + +DiffTaichi is based on the Taichi programming language (Hu et al., 2019a). Taichi is an imperative programming language embedded in $\mathrm { C } { + } { + } 1 4$ . It delivers both high performance and high productivity on modern hardware. The key design that distinguishes Taichi from other imperative programming languages such as $\mathrm { C + + / C U D A }$ is the decoupling of computation from data structures. This allows programmers to easily switch between different data layouts and access data structures with indices (i.e. $\times [ \mathfrak { i } , \ \mathfrak { j } , \ \mathsf { k } ] ,$ , as if they are normal dense arrays, regardless of the underlying layout. The Taichi compiler then takes both the data structure and algorithm information to apply performance optimizations. Taichi provides “parallel-for" loops as a first-class construct. These designs make Taichi especially suitable for writing high-performance physical simulators. For more details, readers are referred to Hu et al. (2019a). + +The DiffTaichi language frontend is embedded in Python, and a Python AST transformer compiles DiffTaichi code to Taichi intermediate representation (IR). Unlike Python, the DiffTaichi language is compiled, statically-typed, parallel, and differentiable. We extend the Taichi compiler to further compile and automatically differentiate the generated Taichi IR into forward and backward executables. + +We demonstrate the language using a mass-spring simulator, with three springs and three mass points, as shown right. In this section we introduce the forward simulator using the DiffTaichi frontend of Taichi, which is an easier-to-use wrapper of the Taichi $\mathrm { C } { + } { + } 1 4$ frontend. + +Allocating Global Variables Firstly we allocate a set of global tensors to store the simulation state. These tensors include a scalar loss of type float32, 2D tensors $\times , ~ \lor$ , force of size steps $\times { \mathsf n _ { - } }$ springs and type float $3 2 \times 2$ , and 1D arrays of size n_spring for spring properties: spring_anchor_a (int32), spring_anchor_b (int32), spring_length (float32). + +Defining Kernels A mass-spring system is modeled by Hooke’s law $\textbf { F } = \ k ( \| \mathbf { x } _ { a } - \mathbf { x } _ { b } \| _ { 2 } \ - $ $l _ { 0 } ) \frac { \mathbf { x } _ { a } - \mathbf { x } _ { b } } { \lVert \mathbf { x } _ { a } - \mathbf { x } _ { b } \rVert _ { 2 } }$ where $k$ is the spring stiffness, $\mathbf { F }$ is spring force, $\mathbf { x } _ { a }$ and $\mathbf { x } _ { b }$ are the positions of two mass points, and $l _ { 0 }$ is the rest length. The following kernel loops over all the springs and scatters forces to mass points: + +@ti.kernel +def apply_spring_force(t: ti.i32): # Kernels can have parameters. Here t is a parameter with type int32. for i in range(n_springs): # A parallel for, preferably on GPU a, $\textrm { b } =$ spring_anchor_a[i], spring_anchor_b[i] $\times \_ a$ , ${ \bf \sf x _ { - } b } ~ = ~ { \bf \sf x _ { \bar { \tau } } t } ~ - ~ { \bf \epsilon } _ { 1 }$ , a], $\times \left[ { \ t { \mathrm { ~ \ - ~ } } 1 } \right.$ , b] dist $= \mathbf { \nabla } \times \_ \mathsf { a } - \mathbf { \nabla } \times \_ \mathsf { b }$ length $=$ dist.norm() + 1e-4 $\begin{array} { r l } { \mathsf { F } } & { { } = } \end{array}$ (length - spring_length[i]) $\star$ spring_stiffness $\star$ dist / length # Apply spring impulses to mass points. force[t, a] $+ = - \mathsf { F } \sharp \mathsf { \Gamma } ^ { \prime \prime } { + } = { \mathrm { \Omega } } ^ { \prime \prime }$ is atomic by default force[t, b] $\mathrm { \Sigma } + + + \mathrm { \Sigma } \mathsf { F }$ + +For each particle $i$ , we use semi-implicit Euler time integration with damping: $\begin{array} { r } { { \pmb v } _ { t , i } = e ^ { - \Delta t \alpha } { \pmb v } _ { t - 1 , i } + } \end{array}$ $\begin{array} { r } { \frac { \Delta t } { m _ { i } } { \bf F } _ { t , i } , { \bf x } _ { t , i } = { \bf x } _ { t - 1 , i } + \Delta t { \bf v } _ { t , i } } \end{array}$ , where $\pmb { v } _ { t , i } , \mathbf { x } _ { t , i } , m _ { i }$ are the velocity, position and mass of particle $i$ at time step $t$ , respectively. $\alpha$ is a damping factor. The kernel is as follows: + +@ti.kernel +def time_integrate(t: ti.i32): for i in range(n_objects): $\qquad \mathsf { s } \quad \mathsf { = }$ math.exp(-dt $\star$ damping) $\#$ Compile-time evaluation since dt and damping are constants $\mathsf { v } [ \mathsf { t } , \mathsf { ~ \mathsf { ~ \mathsf { \Sigma } ~ } } _ { } { \mathsf { \Sigma } } ] = \mathsf { s } \star \mathsf { v } [ \mathsf { t } - \mathsf { ~ \mathsf { \Sigma } ~ } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } \mathsf { i } ] + \mathsf { d } \mathsf { t } \star \mathsf { \Sigma } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } _ { } { \mathsf { \Sigma } } _ { }$ force[t, i] / mass $\#$ mass $= ~ 1$ in this example x[t, $\dot { \bf ~ l } ] ~ = ~ { \bf \nabla } \times [ \dot { \bf t } ~ - ~ { \bf 1 }$ , i] $^ +$ dt $\star$ v[t, i] + +Assembling the Forward Simulator With these components, we define the forward time integration: + +def forward(): for t in range(1, steps): apply_spring_force(t) time_integrate(t) + +# 3 AUTOMATICALLY DIFFERENTIATING PHYSICAL SIMULATORS IN TAICHI + +The main goal of DiffTaichi’s automatic differentiation (AD) system is to generate gradient simulators automatically with minimal code changes to the traditional forward simulators. + +Design Decision Source Code Transformation (SCT) (Griewank & Walther, 2008) and Tracing (Wengert, 1964) are common choices when designing AD systems. In our setting, using SCT to differentiate a whole simulator with thousands of time steps, results in high performance yet poor flexibility and long compilation time. On the other hand, naively adopting tracing provides flexibility yet poor performance, since the “megakernel" structure is not preserved during backpropagation. To get both performance and flexibility, we developed a two-scale automatic differentiation system (Figure 2): we use SCT for differentiating within kernels, and use a light-weight tape that only stores function pointers and arguments for end-to-end simulation differentiation. The global tensors are natural checkpoints for gradient evaluation. + +![](images/d619c6452f4310c09e17907c3247b30a16687a9cc6a3b7a8d7f36ee4d41df87c.jpg) +Figure 2: Left: The DiffTaichi system. We reuse some infrastructure (white boxes) from Taichi, while the blue boxes are our extensions for differentiable programming. Right: The tape records kernel launches and replays the gradient kernels in reverse order during backpropagation. + +Assumption Unlike functional programming languages where immutable output buffers are generated, imperative programming allows programmers to freely modify global tensors. To make automatic differentiation well-defined under this setting, we make the following assumption on imperative kernels: + +# Global Data Access Rules: + +1) If a global tensor element is written more than once, then starting from the second write, the write must come in the form of an atomic add (“accumulation”). 2) No read accesses happen to a global tensor element, until its accumulation is done. + +In forward simulators, programmers may make subtle changes to satisfy the rules. For instance, in the mass-spring simulation example, we record the whole history of $\times$ and $\vee$ , instead of keeping only the latest values. The memory consumption issues caused by this can be alleviated via checkpointing, as discussed later in Appendix D. + +With these assumptions, kernels will not overwrite the outputs of each other, and the goal of AD is clear: given a primal kernel $f$ that takes as input $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ and outputs (or accumulates to) $Y _ { 1 } , Y _ { 2 } , \dots , Y _ { m }$ , the generated gradient (adjoint) kernel $f ^ { * }$ should take as input $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ and $Y _ { 1 } ^ { * }$ $^ { \prime * } _ { 1 } , Y _ { 2 } ^ { * } , \ldots , Y _ { m } ^ { * }$ and accumulate gradient contributions to $X _ { 1 } ^ { * } , X _ { 2 } ^ { * } , \ldots , X _ { m } ^ { * }$ , where each $X _ { i } ^ { * }$ is an adjoint of $X _ { i }$ , i.e. $\partial ( \mathrm { I o s s } ) / \partial X _ { i }$ . + +Storage Control of Adjoint Tensors Users can specify the storage of adjoint tensors using the Taichi data structure description language (Hu et al., 2019a), as if they are primal tensors. We also provide ti.root.lazy_grad() to automatically place the adjoint tensors following the layout of their primals. + +# 3.1 LOCAL AD: DIFFERENTIATING TAICHI KERNELS USING SOURCE CODE TRANSFORMS + +A typical Taichi kernel consists of multiple levels of for loops and a body block. To make later AD easier, we introduce two basic code transforms to simplify the loop body, as detailed below. + +![](images/1b4c20ad358307d39d56d6c1309b5853a1902c2d261bda590bc40d6f691560eb.jpg) +Figure 3: Simple IR preprocessing before running the AD source code transform (left to right). Demonstrated in ${ \mathsf { C } } { + } { + }$ . The actual Taichi IR is often more complex. Containing loops are ignored. + +Flatten Branching In physical simulation branches are common, e.g., when implementing boundary conditions and collisions. To simplify the reverse-mode AD pass, we first replace “if” statements with ternary operators select(cond, value_if_true, value_if_false), whose gradients are clearly defined (Fig. 3, middle). This is a common transformation in program vectorization (e.g. Karrenberg & Hack (2011); Pharr & Mark (2012)). + +Eliminate Mutable Local Variables After removing branching, we end up with straight-line loop bodies. To further simplify the IR and make the procedure truly single-assignment, we apply a series of local variable store forwarding transforms, until the mutable local variables can be fully eliminated (Fig. 3, right). + +After these two custom IR simplification transforms, DiffTaichi only has to differentiate the straightline code without mutable variables, which it achieves with reverse-mode AD, using a standard source code transformation (Griewank & Walther, 2008). More details on this transform are in Appendix B. + +Loops Most loops in physical simulation are parallel loops, and during AD we preserve the parallel loop structures. For loops that are not explicitly marked as parallel, we reverse the loop order during AD transforms. We do not support loops that carry a mutating local variable since that would require a complex and costly run-time stack to maintain the history of local variables. Instead, users are instructed to employ global variables that satisfy the global data access rules. + +Parallelism and Thread Safety For forward simulation, we inherit the “parallel-for" construct from Taichi to map each loop iteration onto CPU/GPU threads. Programmers use atomic operations for thread safety. Our system can automatically differentiate these atomic operations. Gradient contributions in backward kernels are accumulated to the adjoint tensors via atomic adds. + +# 3.2 GLOBAL AD: END-TO-END BACKPROPAGATION USING A LIGHT-WEIGHT TAPE + +We construct a tape (Fig. 2, right) of the kernel execution so that gradient kernels can be replayed in a reversed order. The tape is very light-weight: since the intermediate results are stored in global tensors, during forward simulation the tape only records kernel names and the (scalar) input parameters, unlike other differentiable functional array systems where all the intermediate buffers have to be recorded by the tape. Whenever a DiffTaichi kernel is launched, we append the kernel function pointer and parameters to the tape. When evaluating gradients, we traverse the reversed tape, and invoke the gradient kernels with the recorded parameters. Note that DiffTaichi AD is evaluating gradients with respect to input global tensors instead of the input parameters. + +Learning/Optimization with Gradients Now we revisit the mass-spring example and make it differentiable for optimization. Suppose the goal is to optimize the rest lengths of the springs so that the triangle area formed by the three springs becomes 0.2 at the end of the simulation. We first define the loss function: + +![](images/d0cff8a0fd4c191b5c98eac383ef9116460f5f43114bbe3a5b620479de17ef41.jpg) + +Goal: Adjust the spring +rest lengths,so that thisarea=0.2 +after 1024 time steps +(initialarea $\mathbf { \tau } = \mathbf { 0 . 0 0 5 }$ ） + +The programmer uses ti.Tape to memorize forward kernel launches. It automatically replays the gradients of these kernels in reverse for backpropagation. Initially the springs have lengths [0.1, 0.1, 0.14], and after optimization the rest lengths are [0.600, 0.600, 0.529]. This means the springs will expand the triangle according to Hooke’s law and form a larger triangle: [Reproduce: mass_spring_simple.py] + +![](images/f162e45e31d7137450ef8894d248c79886fa27ae9df52a7e2dac14f42c14ae6e.jpg) + +Complex Kernels Sometimes the user may want to override the gradients provided by the compiler. For example, when differentiating a 3D singular value decomposition done with an iterative solver, it is better to use a manually engineered SVD derivative subroutine for better stability. We provide two more decorators ti.complex_kernel and ti.complex_kernel_grad to overwrite the default automatic differentiation, as detailed in Appendix C. Apart from custom gradients, complex kernels can also be used to implement checkpointing, as detailed in Appendix D. + +# 4 EVALUATION + +We evaluate DiffTaichi on 10 different physical simulators covering large-scale continuum and small-scale rigid body simulations. All results can be reproduced with the provided script. The dynamic/optimization processes are visualized in the supplemental video. In this section we focus our discussions on three simulators. More details on the simulators are in Appendix E. + +# 4.1 DIFFERENTIABLE CONTINUUM MECHANICS FOR ELASTIC OBJECTS [diffmpm] + +First, we build a differentiable continuum simulation for soft robotics applications. The physical system is governed by momentum and mass conservation, i.e. $\begin{array} { r } { \rho \frac { D \mathbf { v } } { D t } = \nabla \cdot \boldsymbol { \sigma } + \rho \mathbf { g } } \end{array}$ , $\begin{array} { r } { \frac { D \rho } { D t } + \rho \nabla \cdot \mathbf { v } = } \end{array}$ 0. We follow ChainQueen’s implementation (Hu et al., 2019b) and use the moving least squares material point method (Hu et al., 2018) to simulate the system. We were able to easily translate the original CUDA simulator into DiffTaichi syntax. Using this simulator and an open-loop controller, we can easily train a soft robot to move forward (Fig. 1, diffmpm). + +Performance and Productivity Compared with manual gradient implementations in (Hu et al., 2019b), getting gradients in DiffTaichi is effortless. As a result, the DiffTaichi implementation is $4 . 2 \times$ shorter in terms of lines of code, and runs almost as fast; compared with TensorFlow, DiffTaichi code is $1 . 7 \times$ shorter and $1 8 8 \times$ faster (Table 1). The Tensorflow implementation is verbose due to the heavy use of tf.gather_nd/scatter_nd and array transposing and broadcasting. + +Table 1: diffmpm performance comparison on an NVIDIA GTX 1080 Ti GPU. We benchmark in 2D using 6.4K particles. For the lines of code, we only include the essential implementation, excluding boilerplate code. [Reproduce: python3 diffmpm_benchmark.py] + +
ApproachForward TimeBackward TimeTotal Time# Lines of Code
TensorFlow13.20 ms35.70 ms48.90 ms (188.×)190
CUDA0.10 ms0.14 ms0.24 ms (0.92×)460
DiffTaichi0.11 ms0.15 ms0.26 ms (1.00×)110
+ +# 4.2 DIFFERENTIABLE INCOMPRESSIBLE FLUID SIMULATOR [smoke] + +We implemented a smoke simulator (Fig. 1, smoke) with semi-Lagrangian advection (Stam, 1999) and implicit pressure projection, following the example in Autograd (Maclaurin et al., 2015). Using gradient descent optimization on the initial velocity field, we are able to find a velocity field that changes the pattern of the fluid to a target image (Fig. 7a in Appendix). We compare the performance of our system against PyTorch, Autograd, and JAX in Table 2. Note that as an example from the + +Table 2: smoke benchmark against Autograd, PyTorch, and JAX. We used a $1 1 0 ~ \times ~ 1 1 0$ grid and 100 time steps, each with 6 Jacobi pressure projections. [Reproduce: python3 smoke_[autograd/pytorch/jax/taichi_cpu/taichi_gpu].py]. Note that the Autograd program uses float64 precision, which approximately doubles the run time. + +
ApproachForward TimeBackward TimeTotal Time#Essential LoC
PyTorch (CPU, f32)405 ms328 ms733 ms (13.8×)74
PyTorch (GPU, f32)254 ms457 ms711 ms (13.4×)74
Autograd (CPU, f64)307 ms1197 ms1504 ms (28.4×)51
JAX (GPU, f32)24 ms75 ms99 ms (1.9×)90
DiffTaichi (CPU, f32)66 ms132 ms198 ms (3.7x)75
DiffTaichi (GPU, f32)24 ms29 ms53 ms (1.0×)75
+ +Autograd library, this grid-based simulator is intentionally simplified to suit traditional array-based programs. For example, a periodic boundary condition is used so that Autograd can represent it using numpy.roll, without any branching. Still, Taichi delivers higher performance than these arraybased systems. The whole program takes 10 seconds to run in DiffTaichi on a GPU, and 2 seconds are spent on JIT. JAX JIT compilation takes 2 minutes. + +# 4.3 DIFFERENTIABLE RIGID BODY SIMULATORS [rigid_body] + +We built an impulse-based (Catto, 2009) differentiable rigid body simulator (Fig. 1, rigid_body) for optimizing robot controllers. This simulator supports rigid body collision and friction, spring forces, joints, and actuation. The simulation is end-to-end differentiable except for a countable number of discontinuities. Interestingly, although the forward simulator works well, naively differentiating it with DiffTaichi leads to completely misleading gradients, due to the rigid body collisions. We discuss the cause and solution of this issue below. + +Improving collision gradients Consider the rigid ball example in Fig. 4 (left), where a rigid ball collides with a friction-less ground. Gravity is ignored, and due to conservation of kinetic energy the ball keeps a constant speed even after this elastic collision. + +In the forward simulation, using a small $\Delta t$ often leads to a reasonable result, as done in many physics simulators. Lowering the initial ball height will increase the final ball height, since there is less distance to travel before the ball hits the ground and more after (see the loss curves in Fig.4, middle right). However, using a naive time integrator, no matter how small $\Delta t$ is, the evaluated gradient of final height w.r.t. initial height will be 1 instead of $- 1$ . This counter-intuitive behavior is due to the fact that time discretization itself is not differentiated by the compiler. Fig. 4 explains this effect in greater detail. + +![](images/8f7ca48f4fbe280ccf625c8aaa38b729d47690edbb0c833a8083c4b877937aa9.jpg) +Figure 4: How gradients can go wrong with naive time integrators. For clarity we use a large $\Delta t$ here. Left: Since collision detection only happens at multiples of $\Delta t$ ( $2 \Delta t$ in this case), lowering the initial position of the ball (light blue) leads to a lowered final position. Middle Left: By improving the time integrator to support continuous time of impact (TOI), collisions can be detected at any time, e.g. $1 . 9 \Delta t$ (light red). Now the blue ball ends up higher than the green one. Middle Right: Although the two time integration techniques lead to almost identical forward results (in practice $\Delta t$ is small), the naive time integrator gives an incorrect gradient of 1, but adding TOI yields the correct gradient. Please see our supplemental video for a better demonstration. [Reproduce: python3 rigid_body_toi.py] Right: When zooming in, the loss of the naive integrator is decreasing, and the saw-tooth pattern explains the positive gradients. [Reproduce: python3 rigid_body_toi.py zoom] + +![](images/c11f80cbb1f40de4474cb8c3aaa6f3370d1eb31a4e45e82731191a2044cbfddb.jpg) +Figure 5: Adding TOI greatly improves gradient and optimization quality. Each experiment is repeated five times. [Reproduce: python3 [mass_spring/rigid_body.py] [1/2] plot && python3 plot_losses.py] + +We propose a simple solution of adding continuous collision resolution (see, for example, Redon et al. (2002)), which considers precise time of impact (TOI), to the forward program (Fig. 4, middle left). Although it barely improves the forward simulation (Fig. 4, middle right), the gradient will be corrected effectively (Fig. 4, right). The details of continuous collision detection are in Appendix F. In real-world simulators, we find the TOI technique leads to significant improvement in gradient quality in controller optimization tasks (Fig. 5). Having TOI or not barely affects forward simulation: in the supplemental video, we show that a robot controller optimized in a simulator with TOI, actually works well in a simulator without TOI. + +The takeaway is, differentiating physical simulators does not always yield useful gradients of the physical system being simulated, even if the simulator does forward simulation well. In Appendix G, we discuss some additional gradient issues we have encountered. + +# 5 RELATED WORK + +Differentiable programming The recent rise of deep learning has motivated the development of differentiable programming libraries for deep NNs, most notably auto-differentiation frameworks such as Theano (Bergstra et al., 2010), TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2017). However, physical simulation requires complex and customizable operations due to the intrinsic computational irregularity. Using the aforementioned frameworks, programmers have to compose these coarse-grained basic operations into desired complex operations. Doing so often leads to unsatisfactory performance. + +Earlier work on automatic differentiation focuses on transforming existing scalar code to obtain derivatives (e.g. Utke et al. (2008), Hascoet & Pascual (2013), Pearlmutter & Siskind (2008)). A recent trend has emerged for modern programming languages to support differentiable function transformations through annotation (e.g. Innes et al. (2019), Wei et al. (2019)). These frameworks enable differentiating general programming languages, yet they provide limited parallelism. + +Differentiable array programming languages such as Halide (Ragan-Kelley et al., 2013; Li et al., 2018b), Autograd (Maclaurin et al., 2015), JAX (Bradbury et al., 2018), and Enoki (Jakob, 2019) operate on arrays instead of scalars to utilize parallelism. Instead of operating on arrays that are immutable, DiffTaichi uses an imperative style with flexible indexing to make porting existing physical simulation algorithms easier. + +Differentiable Physical Simulators Building differentiable simulators for robotics and machine learning has recently increased in popularity. Without differentiable programming, Battaglia et al. (2016), Chang et al. (2016) and Mrowca et al. (2018) used NNs to approximate the physical process and used the NN gradients as the approximate simulation gradients. Degrave et al. (2016) and de Avila Belbute-Peres et al. (2018b) used Theano and PyTorch respectively to build differentiable rigid body simulators. Schenck & Fox (2018) differentiates position-based fluid using custom CUDA kernels. Popovic et al. ´ (2000) used a differentiable rigid body simulator for manipulating physically based animations. The ChainQueen differentiable elastic object simulator (Hu et al., 2019b) implements forward and gradient versions of continuum mechanics in hand-written CUDA kernels, leading to performance that is two orders of magnitude higher than a pure TensorFlow implementation. Liang et al. (2019) built a differentiable cloth simulator for material estimation and motion control. The deep learning community also often incorporates differentiable rendering operations (OpenDR (Loper & Black, 2014), N3MR (Kato et al., 2018), redner (Li et al., 2018a), Mitsuba 2 (Nimier-David et al., 2019)) to learn from 3D scenes. + +# 6 CONCLUSION + +We have presented DiffTaichi, a new differentiable programming language designed specifically for building high-performance differentiable physical simulators. Motivated by the need for supporting megakernels, imperative programming, and flexible indexing, we developed a tailored two-scale automatic differentiation system. We used DiffTaichi to build 10 simulators and integrated them into deep neural networks, which proved the performance and productivity of DiffTaichi over existing systems. We hope our programming language can greatly lower the barrier of future research on differentiable physical simulation in the machine learning and robotics communities. + +# BIBLIOGRAPHY + +Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for largescale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), pp. 265–283, 2016. + +Peter W. Battaglia, Razvan Pascanu, Matthew Lai, Danilo Rezende, and Koray Kavukcuoglu. Interaction networks for learning about objects, relations and physics. 2016. + +James Bergstra, Olivier Breuleux, Frédéric Bastien, Pascal Lamblin, Razvan Pascanu, Guillaume Desjardins, Joseph Turian, David Warde-Farley, and Yoshua Bengio. Theano: A cpu and gpu math compiler in python. In Proc. 9th Python in Science Conf, volume 1, pp. 3–10, 2010. + +James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, and Skye Wanderman-Milne. JAX: composable transformations of Python+NumPy programs, 2018. URL http://github.com/google/jax. + +Erin Catto. Modeling and solving constraints. In Game Developers Conference, pp. 16, 2009. + +Michael B Chang, Tomer Ullman, Antonio Torralba, and Joshua B Tenenbaum. A compositional object-based approach to learning physical dynamics. ICLR, 2016. + +Filipe de Avila Belbute-Peres, Kevin Smith, Kelsey Allen, Josh Tenenbaum, and J Zico Kolter. End-to-end differentiable physics for learning and control. In Advances in Neural Information Processing Systems, pp. 7178–7189, 2018a. + +Filipe de Avila Belbute-Peres, Kevin A Smith, Kelsey Allen, Joshua B Tenenbaum, and J Zico Kolter. End-to-end differentiable physics for learning and control. 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A moving least squares material point method with displacement discontinuity and two-way rigid body coupling. 37(4):150, 2018. + +Yuanming Hu, Tzu-Mao Li, Luke Anderson, Jonathan Ragan-Kelley, and Frédo Durand. Taichi: A language for high-performance computation on spatially sparse data structures. In SIGGRAPH Asia 2019 Technical Papers, pp. 201. ACM, 2019a. + +Yuanming Hu, Jiancheng Liu, Andrew Spielberg, Joshua B Tenenbaum, William T Freeman, Jiajun Wu, Daniela Rus, and Wojciech Matusik. Chainqueen: A real-time differentiable physical simulator for soft robotics. Proceedings of IEEE International Conference on Robotics and Automation (ICRA), 2019b. + +Mike Innes, Alan Edelman, Keno Fischer, Chris Rackauckus, Elliot Saba, Viral B Shah, and Will Tebbutt. Zygote: A differentiable programming system to bridge machine learning and scientific computing. arXiv preprint arXiv:1907.07587, 2019. + +Wenzel Jakob. Enoki: structured vectorization and differentiation on modern processor architectures, 2019. https://github.com/mitsuba-renderer/enoki. + +Chenfanfu Jiang. The material point method for the physics-based simulation of solids and fluids. University of California, Los Angeles, 2015. + +Ralf Karrenberg and Sebastian Hack. Whole-function vectorization. In Code Generation and Optimization, pp. 141–150, 2011. + +Hiroharu Kato, Yoshitaka Ushiku, and Tatsuya Harada. Neural 3d mesh renderer. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3907–3916, 2018. + +Stig Larsson and Vidar Thomée. Partial differential equations with numerical methods, volume 45. Springer Science & Business Media, 2008. + +Tzu-Mao Li, Miika Aittala, Frédo Durand, and Jaakko Lehtinen. Differentiable monte carlo ray tracing through edge sampling. In SIGGRAPH Asia 2018 Technical Papers, pp. 222. ACM, 2018a. + +Tzu-Mao Li, Michaël Gharbi, Andrew Adams, Frédo Durand, and Jonathan Ragan-Kelley. Differentiable programming for image processing and deep learning in Halide. ACM Transactions on Graphics (TOG), 37(4):139, 2018b. + +Junbang Liang, Ming C Lin, and Vladlen Koltun. Differentiable cloth simulation for inverse problems. Advances in Neural Information Processing Systems, 2019. + +Matthew M Loper and Michael J Black. Opendr: An approximate differentiable renderer. In European Conference on Computer Vision, pp. 154–169. Springer, 2014. + +Dougal Maclaurin, David Duvenaud, and Ryan P Adams. Autograd: Effortless gradients in numpy. In ICML 2015 AutoML Workshop, volume 238, 2015. + +Aleka McAdams, Andrew Selle, Rasmus Tamstorf, Joseph Teran, and Eftychios Sifakis. Computing the singular value decomposition of 3x3 matrices with minimal branching and elementary floating point operations. Technical report, University of Wisconsin-Madison Department of Computer Sciences, 2011. + +Damian Mrowca, Chengxu Zhuang, Elias Wang, Nick Haber, Li Fei-Fei, Joshua B Tenenbaum, and Daniel LK Yamins. Flexible neural representation for physics prediction. 1806.08047, 2018. + +Merlin Nimier-David, Delio Vicini, Tizian Zeltner, and Wenzel Jakob. Mitsuba 2: A retargetable forward and inverse renderer. Transactions on Graphics (Proceedings of SIGGRAPH Asia), 38 (6), November 2019. doi: 10.1145/3355089.3356498. + +Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017. + +Barak A. Pearlmutter and Jeffrey Mark Siskind. Reverse-mode AD in a functional framework: Lambda the ultimate backpropagator. ACM Transactions on Programming Languages and Systems, 30(2):7:1–7:36, 2008. + +Matt Pharr and William R Mark. ispc: A spmd compiler for high-performance cpu programming. In Innovative Parallel Computing, pp. 1–13, 2012. + +Jovan Popovic, Steven M Seitz, Michael Erdmann, Zoran Popovi ´ c, and Andrew Witkin. Interactive ´ manipulation of rigid body simulations. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pp. 209–217. ACM Press/Addison-Wesley Publishing Co., 2000. + +Jonathan Ragan-Kelley, Connelly Barnes, Andrew Adams, Sylvain Paris, Frédo Durand, and Saman Amarasinghe. Halide: A language and compiler for optimizing parallelism, locality, and recomputation in image processing pipelines. SIGPLAN Not., 48(6):519–530, jun 2013. + +Stéphane Redon, Abderrahmane Kheddar, and Sabine Coquillart. Fast continuous collision detection between rigid bodies. In Computer graphics forum, volume 21, pp. 279–287. Wiley Online Library, 2002. + +Connor Schenck and Dieter Fox. Spnets: Differentiable fluid dynamics for deep neural networks. arXiv preprint arXiv:1806.06094, 2018. + +Jos Stam. Stable fluids. In Siggraph, volume 99, pp. 121–128, 1999. + +Andre Pradhana Tampubolon, Theodore Gast, Gergely Klár, Chuyuan Fu, Joseph Teran, Chenfanfu Jiang, and Ken Museth. Multi-species simulation of porous sand and water mixtures. ACM Transactions on Graphics (TOG), 36(4):105, 2017. + +Jean Utke, Uwe Naumann, Mike Fagan, Nathan Tallent, Michelle Strout, Patrick Heimbach, Chris Hill, and Carl Wunsch. Openad/f: A modular open-source tool for automatic differentiation of fortran codes. 34(4):18, 2008. + +Jui-Hsien Wang, Ante Qu, Timothy R Langlois, and Doug L James. Toward wave-based sound synthesis for computer animation. ACM Trans. Graph., 37(4):109–1, 2018. + +Richard Wei, Marc Rasi Dan Zheng, and Bart Chrzaszcz. Differentiable programming mega-proposal. https://github.com/apple/swift/blob/master/docs/ DifferentiableProgramming.md, 2019. Accessed: 2019-09-25. + +R. E. Wengert. A simple automatic derivative evaluation program. Communications of the ACM, 7 (8):463–464, aug 1964. + +# A COMPARISON WITH EXISTING SYSTEMS + +Workload differences between deep learning and differentiable physical simulation Existing differentiable programming tools for deep learning are typically centered around large data blobs. For example, in AlexNet, the second convolution layer has size $2 7 \times 2 7 \times 1 2 8 \times 1 2 8$ . These tools usually provide users with both low-level operations such as tensor add and mul, and high-level operations such as convolution. The bottleneck of typical deep-learning-based computer vision tasks are convolutions, so the provided high-level operations, with very high arithmetic intensity2, can fully exploit hardware capability. However, the provided operations are “atoms” of these differentiable programming tools, and cannot be further customized. Users often have to use low-level operations to compose their desired high-level operations. This introduces a lot of temporary buffers, and potentially excessive GPU kernel launches. As shown in Hu et al. (2019b), a pure TensorFlow implementation of a complex physical simulator is $1 3 2 \times$ slower than a CUDA implementation, due to excessive GPU kernel launches and the lack of producer-consumer locality3. + +The table below compares DiffTaichi with existing tools for build differentiable physical simulators. + +Table 3: Comparisons between DiffTaichi and other differentiable programming tools. Note that this table only discusses features related to differentiable physical simulation, and the other tools may not have been designed for this purpose. For example, PyTorch and TensorFlow are designed for classical deep learning tasks and have proven successful in their target domains. Also note that the XLA backend of TensorFlow and JIT feature of PyTorch allow them to fuse operators to some extent, but for simulation we want complete operator fusion within megakernels. “Swift” AD (Wei et al., 2019) is partially implemented as of November 2019. “Julia” refers to Innes et al. (2019). + +
FeatureDiffTaichiPyTorchTensorFlowEnokiJAXHalideJuliaSwift
GPUMegakernels<A△<
Imperative Scheme【【
Parallelism?
Flexible Indexing【【
+ +# B DIFFERENTATING STRAIGHT-LINE TAICHI KERNELS USING SOURCE CODE TRANSFORM + +Primal and adjoint kernels Recall that in DiffTaichi, (primal) kernels are operators that take as input multiple tensors (e.g., $X , Y )$ and output another set of tensors. Mathematically, kernel $f$ has the form + +$$ +f ( X _ { 0 } , X _ { 1 } , . . , X _ { n } ) = Y _ { 0 } , Y _ { 1 } , . . . , Y _ { m } . +$$ + +Kernels usually execute uniform operations on these tensors. When it comes to differentiable programming, a loss function is defined on the final output tensors. The gradients of the loss function “ $L ^ { \prime \prime }$ with respect to each tensor are stored in adjoint tensors and computed via adjoint kernels. + +The adjoint tensor of (primal) tensor $X _ { i j k }$ is denoted as $X _ { i j k } ^ { * }$ . Its entries are defined by $X _ { i j k } ^ { * } =$ $\partial L / \partial X _ { i j k }$ . At a high level, our automatic differentiation (AD) system transforms a primal kernel into its adjoint form. Mathematically, + +Reverse-Mode Automatic Differentiation + +$$ +( \mathbf { a d j o i n t } ) f ^ { * } ( X _ { 0 } , X _ { 1 } , . . , X _ { n } , Y _ { 0 } ^ { * } , Y _ { 1 } ^ { * } , . . . , Y _ { m } ^ { * } ) = X _ { 0 } ^ { * } , X _ { 1 } ^ { * } , . . , X _ { n } ^ { * } . +$$ + +Differentiating within kernels: The “make_adjoint” pass (reverse-mode AD) After the preprocessing passes, which flatten branching and eliminate mutable local variables, the “make_adjoint” pass transforms a forward evaluation (primal) kernel into its gradient accumulation (“adjoint”) kernel. It takes straight-line code directly and operates on the hierarchical intermediate representation (IR) of Taichi4 . Multiple outer for loops are allowed for the primal kernel. The Taichi compiler will distribute these parallel iterations onto CPU/GPU threads. + +During the “make_adjoint” pass, for each SSA instruction, a local adjoint variable will be allocated for gradient contribution accumulation. The compiler will traverse the statements in reverse order, and accumulate the gradients to the corresponding adjoint local variable. + +For example, a 1D array operation $y _ { i } = \sin x _ { i } ^ { 2 }$ has its IR representation as follows: + +
for i ∈ range(0, 16, step 1) do
%1=load x[i]
%2 = mul %1, %1
%3= sin(%2)
store y[i] = %3
end for
+ +The above primal kernel will be transformed into the following adjoint kernel: + +
for i in range(0,16, step 1) do
/ adjoint variables
%1adj = alloca 0.0
%2adj = alloca 0.0
%3adj = alloca 0.0
// original forward computation
%1=load x[] %2 = mul %1, %1
%3= sin(%2)
/ reverse accumulation
%4 = load y_adj[i]
%3adj += %4
%5 = cos(%2)
%2adj += %3adj * %5
%1adj += 2 * %1 * %2adj
atomic add x_adj[i],%1adj end for
+ +Note that for clarity the transformed code is not strictly SSA here. The actual IR has more instructions. A following simplification pass will simplify redundant instructions generated by the AD pass. + +# C COMPLEX KERNELS + +Here we demonstrated how to use complex kernels to override the automatic differentiation system. We use singular value decomposition (SVD) of $3 \times 3$ matrices $\mathbf { M } = \mathbf { U } \pmb { \Sigma } \mathbf { V } ^ { * }$ ) as an example. Fast SVD solvers used in physical simulation are often iterative, yet directly evaluate the gradient of this iterative process is likely numerically unstable. Suppose we use McAdams et al. (2011) as the forward SVD solver, and use the method in Jiang (2015) (Section 2.1.1.2) to evalute the gradients, the complex kernels are used as follows: + +# Do Singular Value Decomposition (SVD) on n matrices +@ti.kernel +def iterative_svd(num_iterations: ti.f32): for i in range(n): input $=$ matrix_M[i] for iter in range(num_iterations): ... iteratively solve SVD using McAdams et al. 2011 matrix_U[i] $=$ ... matrix_Sigma[i] $=$ matrix_V[i] $=$ ... +@ti.kernel +def svd_gradient(): for i in range(n): ... Implement, for example, section 2.1.1.2 of Jiang (2015) . +# A complex kernel that is registered as the svd_forward complex kernel +@ti.complex_kernel_grad(svd_forward) +def svd_backward(num_iterations): # differentiave SVD svd_gradient() + +# D CHECKPOINTING + +In this section we demonstrate how to use checkpointing via complex kernels. The goal of checkpointing is to use recomputation to save memory space. We demonstrate this using the diffmpm example, whose simulation cycle consists of particle to grid transform (p2g), grid boundary conditions (grid_op), and grid to particle transform $( \mathtt { g } 2 \mathsf { p } )$ . We assume the simulation has $O ( n )$ time steps. + +# D.1 RECOMPUTATION WITHIN TIME STEPS + +A naive implementation without checkpointing allocates $O ( n )$ copied of the simulation grid, which can cost a lot of memory space. Actually, if we recompute the grid states during the backward simulation time step by redoing ${ \mathsf { p } } 2 { \mathsf { g } }$ and grid_op, we can reused the grid states and allocate only one copy. This checkpointing optimization is demonstrated in the code below: + +
@ti.complex_kernel
def advance(s):
clear_grid()
compute_actuation(s)
p2g(s)
grid_op()
g2p(s)
@ti.complex_kernel_grad(advance)
def advance_grad(s):
clear_grid() p2g(s)
grid_op() # recompute the grid
g2p.grad(s)
grid_op.grad()
p2g.grad(s) compute_actuation.grad(s)
+ +# D.2 SEGMENT-WISE RECOMPUTATION + +Given a simulation with $O ( n )$ time steps, if all simulation steps are recorded, the space consumption is $O ( n )$ . This linear space consumption is sometimes too large for high-resolution simulations with long time horizon. Fortunately, we can reduce the space consumption using a segment-wise checkpointing trick: We split the simulation into segments of $S$ steps, and in forward simulation store only the first simulation state in each segment. During backpropagation when we need the remaining simulation states in a segment, we recompute them based on the first state in that segment. + +Note that if the segment size is $O ( S )$ , then we only need to store $O ( n / S )$ simulation steps for checkpoints and $\bar { O ( S ) }$ reusable simulation steps for backpropagation within segments. The total√ space consumption is √ $O ( S + n / S )$ . Setting ${ \cal S } \doteq { \cal O } ( \sqrt { n } )$ reduces memory consumption from $O ( n )$ to $O ( { \sqrt { n } } )$ . The time complexity remains $O ( n )$ . + +# E DETAILS ON 10 DIFFERENTIABLE SIMULATORS + +E.1 DIFFERENTIABLE CONTINUUM MECHANICS FOR ELASTIC OBJECTS [diffmpm] + +![](images/981f05ea68d2115e11a3dd53e1bd3fa25dcdf8b91cb3428f0f8ba3219819063d.jpg) +Figure 6: Controller optimization with our differentiable continuum simulators. Left: the 2D robot with four muscles. Middle: A 3D robot with 16 muscles and 30K particles crawling on the ground. [Reproduce: python3 [diffmpm/diffmpm3d].py] Right: We couple the robot (30K particles) and the liquid simulator (13K particles), and optimize its open-loop controller in this difficult situation.[Reproduce: python3 liquid.py] + +# E.2 DIFFERENTIABLE LIQUID SIMULATOR [liquid] + +We follow the weakly compressible fluid model in Tampubolon et al. (2017) and implemented a 3D differentiable liquid simulator within the [diffmpm3d] framework. Our liquid simulation can be two-way coupled with elastic object simulation (Figure 6, right). + +# E.3 DIFFERENTIABLE INCOMPRESSIBLE FLUID SIMULATOR [smoke] + +![](images/032fd5e7cb773f9949bec038a3eaf43d815115cd2c63f7d61cc1a2a2e1ce9f67.jpg) +Figure 7: (a): (Left to right) with an optimized initial smoke velocity field, the fluid changes its pattern to a “Taichi" symbol. [Reproduce: python3 smoke_taichi.py] (b): Unoptimized (top three) and optimized (bottom three) waves at time step 3, 189, and 255. [Reproduce: python3 wave.py] + +Backpropagating Through Pressure Projection We followed the baseline implementation in Autograd, and used 10 Jacobi iterations for pressure projection. Technically, 10 Jacobi iterations are not sufficient to make the velocity field fully divergence-free. However, in this example, it does a decent job, and we are able to successfully backpropagate through the unrolled 10 Jacobi iterations. + +In larger-scale simulations, 10 Jacobi iterations are likely not sufficient. Assuming the Poisson solve is done by an iterative solver (e.g. multigrid preconditioned conjugate gradients, MGPCG) with 5 multigrid levels and 50 conjugate gradient iterations, then automatic differentiation will likely not be able to provide gradients with sufficient numerical accuracy across this long iterative process. The accuracy is likely worse when conjugate gradients present, as they are known to numerically drift as the number of iterations increases. In this case, the user can still use DiffTaichi to implement the forward MGPCG solver, while implementing the backward part of the Poisson solve manually, likely using adjoint methods (Errico, 1997). DiffTaichi provides “complex kernels” to override the built-in AD system, as shown in Appendix C. + +E.4 DIFFERENTIABLE HEIGHT FIELD SHALLOW WATER SIMULATOR [wave] + +We adopt the wave equation in Wang et al. (2018) to model shallow water height field evolution: + +$$ +\ddot { u } = c ^ { 2 } \nabla ^ { 2 } u + c \alpha \nabla ^ { 2 } \dot { u } , +$$ + +where $u$ is the height of shallow water, $c$ is the “speed of sound” and $\alpha$ is a damping coefficient. We use the $\dot { u }$ and $\ddot { u }$ notations for the first and second order partial derivatives of $u$ w.r.t time $t$ respectively. + +Wang et al. (2018) used the finite different time-domain (FDTD) method (Larsson & Thomée, 2008) to discretize Eqn. 1, yielding an update scheme: + +where + +$$ +\begin{array} { r l } & { \mathrm { \quad } _ { t , i , j } = 2 u _ { t - 1 , i , j } + ( c ^ { 2 } \dot { \Delta t ^ { 2 } } + c \alpha \Delta t ) ( \nabla ^ { 2 } u ) _ { t - 1 , i , j } - p _ { t - 2 , i , j } - c \alpha \Delta t ( \nabla ^ { 2 } u ) _ { t - 2 , i , j } , } \\ & { \mathrm { \quad } ( \nabla ^ { 2 } u ) _ { t , i , j } = \frac { - 4 u _ { t , i , j } + u _ { t , i , j + 1 } + u _ { t , i , j - 1 } + u _ { t , i + 1 , j } + u _ { t , i - 1 , j } } { { \Delta x ^ { 2 } } } . } \end{array} +$$ + +We implemented this wave simulator in DiffTaichi to simulate shallow water. We used a grid of resolution $1 2 8 \times 1 2 8$ and 256 time steps. The loss function is defined as + +$$ +L = \sum _ { i , j } \Delta x ^ { 2 } ( u _ { T , i , j } - \hat { u } _ { i , j } ) ^ { 2 } +$$ + +where $T$ is the final time step, and $\hat { u }$ is the target height field. 200 gradient descent iterations are then used to optimize the initial height field. We set $\hat { u }$ to be the pattern “Taichi", and Fig. 7b shows the unoptimized and optimized wave evolution. + +We set the “Taichi" symbol as the target pattern. Fig. 7b shows the unoptimized and optimized final wave patterns. More details on discretization is in Appendix E. + +# E.5 DIFFERENTIABLE MASS-SPRING SYSTEM [mass_spring] + +We extend the mass-spring system in the main text with ground collision and a NN controller. The time-of-impact fix is implemented for improved gradients. The optimization goal is to maximize the distance moved forward with 2048 time steps. We designed three mass-spring robots as shown in Fig. 8 (left). + +# E.6 DIFFERENTIABLE BILLIARD SIMULATOR [billiards] + +A differentiable rigid body simulator is built for optimizing a billiards strategy (Fig. 8, middle). We used forward Euler for the billiard ball motion and conservation of momentum and kinetic energy for collision resolution. + +![](images/f3763ea7f88df93ff82ce6e0a540a6a0205821642b84f97bf494643d6ae4ae94.jpg) +Figure 8: Left: Three mass-spring robots. The red and blue springs are actuated. A two layer NN is used as controller. [Reproduce: python3 mass_spring.py [1/2/3] train]. Middle: Optimizing billiards. The optimizer adjusts the initial position and velocity of the white ball, so that the blue ball will reach the target destination (black dot). [Reproduce: python3 billiards.py] Right: Optimizing a robot walking. The rigid robot is controlled with a NN controller and learned to walk in 20 gradient descent iterations. [Reproduce: python3 rigid_body.py] + +E.7 DIFFERENTIABLE RIGID BODY SIMULATOR [rigid_body] + +Are rigid body collisions differentiable? It is worth noting that discontinuities can happen in rigid body collisions, and at a countable number of discontinuities the objective function is nondifferentiable. However, apart from these discontinuities, the process is still differentiable almost everywhere. The situation of rigid body collision is somewhat similar to the “ReLU” activation function in neural networks: at point $x = 0$ , ReLU is not differentiable (although continuous), yet it is still widely adopted. The rigid body simulation cases are more complex than ReLU, as we have not only non-differentiable points, but also discontinuous points. Based on our experiments, in these impulse-based rigid body simulators, we still find the gradients useful for optimization despite the discontinuities, especially with our time-of-impact fix. + +# E.8 DIFFERENTIABLE WATER RENDERER [water_renderer] + +We implemented differentiable renderers to visualize the refracting water surfaces from wave. We use finite differences to reconstruct the water surface models based on the input height field and refract camera rays to sample the images, using bilinear interpolation for meaningful gradients. To show our system works well with other differentiable programming systems, we use an adversarial optimization goal: fool VGG-16 into thinking that the refracted squirrel image is a goldfish (Fig. 9). + +![](images/421441c6f98e423d61d35851bb943e6b3ea8b7e02cb63c2f433f525d3b714c51.jpg) +Figure 9: This three-stage program (simulation, rendering, recognition) is end-to-end differentiable. Our optimized initial water height field evolves to form a refraction pattern that perturbs the image into one that fools VGG16 $( 9 9 . 9 1 \%$ goldfish). [Reproduce: python3 water_renderer.py] + +E.9 DIFFERENTIABLE VOLUME RENDERER [volume_renderer] + +We implemented a basic volume renderer that simply uses ray marching (we ignore light, scattering, etc.) to integrate a density field over each camera ray. In this task, we render a number of target images from different viewpoints, with the camera rotated around the given volume. The goal is then to optimize for the density field of the volume that would produce these target images: we render candidate images from the same viewpoints and compute an L2 loss between them and the target images, before performing gradient descent on the density field (Fig. 10). Essentially, this demonstrates how to use gradients to reconstruct 3D objects out of $\mathrm { X }$ -ray photos in a brute-force manner. Other approaches to this task include algebraic reconstruction techniques (ART) (Gordon et al., 1970). + +![](images/9aa4dc3d4840c031e419f2d79b7f5a84f44d3106ce41479cbfbd783925778a30.jpg) +Figure 10: Volume rendering of bunny shaped density field. Left: 3 (of the 7) target images. Right: optimized images of the middle bunny after iteration 2, 50, 100. [Reproduce: python3 volume_renderer.py] + +E.10 DIFFERENTIABLE ELECTRIC FIELD SIMULATOR [electric] + +Recall Coulomb’s law: $\mathbf { F } = k { \frac { q _ { 1 } q _ { 2 } } { r ^ { 2 } } } { \hat { \mathbf { r } } }$ . In the right figure, there are eight electrodes carrying static charge. The red ball also carries static charge. The controller, which is a two-layer neural network, tries to manipulate the electrodes so that the red ball follows the path of the blue ball. The bigger the electrode, the more positive charge it carries. + +# F FIXING GRADIENTS WITH TIME OF IMPACT AND CONTINUOUS COLLISION DETECTION + +Here is a naive time integrator in the mass-spring system example: + +@ti.kernel +def advance(t: ti.i32): for $\dot { 7 }$ in range(n_objects): s $=$ math.exp(-dt $\star$ damping) new $\underline { { \boldsymbol { \mathsf { U } } } } \mathrm { ~ \boldsymbol { \mathsf { U } } ~ } = \mathrm { ~ \boldsymbol { \mathsf { S } } ~ } \star$ v[t - 1, i] $^ +$ dt $\star$ gravity $\star$ ti.Vector([0.0, 1.0]) ol $\mathsf { I } \_ { \mathsf { X } } \ = \ \mathsf { x } [ \mathsf { t } \ \textrm { - } \ \mathsf { 1 }$ , i] depth $=$ old_x[1] - ground_height if depth $< ~ \Theta$ and new_ $\iota [ 1 ] ~ < ~ \mathfrak { O }$ : # assuming a sticky ground (infinite coefficient of friction) new_v[0] $= \cdot$ 0 new_v[1] $=$ 0 # Without considering time of impact, we assume the whole dt uses new_v new_x $=$ old_x + dt \* new_v v[t, i] $=$ new_v x[t, i] $=$ new_x + +Implementing TOI in this system is relative straightforward: + +@ti.kernel +def advance_toi(t: ti.i32): for i in range(n_objects): $\qquad \mathsf { s } \quad \mathsf { = }$ math.exp(-dt $\star$ damping) old_v = s \* v[t - 1, i] $^ +$ dt $\star$ gravity $\star$ ti.Vector([0.0, 1.0]) old_x $=$ x[t - 1, i] new_x $=$ old_x $^ +$ dt $\star$ old_v toi $\mathbf { \xi } = \mathbf { \xi } \odot . \Theta$ new_v $=$ old_v if new $\_ x [ 1 ] \ <$ ground_height and old_v[1] < -1e-4: # The 1e-4 safe guard is important for numerical stability toi $=$ -(old_x[1] - ground_height) / old_v[1] # Compute the time of impact new_ $. { } v = { }$ ti.Vector([0.0, 0.0]) # Note that with time of impact, dt is divided into two parts, # the first part using old_v, and second part using new_v new $\underline { { \boldsymbol { \mathsf { X } } } } \ = \ \mathsf { o l d } _ { - } \mathsf { { X } } \ +$ toi $\star$ old_v $^ +$ (dt - toi) $\star$ new_v v[t, i] $=$ new_v x[t, i] $=$ new_x In rigid body simulation, the implementation follows the same idea yet is slightly more complex. +Please refer to rigid_body.py for more details. + +# G ADDITIONAL TIPS ON GRADIENT BEHAVIORS + +Initialization matters: flat lands and local minima in physical processes A trivial example of objective flat land is in billiards. Without proper initialization, gradient descent will make no progress since gradients are zero (Fig. 11). Also note the local minimum near $\left( - 5 , 0 . 0 3 \right)$ . + +![](images/6abbebd93d07e719391b6547b105423faccf65fd06c9d389daa30882fcaf7f37.jpg) +Figure 11: Left: Scanning initial velocity in the billiard example. Middle: Most initial angles yield a flat objective (final distance between the blue ball and black destination) of 0.065, since the white ball does not collide with any other balls and imposes no effect on the pink ball via the chain reaction. Right: A zoomed-in view of the middle figure. The complex collisions lead to a lot of local minimums. [Reproduce: python3 billiards.py 1.0/0.23] + +In mass_spring and rigid_body, once the robot falls down, gradient descent will quickly become trapped. A robot on the ground will make no further progress, no matter how it changes its controller. This leads to a more non-trivial local minimum and zero gradient case. + +Ideal physical models are only “ideal”: discontinuities and singularities Real-world macroscopic physical processes are usually continuous. However, building upon ideal physical models, even in the forward physical simulation results can contain discontinuities. For example, in a rigid body model with friction, changing the initial rotation of the box can lead to different corners hitting the ground first, and result in a discontinuity (Fig. 12). In electric and mass_spring, due to the $\textstyle { \frac { 1 ^ { - } } { r ^ { 2 } } }$ and $\textstyle { \frac { 1 } { r } }$ terms, when $r 0$ , gradients can be very inaccurate due to numerical precision issues. Note that $\dot { d } ( 1 / r ) / d r = - 1 / r ^ { 2 }$ , and the gradient is more numerically problematic than the primal for a small $r$ . Safeguarding $r$ is critically important for gradient stability. + +![](images/72565f7978d7888570984925767043a72fe68b5f1e827cd467d3ff4fe1b285f9.jpg) +Figure 12: Friction in rigid body with collision is a common source of discontinuity. In this scene a rigid body hits the ground. Slightly rotating the rigid body changes which corner (A/B) hits the ground first, and different normal/friction impulses will be applied to the rigid body. This leads to a discontinuity in its final position $\mathrm { l o s s } \mathrm { = }$ final y coordinate). [Reproduce: python3 rigid_body_discontinuity.py] Please see our supplemental video for more details. \ No newline at end of file diff --git a/parse/train/B1eB5xSFvr/B1eB5xSFvr_content_list.json b/parse/train/B1eB5xSFvr/B1eB5xSFvr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d6bfd92bceb90209f6640cea9b254356babe7c18 --- /dev/null +++ b/parse/train/B1eB5xSFvr/B1eB5xSFvr_content_list.json @@ -0,0 +1,2314 @@ +[ + { + "type": "text", + "text": "DIFFTAICHI: DIFFERENTIABLE PROGRAMMING FORPHYSICAL SIMULATION", + "text_level": 1, + "bbox": [ + 174, + 98, + 795, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yuanming $\\mathbf { H } \\mathbf { u } ^ { \\dag }$ , Luke Anderson†, Tzu-Mao $\\mathbf { L i } ^ { * }$ , Qi $\\mathbf { S u n } ^ { \\ddagger }$ , Nathan $\\mathbf { C a r r } ^ { \\dagger }$ , \nJonathan Ragan-Kelley∗, Frédo Durand† \n†MIT CSAIL {yuanming,lukea,fredo}@mit.edu \n‡Adobe Research {qisu,ncarr}@adobe.com \n∗UC Berkeley {tzumao,jrk}@berkeley.edu ", + "bbox": [ + 183, + 167, + 684, + 243 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 280, + 544, + 295 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present DiffTaichi, a new differentiable programming language tailored for building high-performance differentiable physical simulators. Based on an imperative programming language, DiffTaichi generates gradients of simulation steps using source code transformations that preserve arithmetic intensity and parallelism. A light-weight tape is used to record the whole simulation program structure and replay the gradient kernels in a reversed order, for end-to-end backpropagation. We demonstrate the performance and productivity of our language in gradient-based learning and optimization tasks on 10 different physical simulators. For example, a differentiable elastic object simulator written in our language is $4 . 2 \\times$ shorter than the hand-engineered CUDA version yet runs as fast, and is $1 8 8 \\times$ faster than the TensorFlow implementation. Using our differentiable programs, neural network controllers are typically optimized within only tens of iterations. ", + "bbox": [ + 232, + 311, + 764, + 491 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/31d717138f86fae191183842f0f62dd34da1fdd52913d021ce76254056e3dfcd.jpg", + "image_caption": [ + "Figure 1: Left: Our language allows us to seamlessly integrate a neural network (NN) controller and a physical simulation module, and update the weights of the controller or the initial state parameterization (blue). Our simulations typically have $5 1 2 \\sim 2 0 4 8$ time steps, and each time step has up to one thousand parallel operations. Right: 10 differentiable simulators built with DiffTaichi. " + ], + "image_footnote": [], + "bbox": [ + 173, + 551, + 823, + 718 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Differentiable physical simulators are effective components in machine learning systems. For example, de Avila Belbute-Peres et al. (2018a) and Hu et al. (2019b) have shown that controller optimization with differentiable simulators converges one to four orders of magnitude faster than model-free reinforcement learning algorithms. The presence of differentiable physical simulators in the inner loop of these applications makes their performance vitally important. Unfortunately, using existing tools it is difficult to implement these simulators with high performance. ", + "bbox": [ + 174, + 804, + 823, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present DiffTaichi, a new differentiable programming language for high performance physical simulations on both CPU and GPU. It is based on the Taichi programming language (Hu et al., ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2019a). The DiffTaichi automatic differentiation system is designed to suit key language features required by physical simulation, yet often missing in existing differentiable programming tools, as detailed below: ", + "bbox": [ + 176, + 103, + 821, + 145 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Megakernels Our language uses a “megakernel” approach, allowing the programmer to naturally fuse multiple stages of computation into a single kernel, which is later differentiated using source code transformations and just-in-time compilation. Compared to the linear algebra operators in TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2017), DiffTaichi kernels have higher arithmetic intensity and are therefore more efficient for physical simulation tasks. ", + "bbox": [ + 174, + 166, + 825, + 236 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Imperative Parallel Programming In contrast to functional array programming languages that are popular in modern deep learning (Bergstra et al., 2010; Abadi et al., 2016; Li et al., 2018b), most traditional physical simulation programs are written in imperative languages such as Fortran and $\\mathrm { C } { + } { + }$ . DiffTaichi likewise adopts an imperative approach. The language provides parallel loops and control flows (such as “if” statements), which are widely used constructs in physical simulations: they simplify common tasks such as handling collisions, evaluating boundary conditions, and building iterative solvers. Using an imperative style makes it easier to port existing physical simulation code to DiffTaichi. ", + "bbox": [ + 174, + 257, + 825, + 367 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Flexible Indexing Existing parallel differentiable programming systems provide element-wise operations on arrays of the same shape, e.g. c[i, j] $=$ a[i, j] + b[i, j]. However, many physical simulation operations, such as numerical stencils and particle-grid interactions are not elementwise. Common simulation patterns such as y[p[i] $\\star \\ 2$ , $\\mathbf { j } ] \\ = \\ \\times [ \\mathbf { q } [ \\mathbf { j } + \\mathbf { j } ] ]$ can only be expressed with unintuitive scatter/gather operations in these existing systems, which are not only inefficient but also hard to develop and maintain. On the other hand, in DiffTaichi, the programmer directly manipulates array elements via arbitrary indexing, thus allowing partial updates of global arrays and making these common simulation patterns naturally expressible. The explicit indexing syntax also makes it easy for the compiler to perform access optimizations (Hu et al., 2019a). ", + "bbox": [ + 174, + 388, + 825, + 513 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The three requirements motivated us to design a tailored two-scale automatic differentiation system, which makes DiffTaichi especially suitable for developing complex and high-performance differentiable physical simulators, possibly with neural network controllers (Fig. 1, left). Using our language, we are able to quickly implement and automatically differentiate 10 physical simulators1, covering rigid bodies, deformable objects, and fluids (Fig. 1, right). A comprehensive comparison between DiffTaichiand other differentiable programming tools is in Appendix A. ", + "bbox": [ + 174, + 521, + 825, + 604 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND: THE TAICHI PROGRAMMING LANGUAGE ", + "text_level": 1, + "bbox": [ + 176, + 630, + 676, + 646 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "DiffTaichi is based on the Taichi programming language (Hu et al., 2019a). Taichi is an imperative programming language embedded in $\\mathrm { C } { + } { + } 1 4$ . It delivers both high performance and high productivity on modern hardware. The key design that distinguishes Taichi from other imperative programming languages such as $\\mathrm { C + + / C U D A }$ is the decoupling of computation from data structures. This allows programmers to easily switch between different data layouts and access data structures with indices (i.e. $\\times [ \\mathfrak { i } , \\ \\mathfrak { j } , \\ \\mathsf { k } ] ,$ , as if they are normal dense arrays, regardless of the underlying layout. The Taichi compiler then takes both the data structure and algorithm information to apply performance optimizations. Taichi provides “parallel-for\" loops as a first-class construct. These designs make Taichi especially suitable for writing high-performance physical simulators. For more details, readers are referred to Hu et al. (2019a). ", + "bbox": [ + 174, + 665, + 825, + 804 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The DiffTaichi language frontend is embedded in Python, and a Python AST transformer compiles DiffTaichi code to Taichi intermediate representation (IR). Unlike Python, the DiffTaichi language is compiled, statically-typed, parallel, and differentiable. We extend the Taichi compiler to further compile and automatically differentiate the generated Taichi IR into forward and backward executables. ", + "bbox": [ + 174, + 810, + 823, + 880 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We demonstrate the language using a mass-spring simulator, with three springs and three mass points, as shown right. In this section we introduce the forward simulator using the DiffTaichi frontend of Taichi, which is an easier-to-use wrapper of the Taichi $\\mathrm { C } { + } { + } 1 4$ frontend. ", + "bbox": [ + 176, + 103, + 741, + 159 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Allocating Global Variables Firstly we allocate a set of global tensors to store the simulation state. These tensors include a scalar loss of type float32, 2D tensors $\\times , ~ \\lor$ , force of size steps $\\times { \\mathsf n _ { - } }$ springs and type float $3 2 \\times 2$ , and 1D arrays of size n_spring for spring properties: spring_anchor_a (int32), spring_anchor_b (int32), spring_length (float32). ", + "bbox": [ + 174, + 183, + 823, + 239 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Defining Kernels A mass-spring system is modeled by Hooke’s law $\\textbf { F } = \\ k ( \\| \\mathbf { x } _ { a } - \\mathbf { x } _ { b } \\| _ { 2 } \\ - $ $l _ { 0 } ) \\frac { \\mathbf { x } _ { a } - \\mathbf { x } _ { b } } { \\lVert \\mathbf { x } _ { a } - \\mathbf { x } _ { b } \\rVert _ { 2 } }$ where $k$ is the spring stiffness, $\\mathbf { F }$ is spring force, $\\mathbf { x } _ { a }$ and $\\mathbf { x } _ { b }$ are the positions of two mass points, and $l _ { 0 }$ is the rest length. The following kernel loops over all the springs and scatters forces to mass points: ", + "bbox": [ + 173, + 253, + 825, + 316 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "@ti.kernel \ndef apply_spring_force(t: ti.i32): # Kernels can have parameters. Here t is a parameter with type int32. for i in range(n_springs): # A parallel for, preferably on GPU a, $\\textrm { b } =$ spring_anchor_a[i], spring_anchor_b[i] $\\times \\_ a$ , ${ \\bf \\sf x _ { - } b } ~ = ~ { \\bf \\sf x _ { \\bar { \\tau } } t } ~ - ~ { \\bf \\epsilon } _ { 1 }$ , a], $\\times \\left[ { \\ t { \\mathrm { ~ \\ - ~ } } 1 } \\right.$ , b] dist $= \\mathbf { \\nabla } \\times \\_ \\mathsf { a } - \\mathbf { \\nabla } \\times \\_ \\mathsf { b }$ length $=$ dist.norm() + 1e-4 $\\begin{array} { r l } { \\mathsf { F } } & { { } = } \\end{array}$ (length - spring_length[i]) $\\star$ spring_stiffness $\\star$ dist / length # Apply spring impulses to mass points. force[t, a] $+ = - \\mathsf { F } \\sharp \\mathsf { \\Gamma } ^ { \\prime \\prime } { + } = { \\mathrm { \\Omega } } ^ { \\prime \\prime }$ is atomic by default force[t, b] $\\mathrm { \\Sigma } + + + \\mathrm { \\Sigma } \\mathsf { F }$ ", + "bbox": [ + 176, + 340, + 665, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For each particle $i$ , we use semi-implicit Euler time integration with damping: $\\begin{array} { r } { { \\pmb v } _ { t , i } = e ^ { - \\Delta t \\alpha } { \\pmb v } _ { t - 1 , i } + } \\end{array}$ $\\begin{array} { r } { \\frac { \\Delta t } { m _ { i } } { \\bf F } _ { t , i } , { \\bf x } _ { t , i } = { \\bf x } _ { t - 1 , i } + \\Delta t { \\bf v } _ { t , i } } \\end{array}$ , where $\\pmb { v } _ { t , i } , \\mathbf { x } _ { t , i } , m _ { i }$ are the velocity, position and mass of particle $i$ at time step $t$ , respectively. $\\alpha$ is a damping factor. The kernel is as follows: ", + "bbox": [ + 174, + 482, + 823, + 529 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "@ti.kernel \ndef time_integrate(t: ti.i32): for i in range(n_objects): $\\qquad \\mathsf { s } \\quad \\mathsf { = }$ math.exp(-dt $\\star$ damping) $\\#$ Compile-time evaluation since dt and damping are constants $\\mathsf { v } [ \\mathsf { t } , \\mathsf { ~ \\mathsf { ~ \\mathsf { \\Sigma } ~ } } _ { } { \\mathsf { \\Sigma } } ] = \\mathsf { s } \\star \\mathsf { v } [ \\mathsf { t } - \\mathsf { ~ \\mathsf { \\Sigma } ~ } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } \\mathsf { i } ] + \\mathsf { d } \\mathsf { t } \\star \\mathsf { \\Sigma } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } _ { } { \\mathsf { \\Sigma } } _ { }$ force[t, i] / mass $\\#$ mass $= ~ 1$ in this example x[t, $\\dot { \\bf ~ l } ] ~ = ~ { \\bf \\nabla } \\times [ \\dot { \\bf t } ~ - ~ { \\bf 1 }$ , i] $^ +$ dt $\\star$ v[t, i] ", + "bbox": [ + 176, + 532, + 818, + 598 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Assembling the Forward Simulator With these components, we define the forward time integration: ", + "bbox": [ + 173, + 619, + 821, + 648 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "def forward(): for t in range(1, steps): apply_spring_force(t) time_integrate(t) ", + "bbox": [ + 176, + 654, + 364, + 696 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 AUTOMATICALLY DIFFERENTIATING PHYSICAL SIMULATORS IN TAICHI", + "text_level": 1, + "bbox": [ + 169, + 724, + 803, + 741 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The main goal of DiffTaichi’s automatic differentiation (AD) system is to generate gradient simulators automatically with minimal code changes to the traditional forward simulators. ", + "bbox": [ + 171, + 755, + 821, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Design Decision Source Code Transformation (SCT) (Griewank & Walther, 2008) and Tracing (Wengert, 1964) are common choices when designing AD systems. In our setting, using SCT to differentiate a whole simulator with thousands of time steps, results in high performance yet poor flexibility and long compilation time. On the other hand, naively adopting tracing provides flexibility yet poor performance, since the “megakernel\" structure is not preserved during backpropagation. To get both performance and flexibility, we developed a two-scale automatic differentiation system (Figure 2): we use SCT for differentiating within kernels, and use a light-weight tape that only stores function pointers and arguments for end-to-end simulation differentiation. The global tensors are natural checkpoints for gradient evaluation. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/d619c6452f4310c09e17907c3247b30a16687a9cc6a3b7a8d7f36ee4d41df87c.jpg", + "image_caption": [ + "Figure 2: Left: The DiffTaichi system. We reuse some infrastructure (white boxes) from Taichi, while the blue boxes are our extensions for differentiable programming. Right: The tape records kernel launches and replays the gradient kernels in reverse order during backpropagation. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 825, + 181 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assumption Unlike functional programming languages where immutable output buffers are generated, imperative programming allows programmers to freely modify global tensors. To make automatic differentiation well-defined under this setting, we make the following assumption on imperative kernels: ", + "bbox": [ + 174, + 266, + 823, + 323 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Global Data Access Rules: ", + "text_level": 1, + "bbox": [ + 183, + 333, + 369, + 347 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1) If a global tensor element is written more than once, then starting from the second write, the write must come in the form of an atomic add (“accumulation”). 2) No read accesses happen to a global tensor element, until its accumulation is done. ", + "bbox": [ + 181, + 337, + 818, + 390 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In forward simulators, programmers may make subtle changes to satisfy the rules. For instance, in the mass-spring simulation example, we record the whole history of $\\times$ and $\\vee$ , instead of keeping only the latest values. The memory consumption issues caused by this can be alleviated via checkpointing, as discussed later in Appendix D. ", + "bbox": [ + 174, + 398, + 825, + 455 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "With these assumptions, kernels will not overwrite the outputs of each other, and the goal of AD is clear: given a primal kernel $f$ that takes as input $X _ { 1 } , X _ { 2 } , \\ldots , X _ { n }$ and outputs (or accumulates to) $Y _ { 1 } , Y _ { 2 } , \\dots , Y _ { m }$ , the generated gradient (adjoint) kernel $f ^ { * }$ should take as input $X _ { 1 } , X _ { 2 } , \\ldots , X _ { n }$ and $Y _ { 1 } ^ { * }$ $^ { \\prime * } _ { 1 } , Y _ { 2 } ^ { * } , \\ldots , Y _ { m } ^ { * }$ and accumulate gradient contributions to $X _ { 1 } ^ { * } , X _ { 2 } ^ { * } , \\ldots , X _ { m } ^ { * }$ , where each $X _ { i } ^ { * }$ is an adjoint of $X _ { i }$ , i.e. $\\partial ( \\mathrm { I o s s } ) / \\partial X _ { i }$ . ", + "bbox": [ + 173, + 462, + 825, + 531 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Storage Control of Adjoint Tensors Users can specify the storage of adjoint tensors using the Taichi data structure description language (Hu et al., 2019a), as if they are primal tensors. We also provide ti.root.lazy_grad() to automatically place the adjoint tensors following the layout of their primals. ", + "bbox": [ + 173, + 553, + 825, + 608 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 LOCAL AD: DIFFERENTIATING TAICHI KERNELS USING SOURCE CODE TRANSFORMS ", + "text_level": 1, + "bbox": [ + 174, + 631, + 808, + 646 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A typical Taichi kernel consists of multiple levels of for loops and a body block. To make later AD easier, we introduce two basic code transforms to simplify the loop body, as detailed below. ", + "bbox": [ + 173, + 660, + 823, + 689 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/1b4c20ad358307d39d56d6c1309b5853a1902c2d261bda590bc40d6f691560eb.jpg", + "image_caption": [ + "Figure 3: Simple IR preprocessing before running the AD source code transform (left to right). Demonstrated in ${ \\mathsf { C } } { + } { + }$ . The actual Taichi IR is often more complex. Containing loops are ignored. " + ], + "image_footnote": [], + "bbox": [ + 183, + 717, + 813, + 776 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Flatten Branching In physical simulation branches are common, e.g., when implementing boundary conditions and collisions. To simplify the reverse-mode AD pass, we first replace “if” statements with ternary operators select(cond, value_if_true, value_if_false), whose gradients are clearly defined (Fig. 3, middle). This is a common transformation in program vectorization (e.g. Karrenberg & Hack (2011); Pharr & Mark (2012)). ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Eliminate Mutable Local Variables After removing branching, we end up with straight-line loop bodies. To further simplify the IR and make the procedure truly single-assignment, we apply a series of local variable store forwarding transforms, until the mutable local variables can be fully eliminated (Fig. 3, right). ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "After these two custom IR simplification transforms, DiffTaichi only has to differentiate the straightline code without mutable variables, which it achieves with reverse-mode AD, using a standard source code transformation (Griewank & Walther, 2008). More details on this transform are in Appendix B. ", + "bbox": [ + 174, + 166, + 825, + 222 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Loops Most loops in physical simulation are parallel loops, and during AD we preserve the parallel loop structures. For loops that are not explicitly marked as parallel, we reverse the loop order during AD transforms. We do not support loops that carry a mutating local variable since that would require a complex and costly run-time stack to maintain the history of local variables. Instead, users are instructed to employ global variables that satisfy the global data access rules. ", + "bbox": [ + 174, + 251, + 825, + 320 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Parallelism and Thread Safety For forward simulation, we inherit the “parallel-for\" construct from Taichi to map each loop iteration onto CPU/GPU threads. Programmers use atomic operations for thread safety. Our system can automatically differentiate these atomic operations. Gradient contributions in backward kernels are accumulated to the adjoint tensors via atomic adds. ", + "bbox": [ + 174, + 349, + 825, + 405 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 GLOBAL AD: END-TO-END BACKPROPAGATION USING A LIGHT-WEIGHT TAPE", + "text_level": 1, + "bbox": [ + 173, + 435, + 764, + 449 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We construct a tape (Fig. 2, right) of the kernel execution so that gradient kernels can be replayed in a reversed order. The tape is very light-weight: since the intermediate results are stored in global tensors, during forward simulation the tape only records kernel names and the (scalar) input parameters, unlike other differentiable functional array systems where all the intermediate buffers have to be recorded by the tape. Whenever a DiffTaichi kernel is launched, we append the kernel function pointer and parameters to the tape. When evaluating gradients, we traverse the reversed tape, and invoke the gradient kernels with the recorded parameters. Note that DiffTaichi AD is evaluating gradients with respect to input global tensors instead of the input parameters. ", + "bbox": [ + 174, + 465, + 825, + 578 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Learning/Optimization with Gradients Now we revisit the mass-spring example and make it differentiable for optimization. Suppose the goal is to optimize the rest lengths of the springs so that the triangle area formed by the three springs becomes 0.2 at the end of the simulation. We first define the loss function: ", + "bbox": [ + 174, + 606, + 825, + 661 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/d0cff8a0fd4c191b5c98eac383ef9116460f5f43114bbe3a5b620479de17ef41.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 720, + 690, + 799, + 751 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Goal: Adjust the spring \nrest lengths,so that thisarea=0.2 \nafter 1024 time steps \n(initialarea $\\mathbf { \\tau } = \\mathbf { 0 . 0 0 5 }$ ） ", + "bbox": [ + 715, + 752, + 813, + 809 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The programmer uses ti.Tape to memorize forward kernel launches. It automatically replays the gradients of these kernels in reverse for backpropagation. Initially the springs have lengths [0.1, 0.1, 0.14], and after optimization the rest lengths are [0.600, 0.600, 0.529]. This means the springs will expand the triangle according to Hooke’s law and form a larger triangle: [Reproduce: mass_spring_simple.py] ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/f162e45e31d7137450ef8894d248c79886fa27ae9df52a7e2dac14f42c14ae6e.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 176, + 111, + 789, + 228 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Complex Kernels Sometimes the user may want to override the gradients provided by the compiler. For example, when differentiating a 3D singular value decomposition done with an iterative solver, it is better to use a manually engineered SVD derivative subroutine for better stability. We provide two more decorators ti.complex_kernel and ti.complex_kernel_grad to overwrite the default automatic differentiation, as detailed in Appendix C. Apart from custom gradients, complex kernels can also be used to implement checkpointing, as detailed in Appendix D. ", + "bbox": [ + 173, + 261, + 825, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 364, + 315, + 381 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate DiffTaichi on 10 different physical simulators covering large-scale continuum and small-scale rigid body simulations. All results can be reproduced with the provided script. The dynamic/optimization processes are visualized in the supplemental video. In this section we focus our discussions on three simulators. More details on the simulators are in Appendix E. ", + "bbox": [ + 174, + 395, + 825, + 452 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 DIFFERENTIABLE CONTINUUM MECHANICS FOR ELASTIC OBJECTS [diffmpm] ", + "text_level": 1, + "bbox": [ + 174, + 468, + 750, + 483 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "First, we build a differentiable continuum simulation for soft robotics applications. The physical system is governed by momentum and mass conservation, i.e. $\\begin{array} { r } { \\rho \\frac { D \\mathbf { v } } { D t } = \\nabla \\cdot \\boldsymbol { \\sigma } + \\rho \\mathbf { g } } \\end{array}$ , $\\begin{array} { r } { \\frac { D \\rho } { D t } + \\rho \\nabla \\cdot \\mathbf { v } = } \\end{array}$ 0. We follow ChainQueen’s implementation (Hu et al., 2019b) and use the moving least squares material point method (Hu et al., 2018) to simulate the system. We were able to easily translate the original CUDA simulator into DiffTaichi syntax. Using this simulator and an open-loop controller, we can easily train a soft robot to move forward (Fig. 1, diffmpm). ", + "bbox": [ + 173, + 494, + 825, + 580 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Performance and Productivity Compared with manual gradient implementations in (Hu et al., 2019b), getting gradients in DiffTaichi is effortless. As a result, the DiffTaichi implementation is $4 . 2 \\times$ shorter in terms of lines of code, and runs almost as fast; compared with TensorFlow, DiffTaichi code is $1 . 7 \\times$ shorter and $1 8 8 \\times$ faster (Table 1). The Tensorflow implementation is verbose due to the heavy use of tf.gather_nd/scatter_nd and array transposing and broadcasting. ", + "bbox": [ + 173, + 594, + 825, + 665 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/97899956ba3a9476dce612dd80f229b3d181b0a2f5b855a7a2faa40e9e7ce631.jpg", + "table_caption": [ + "Table 1: diffmpm performance comparison on an NVIDIA GTX 1080 Ti GPU. We benchmark in 2D using 6.4K particles. For the lines of code, we only include the essential implementation, excluding boilerplate code. [Reproduce: python3 diffmpm_benchmark.py] " + ], + "table_footnote": [], + "table_body": "
ApproachForward TimeBackward TimeTotal Time# Lines of Code
TensorFlow13.20 ms35.70 ms48.90 ms (188.×)190
CUDA0.10 ms0.14 ms0.24 ms (0.92×)460
DiffTaichi0.11 ms0.15 ms0.26 ms (1.00×)110
", + "bbox": [ + 199, + 732, + 795, + 805 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 DIFFERENTIABLE INCOMPRESSIBLE FLUID SIMULATOR [smoke] ", + "text_level": 1, + "bbox": [ + 176, + 828, + 650, + 843 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We implemented a smoke simulator (Fig. 1, smoke) with semi-Lagrangian advection (Stam, 1999) and implicit pressure projection, following the example in Autograd (Maclaurin et al., 2015). Using gradient descent optimization on the initial velocity field, we are able to find a velocity field that changes the pattern of the fluid to a target image (Fig. 7a in Appendix). We compare the performance of our system against PyTorch, Autograd, and JAX in Table 2. Note that as an example from the ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/309c995483181811a5f14bf06071c17d957ae26e5fe665404c29cbfea4f106c7.jpg", + "table_caption": [ + "Table 2: smoke benchmark against Autograd, PyTorch, and JAX. We used a $1 1 0 ~ \\times ~ 1 1 0$ grid and 100 time steps, each with 6 Jacobi pressure projections. [Reproduce: python3 smoke_[autograd/pytorch/jax/taichi_cpu/taichi_gpu].py]. Note that the Autograd program uses float64 precision, which approximately doubles the run time. " + ], + "table_footnote": [], + "table_body": "
ApproachForward TimeBackward TimeTotal Time#Essential LoC
PyTorch (CPU, f32)405 ms328 ms733 ms (13.8×)74
PyTorch (GPU, f32)254 ms457 ms711 ms (13.4×)74
Autograd (CPU, f64)307 ms1197 ms1504 ms (28.4×)51
JAX (GPU, f32)24 ms75 ms99 ms (1.9×)90
DiffTaichi (CPU, f32)66 ms132 ms198 ms (3.7x)75
DiffTaichi (GPU, f32)24 ms29 ms53 ms (1.0×)75
", + "bbox": [ + 173, + 167, + 830, + 282 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Autograd library, this grid-based simulator is intentionally simplified to suit traditional array-based programs. For example, a periodic boundary condition is used so that Autograd can represent it using numpy.roll, without any branching. Still, Taichi delivers higher performance than these arraybased systems. The whole program takes 10 seconds to run in DiffTaichi on a GPU, and 2 seconds are spent on JIT. JAX JIT compilation takes 2 minutes. ", + "bbox": [ + 174, + 310, + 825, + 380 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 DIFFERENTIABLE RIGID BODY SIMULATORS [rigid_body] ", + "text_level": 1, + "bbox": [ + 174, + 401, + 602, + 415 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We built an impulse-based (Catto, 2009) differentiable rigid body simulator (Fig. 1, rigid_body) for optimizing robot controllers. This simulator supports rigid body collision and friction, spring forces, joints, and actuation. The simulation is end-to-end differentiable except for a countable number of discontinuities. Interestingly, although the forward simulator works well, naively differentiating it with DiffTaichi leads to completely misleading gradients, due to the rigid body collisions. We discuss the cause and solution of this issue below. ", + "bbox": [ + 174, + 426, + 825, + 511 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Improving collision gradients Consider the rigid ball example in Fig. 4 (left), where a rigid ball collides with a friction-less ground. Gravity is ignored, and due to conservation of kinetic energy the ball keeps a constant speed even after this elastic collision. ", + "bbox": [ + 174, + 523, + 825, + 565 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the forward simulation, using a small $\\Delta t$ often leads to a reasonable result, as done in many physics simulators. Lowering the initial ball height will increase the final ball height, since there is less distance to travel before the ball hits the ground and more after (see the loss curves in Fig.4, middle right). However, using a naive time integrator, no matter how small $\\Delta t$ is, the evaluated gradient of final height w.r.t. initial height will be 1 instead of $- 1$ . This counter-intuitive behavior is due to the fact that time discretization itself is not differentiated by the compiler. Fig. 4 explains this effect in greater detail. ", + "bbox": [ + 173, + 571, + 825, + 670 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/8f7ca48f4fbe280ccf625c8aaa38b729d47690edbb0c833a8083c4b877937aa9.jpg", + "image_caption": [ + "Figure 4: How gradients can go wrong with naive time integrators. For clarity we use a large $\\Delta t$ here. Left: Since collision detection only happens at multiples of $\\Delta t$ ( $2 \\Delta t$ in this case), lowering the initial position of the ball (light blue) leads to a lowered final position. Middle Left: By improving the time integrator to support continuous time of impact (TOI), collisions can be detected at any time, e.g. $1 . 9 \\Delta t$ (light red). Now the blue ball ends up higher than the green one. Middle Right: Although the two time integration techniques lead to almost identical forward results (in practice $\\Delta t$ is small), the naive time integrator gives an incorrect gradient of 1, but adding TOI yields the correct gradient. Please see our supplemental video for a better demonstration. [Reproduce: python3 rigid_body_toi.py] Right: When zooming in, the loss of the naive integrator is decreasing, and the saw-tooth pattern explains the positive gradients. [Reproduce: python3 rigid_body_toi.py zoom] " + ], + "image_footnote": [], + "bbox": [ + 179, + 681, + 815, + 771 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c11f80cbb1f40de4474cb8c3aaa6f3370d1eb31a4e45e82731191a2044cbfddb.jpg", + "image_caption": [ + "Figure 5: Adding TOI greatly improves gradient and optimization quality. Each experiment is repeated five times. [Reproduce: python3 [mass_spring/rigid_body.py] [1/2] plot && python3 plot_losses.py] " + ], + "image_footnote": [], + "bbox": [ + 189, + 102, + 808, + 207 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We propose a simple solution of adding continuous collision resolution (see, for example, Redon et al. (2002)), which considers precise time of impact (TOI), to the forward program (Fig. 4, middle left). Although it barely improves the forward simulation (Fig. 4, middle right), the gradient will be corrected effectively (Fig. 4, right). The details of continuous collision detection are in Appendix F. In real-world simulators, we find the TOI technique leads to significant improvement in gradient quality in controller optimization tasks (Fig. 5). Having TOI or not barely affects forward simulation: in the supplemental video, we show that a robot controller optimized in a simulator with TOI, actually works well in a simulator without TOI. ", + "bbox": [ + 173, + 287, + 825, + 400 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The takeaway is, differentiating physical simulators does not always yield useful gradients of the physical system being simulated, even if the simulator does forward simulation well. In Appendix G, we discuss some additional gradient issues we have encountered. ", + "bbox": [ + 174, + 406, + 825, + 449 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 462, + 344, + 478 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Differentiable programming The recent rise of deep learning has motivated the development of differentiable programming libraries for deep NNs, most notably auto-differentiation frameworks such as Theano (Bergstra et al., 2010), TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2017). However, physical simulation requires complex and customizable operations due to the intrinsic computational irregularity. Using the aforementioned frameworks, programmers have to compose these coarse-grained basic operations into desired complex operations. Doing so often leads to unsatisfactory performance. ", + "bbox": [ + 174, + 493, + 825, + 590 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Earlier work on automatic differentiation focuses on transforming existing scalar code to obtain derivatives (e.g. Utke et al. (2008), Hascoet & Pascual (2013), Pearlmutter & Siskind (2008)). A recent trend has emerged for modern programming languages to support differentiable function transformations through annotation (e.g. Innes et al. (2019), Wei et al. (2019)). These frameworks enable differentiating general programming languages, yet they provide limited parallelism. ", + "bbox": [ + 174, + 597, + 823, + 667 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Differentiable array programming languages such as Halide (Ragan-Kelley et al., 2013; Li et al., 2018b), Autograd (Maclaurin et al., 2015), JAX (Bradbury et al., 2018), and Enoki (Jakob, 2019) operate on arrays instead of scalars to utilize parallelism. Instead of operating on arrays that are immutable, DiffTaichi uses an imperative style with flexible indexing to make porting existing physical simulation algorithms easier. ", + "bbox": [ + 174, + 674, + 825, + 744 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Differentiable Physical Simulators Building differentiable simulators for robotics and machine learning has recently increased in popularity. Without differentiable programming, Battaglia et al. (2016), Chang et al. (2016) and Mrowca et al. (2018) used NNs to approximate the physical process and used the NN gradients as the approximate simulation gradients. Degrave et al. (2016) and de Avila Belbute-Peres et al. (2018b) used Theano and PyTorch respectively to build differentiable rigid body simulators. Schenck & Fox (2018) differentiates position-based fluid using custom CUDA kernels. Popovic et al. ´ (2000) used a differentiable rigid body simulator for manipulating physically based animations. The ChainQueen differentiable elastic object simulator (Hu et al., 2019b) implements forward and gradient versions of continuum mechanics in hand-written CUDA kernels, leading to performance that is two orders of magnitude higher than a pure TensorFlow implementation. Liang et al. (2019) built a differentiable cloth simulator for material estimation and motion control. The deep learning community also often incorporates differentiable rendering operations (OpenDR (Loper & Black, 2014), N3MR (Kato et al., 2018), redner (Li et al., 2018a), Mitsuba 2 (Nimier-David et al., 2019)) to learn from 3D scenes. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 152, + 318, + 169 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have presented DiffTaichi, a new differentiable programming language designed specifically for building high-performance differentiable physical simulators. Motivated by the need for supporting megakernels, imperative programming, and flexible indexing, we developed a tailored two-scale automatic differentiation system. We used DiffTaichi to build 10 simulators and integrated them into deep neural networks, which proved the performance and productivity of DiffTaichi over existing systems. 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", + "bbox": [ + 173, + 410, + 823, + 438 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A COMPARISON WITH EXISTING SYSTEMS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 545, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Workload differences between deep learning and differentiable physical simulation Existing differentiable programming tools for deep learning are typically centered around large data blobs. For example, in AlexNet, the second convolution layer has size $2 7 \\times 2 7 \\times 1 2 8 \\times 1 2 8$ . These tools usually provide users with both low-level operations such as tensor add and mul, and high-level operations such as convolution. The bottleneck of typical deep-learning-based computer vision tasks are convolutions, so the provided high-level operations, with very high arithmetic intensity2, can fully exploit hardware capability. However, the provided operations are “atoms” of these differentiable programming tools, and cannot be further customized. Users often have to use low-level operations to compose their desired high-level operations. This introduces a lot of temporary buffers, and potentially excessive GPU kernel launches. As shown in Hu et al. (2019b), a pure TensorFlow implementation of a complex physical simulator is $1 3 2 \\times$ slower than a CUDA implementation, due to excessive GPU kernel launches and the lack of producer-consumer locality3. ", + "bbox": [ + 174, + 133, + 825, + 300 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The table below compares DiffTaichi with existing tools for build differentiable physical simulators. ", + "bbox": [ + 176, + 308, + 820, + 321 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Table 3: Comparisons between DiffTaichi and other differentiable programming tools. Note that this table only discusses features related to differentiable physical simulation, and the other tools may not have been designed for this purpose. For example, PyTorch and TensorFlow are designed for classical deep learning tasks and have proven successful in their target domains. Also note that the XLA backend of TensorFlow and JIT feature of PyTorch allow them to fuse operators to some extent, but for simulation we want complete operator fusion within megakernels. “Swift” AD (Wei et al., 2019) is partially implemented as of November 2019. “Julia” refers to Innes et al. (2019). ", + "bbox": [ + 173, + 337, + 825, + 435 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/25163e378db244caad0ad719b9ab3ea29c927a4154255840dcbf0e96f2789e55.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
FeatureDiffTaichiPyTorchTensorFlowEnokiJAXHalideJuliaSwift
GPUMegakernels<A△<
Imperative Scheme【【
Parallelism?
Flexible Indexing【【
", + "bbox": [ + 176, + 448, + 825, + 523 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B DIFFERENTATING STRAIGHT-LINE TAICHI KERNELS USING SOURCE CODE TRANSFORM ", + "text_level": 1, + "bbox": [ + 174, + 550, + 772, + 584 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Primal and adjoint kernels Recall that in DiffTaichi, (primal) kernels are operators that take as input multiple tensors (e.g., $X , Y )$ and output another set of tensors. Mathematically, kernel $f$ has the form ", + "bbox": [ + 173, + 599, + 825, + 642 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/94a874ecb45bf205dfe9e9ec39d8ffd3c4dd5eafb86bc11d895d088dae461c9f.jpg", + "text": "$$\nf ( X _ { 0 } , X _ { 1 } , . . , X _ { n } ) = Y _ { 0 } , Y _ { 1 } , . . . , Y _ { m } .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 642, + 620, + 660 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Kernels usually execute uniform operations on these tensors. When it comes to differentiable programming, a loss function is defined on the final output tensors. The gradients of the loss function “ $L ^ { \\prime \\prime }$ with respect to each tensor are stored in adjoint tensors and computed via adjoint kernels. ", + "bbox": [ + 174, + 671, + 825, + 713 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The adjoint tensor of (primal) tensor $X _ { i j k }$ is denoted as $X _ { i j k } ^ { * }$ . Its entries are defined by $X _ { i j k } ^ { * } =$ $\\partial L / \\partial X _ { i j k }$ . At a high level, our automatic differentiation (AD) system transforms a primal kernel into its adjoint form. Mathematically, ", + "bbox": [ + 176, + 719, + 825, + 765 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Reverse-Mode Automatic Differentiation ", + "bbox": [ + 346, + 808, + 627, + 828 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7936100fd4cf2a40fb9a26e72743746630fb23e7b3b1a76a1db75c664209fd62.jpg", + "text": "$$\n( \\mathbf { a d j o i n t } ) f ^ { * } ( X _ { 0 } , X _ { 1 } , . . , X _ { n } , Y _ { 0 } ^ { * } , Y _ { 1 } ^ { * } , . . . , Y _ { m } ^ { * } ) = X _ { 0 } ^ { * } , X _ { 1 } ^ { * } , . . , X _ { n } ^ { * } .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 847, + 714, + 863 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Differentiating within kernels: The “make_adjoint” pass (reverse-mode AD) After the preprocessing passes, which flatten branching and eliminate mutable local variables, the “make_adjoint” pass transforms a forward evaluation (primal) kernel into its gradient accumulation (“adjoint”) kernel. It takes straight-line code directly and operates on the hierarchical intermediate representation (IR) of Taichi4 . Multiple outer for loops are allowed for the primal kernel. The Taichi compiler will distribute these parallel iterations onto CPU/GPU threads. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "During the “make_adjoint” pass, for each SSA instruction, a local adjoint variable will be allocated for gradient contribution accumulation. The compiler will traverse the statements in reverse order, and accumulate the gradients to the corresponding adjoint local variable. ", + "bbox": [ + 174, + 194, + 825, + 236 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For example, a 1D array operation $y _ { i } = \\sin x _ { i } ^ { 2 }$ has its IR representation as follows: ", + "bbox": [ + 174, + 242, + 714, + 258 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/b1e023b08421ea3290cc79e3ba96a4c4b0ba9a2925d5fa5c4858479a63511b3d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
for i ∈ range(0, 16, step 1) do
%1=load x[i]
%2 = mul %1, %1
%3= sin(%2)
store y[i] = %3
end for
", + "bbox": [ + 174, + 275, + 825, + 368 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The above primal kernel will be transformed into the following adjoint kernel: ", + "bbox": [ + 176, + 393, + 686, + 407 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/04e0c4cf294ba438bcb7f620510cc5e911d75f2d1521877e2505e8b9acbb5e7a.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
for i in range(0,16, step 1) do
/ adjoint variables
%1adj = alloca 0.0
%2adj = alloca 0.0
%3adj = alloca 0.0
// original forward computation
%1=load x[] %2 = mul %1, %1
%3= sin(%2)
/ reverse accumulation
%4 = load y_adj[i]
%3adj += %4
%5 = cos(%2)
%2adj += %3adj * %5
%1adj += 2 * %1 * %2adj
atomic add x_adj[i],%1adj end for
", + "bbox": [ + 178, + 425, + 823, + 669 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Note that for clarity the transformed code is not strictly SSA here. The actual IR has more instructions. A following simplification pass will simplify redundant instructions generated by the AD pass. ", + "bbox": [ + 173, + 694, + 825, + 736 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C COMPLEX KERNELS ", + "text_level": 1, + "bbox": [ + 176, + 757, + 379, + 773 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we demonstrated how to use complex kernels to override the automatic differentiation system. We use singular value decomposition (SVD) of $3 \\times 3$ matrices $\\mathbf { M } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { * }$ ) as an example. Fast SVD solvers used in physical simulation are often iterative, yet directly evaluate the gradient of this iterative process is likely numerically unstable. Suppose we use McAdams et al. (2011) as the forward SVD solver, and use the method in Jiang (2015) (Section 2.1.1.2) to evalute the gradients, the complex kernels are used as follows: ", + "bbox": [ + 173, + 790, + 825, + 873 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "# Do Singular Value Decomposition (SVD) on n matrices \n@ti.kernel \ndef iterative_svd(num_iterations: ti.f32): for i in range(n): input $=$ matrix_M[i] for iter in range(num_iterations): ... iteratively solve SVD using McAdams et al. 2011 matrix_U[i] $=$ ... matrix_Sigma[i] $=$ matrix_V[i] $=$ ... \n@ti.kernel \ndef svd_gradient(): for i in range(n): ... Implement, for example, section 2.1.1.2 of Jiang (2015) . \n# A complex kernel that is registered as the svd_forward complex kernel \n@ti.complex_kernel_grad(svd_forward) \ndef svd_backward(num_iterations): # differentiave SVD svd_gradient() ", + "bbox": [ + 176, + 125, + 599, + 227 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 178, + 286, + 638, + 328 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 178, + 337, + 665, + 390 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D CHECKPOINTING ", + "text_level": 1, + "bbox": [ + 176, + 417, + 352, + 434 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section we demonstrate how to use checkpointing via complex kernels. The goal of checkpointing is to use recomputation to save memory space. We demonstrate this using the diffmpm example, whose simulation cycle consists of particle to grid transform (p2g), grid boundary conditions (grid_op), and grid to particle transform $( \\mathtt { g } 2 \\mathsf { p } )$ . We assume the simulation has $O ( n )$ time steps. ", + "bbox": [ + 173, + 449, + 825, + 520 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D.1 RECOMPUTATION WITHIN TIME STEPS ", + "text_level": 1, + "bbox": [ + 176, + 535, + 483, + 550 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A naive implementation without checkpointing allocates $O ( n )$ copied of the simulation grid, which can cost a lot of memory space. Actually, if we recompute the grid states during the backward simulation time step by redoing ${ \\mathsf { p } } 2 { \\mathsf { g } }$ and grid_op, we can reused the grid states and allocate only one copy. This checkpointing optimization is demonstrated in the code below: ", + "bbox": [ + 174, + 561, + 825, + 618 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/3d12a099f276f1150b3e2e82e70dcede1056588643ada2458447347b9304ac17.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
@ti.complex_kernel
def advance(s):
clear_grid()
compute_actuation(s)
p2g(s)
grid_op()
g2p(s)
@ti.complex_kernel_grad(advance)
def advance_grad(s):
clear_grid() p2g(s)
grid_op() # recompute the grid
g2p.grad(s)
grid_op.grad()
p2g.grad(s) compute_actuation.grad(s)
", + "bbox": [ + 174, + 643, + 821, + 833 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D.2 SEGMENT-WISE RECOMPUTATION ", + "text_level": 1, + "bbox": [ + 176, + 856, + 455, + 869 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Given a simulation with $O ( n )$ time steps, if all simulation steps are recorded, the space consumption is $O ( n )$ . This linear space consumption is sometimes too large for high-resolution simulations with long time horizon. Fortunately, we can reduce the space consumption using a segment-wise checkpointing trick: We split the simulation into segments of $S$ steps, and in forward simulation store only the first simulation state in each segment. During backpropagation when we need the remaining simulation states in a segment, we recompute them based on the first state in that segment. ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 825, + 146 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Note that if the segment size is $O ( S )$ , then we only need to store $O ( n / S )$ simulation steps for checkpoints and $\\bar { O ( S ) }$ reusable simulation steps for backpropagation within segments. The total√ space consumption is √ $O ( S + n / S )$ . Setting ${ \\cal S } \\doteq { \\cal O } ( \\sqrt { n } )$ reduces memory consumption from $O ( n )$ to $O ( { \\sqrt { n } } )$ . The time complexity remains $O ( n )$ . ", + "bbox": [ + 174, + 152, + 825, + 209 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E DETAILS ON 10 DIFFERENTIABLE SIMULATORS ", + "text_level": 1, + "bbox": [ + 174, + 228, + 602, + 244 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E.1 DIFFERENTIABLE CONTINUUM MECHANICS FOR ELASTIC OBJECTS [diffmpm] ", + "bbox": [ + 176, + 260, + 750, + 275 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/981f05ea68d2115e11a3dd53e1bd3fa25dcdf8b91cb3428f0f8ba3219819063d.jpg", + "image_caption": [ + "Figure 6: Controller optimization with our differentiable continuum simulators. Left: the 2D robot with four muscles. Middle: A 3D robot with 16 muscles and 30K particles crawling on the ground. [Reproduce: python3 [diffmpm/diffmpm3d].py] Right: We couple the robot (30K particles) and the liquid simulator (13K particles), and optimize its open-loop controller in this difficult situation.[Reproduce: python3 liquid.py] " + ], + "image_footnote": [], + "bbox": [ + 191, + 291, + 820, + 347 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E.2 DIFFERENTIABLE LIQUID SIMULATOR [liquid] ", + "text_level": 1, + "bbox": [ + 178, + 455, + 532, + 469 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We follow the weakly compressible fluid model in Tampubolon et al. (2017) and implemented a 3D differentiable liquid simulator within the [diffmpm3d] framework. Our liquid simulation can be two-way coupled with elastic object simulation (Figure 6, right). ", + "bbox": [ + 174, + 481, + 825, + 522 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E.3 DIFFERENTIABLE INCOMPRESSIBLE FLUID SIMULATOR [smoke] ", + "text_level": 1, + "bbox": [ + 174, + 540, + 651, + 554 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/032fd5e7cb773f9949bec038a3eaf43d815115cd2c63f7d61cc1a2a2e1ce9f67.jpg", + "image_caption": [ + "Figure 7: (a): (Left to right) with an optimized initial smoke velocity field, the fluid changes its pattern to a “Taichi\" symbol. [Reproduce: python3 smoke_taichi.py] (b): Unoptimized (top three) and optimized (bottom three) waves at time step 3, 189, and 255. [Reproduce: python3 wave.py] " + ], + "image_footnote": [], + "bbox": [ + 179, + 570, + 818, + 693 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Backpropagating Through Pressure Projection We followed the baseline implementation in Autograd, and used 10 Jacobi iterations for pressure projection. Technically, 10 Jacobi iterations are not sufficient to make the velocity field fully divergence-free. However, in this example, it does a decent job, and we are able to successfully backpropagate through the unrolled 10 Jacobi iterations. ", + "bbox": [ + 174, + 762, + 825, + 819 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In larger-scale simulations, 10 Jacobi iterations are likely not sufficient. Assuming the Poisson solve is done by an iterative solver (e.g. multigrid preconditioned conjugate gradients, MGPCG) with 5 multigrid levels and 50 conjugate gradient iterations, then automatic differentiation will likely not be able to provide gradients with sufficient numerical accuracy across this long iterative process. The accuracy is likely worse when conjugate gradients present, as they are known to numerically drift as the number of iterations increases. In this case, the user can still use DiffTaichi to implement the forward MGPCG solver, while implementing the backward part of the Poisson solve manually, likely using adjoint methods (Errico, 1997). DiffTaichi provides “complex kernels” to override the built-in AD system, as shown in Appendix C. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E.4 DIFFERENTIABLE HEIGHT FIELD SHALLOW WATER SIMULATOR [wave] ", + "bbox": [ + 173, + 147, + 707, + 164 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We adopt the wave equation in Wang et al. (2018) to model shallow water height field evolution: ", + "bbox": [ + 171, + 174, + 802, + 189 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/07ab8b3c6aa8fcdf7f1a2381afb5d3e5df30d0b5b0a6a5fc517f9d38e009e8f7.jpg", + "text": "$$\n\\ddot { u } = c ^ { 2 } \\nabla ^ { 2 } u + c \\alpha \\nabla ^ { 2 } \\dot { u } ,\n$$", + "text_format": "latex", + "bbox": [ + 423, + 191, + 573, + 209 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $u$ is the height of shallow water, $c$ is the “speed of sound” and $\\alpha$ is a damping coefficient. We use the $\\dot { u }$ and $\\ddot { u }$ notations for the first and second order partial derivatives of $u$ w.r.t time $t$ respectively. ", + "bbox": [ + 174, + 213, + 828, + 256 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Wang et al. (2018) used the finite different time-domain (FDTD) method (Larsson & Thomée, 2008) to discretize Eqn. 1, yielding an update scheme: ", + "bbox": [ + 173, + 262, + 821, + 291 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 174, + 316, + 217, + 330 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/f09174162393aa18666d7f419aba2cada6b04a8df2c7aa4a6f0a0ab6ef5dea48.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { \\quad } _ { t , i , j } = 2 u _ { t - 1 , i , j } + ( c ^ { 2 } \\dot { \\Delta t ^ { 2 } } + c \\alpha \\Delta t ) ( \\nabla ^ { 2 } u ) _ { t - 1 , i , j } - p _ { t - 2 , i , j } - c \\alpha \\Delta t ( \\nabla ^ { 2 } u ) _ { t - 2 , i , j } , } \\\\ & { \\mathrm { \\quad } ( \\nabla ^ { 2 } u ) _ { t , i , j } = \\frac { - 4 u _ { t , i , j } + u _ { t , i , j + 1 } + u _ { t , i , j - 1 } + u _ { t , i + 1 , j } + u _ { t , i - 1 , j } } { { \\Delta x ^ { 2 } } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 290, + 772, + 357 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We implemented this wave simulator in DiffTaichi to simulate shallow water. We used a grid of resolution $1 2 8 \\times 1 2 8$ and 256 time steps. The loss function is defined as ", + "bbox": [ + 171, + 363, + 823, + 392 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/284b5abfd336d90309bc4bad734d0c252d1e0b9a090341ca6973d05ef9429c3d.jpg", + "text": "$$\nL = \\sum _ { i , j } \\Delta x ^ { 2 } ( u _ { T , i , j } - \\hat { u } _ { i , j } ) ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 401, + 407, + 596, + 443 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $T$ is the final time step, and $\\hat { u }$ is the target height field. 200 gradient descent iterations are then used to optimize the initial height field. We set $\\hat { u }$ to be the pattern “Taichi\", and Fig. 7b shows the unoptimized and optimized wave evolution. ", + "bbox": [ + 174, + 450, + 826, + 493 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We set the “Taichi\" symbol as the target pattern. Fig. 7b shows the unoptimized and optimized final wave patterns. More details on discretization is in Appendix E. ", + "bbox": [ + 174, + 500, + 823, + 530 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E.5 DIFFERENTIABLE MASS-SPRING SYSTEM [mass_spring] ", + "text_level": 1, + "bbox": [ + 174, + 545, + 596, + 560 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We extend the mass-spring system in the main text with ground collision and a NN controller. The time-of-impact fix is implemented for improved gradients. The optimization goal is to maximize the distance moved forward with 2048 time steps. We designed three mass-spring robots as shown in Fig. 8 (left). ", + "bbox": [ + 173, + 570, + 825, + 627 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E.6 DIFFERENTIABLE BILLIARD SIMULATOR [billiards] ", + "text_level": 1, + "bbox": [ + 173, + 643, + 578, + 659 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A differentiable rigid body simulator is built for optimizing a billiards strategy (Fig. 8, middle). We used forward Euler for the billiard ball motion and conservation of momentum and kinetic energy for collision resolution. ", + "bbox": [ + 174, + 670, + 825, + 712 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/f3763ea7f88df93ff82ce6e0a540a6a0205821642b84f97bf494643d6ae4ae94.jpg", + "image_caption": [ + "Figure 8: Left: Three mass-spring robots. The red and blue springs are actuated. A two layer NN is used as controller. [Reproduce: python3 mass_spring.py [1/2/3] train]. Middle: Optimizing billiards. The optimizer adjusts the initial position and velocity of the white ball, so that the blue ball will reach the target destination (black dot). [Reproduce: python3 billiards.py] Right: Optimizing a robot walking. The rigid robot is controlled with a NN controller and learned to walk in 20 gradient descent iterations. [Reproduce: python3 rigid_body.py] " + ], + "image_footnote": [], + "bbox": [ + 186, + 736, + 812, + 825 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E.7 DIFFERENTIABLE RIGID BODY SIMULATOR [rigid_body] ", + "bbox": [ + 173, + 103, + 602, + 118 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Are rigid body collisions differentiable? It is worth noting that discontinuities can happen in rigid body collisions, and at a countable number of discontinuities the objective function is nondifferentiable. However, apart from these discontinuities, the process is still differentiable almost everywhere. The situation of rigid body collision is somewhat similar to the “ReLU” activation function in neural networks: at point $x = 0$ , ReLU is not differentiable (although continuous), yet it is still widely adopted. The rigid body simulation cases are more complex than ReLU, as we have not only non-differentiable points, but also discontinuous points. Based on our experiments, in these impulse-based rigid body simulators, we still find the gradients useful for optimization despite the discontinuities, especially with our time-of-impact fix. ", + "bbox": [ + 173, + 136, + 825, + 261 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E.8 DIFFERENTIABLE WATER RENDERER [water_renderer] ", + "text_level": 1, + "bbox": [ + 173, + 294, + 584, + 308 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We implemented differentiable renderers to visualize the refracting water surfaces from wave. We use finite differences to reconstruct the water surface models based on the input height field and refract camera rays to sample the images, using bilinear interpolation for meaningful gradients. To show our system works well with other differentiable programming systems, we use an adversarial optimization goal: fool VGG-16 into thinking that the refracted squirrel image is a goldfish (Fig. 9). ", + "bbox": [ + 173, + 325, + 825, + 396 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/421441c6f98e423d61d35851bb943e6b3ea8b7e02cb63c2f433f525d3b714c51.jpg", + "image_caption": [ + "Figure 9: This three-stage program (simulation, rendering, recognition) is end-to-end differentiable. Our optimized initial water height field evolves to form a refraction pattern that perturbs the image into one that fools VGG16 $( 9 9 . 9 1 \\%$ goldfish). [Reproduce: python3 water_renderer.py] " + ], + "image_footnote": [], + "bbox": [ + 176, + 426, + 823, + 513 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E.9 DIFFERENTIABLE VOLUME RENDERER [volume_renderer] ", + "bbox": [ + 174, + 609, + 604, + 623 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We implemented a basic volume renderer that simply uses ray marching (we ignore light, scattering, etc.) to integrate a density field over each camera ray. In this task, we render a number of target images from different viewpoints, with the camera rotated around the given volume. The goal is then to optimize for the density field of the volume that would produce these target images: we render candidate images from the same viewpoints and compute an L2 loss between them and the target images, before performing gradient descent on the density field (Fig. 10). Essentially, this demonstrates how to use gradients to reconstruct 3D objects out of $\\mathrm { X }$ -ray photos in a brute-force manner. Other approaches to this task include algebraic reconstruction techniques (ART) (Gordon et al., 1970). ", + "bbox": [ + 173, + 640, + 825, + 766 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/9aa4dc3d4840c031e419f2d79b7f5a84f44d3106ce41479cbfbd783925778a30.jpg", + "image_caption": [ + "Figure 10: Volume rendering of bunny shaped density field. Left: 3 (of the 7) target images. Right: optimized images of the middle bunny after iteration 2, 50, 100. [Reproduce: python3 volume_renderer.py] " + ], + "image_footnote": [], + "bbox": [ + 210, + 792, + 802, + 866 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E.10 DIFFERENTIABLE ELECTRIC FIELD SIMULATOR [electric] ", + "bbox": [ + 174, + 103, + 627, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Recall Coulomb’s law: $\\mathbf { F } = k { \\frac { q _ { 1 } q _ { 2 } } { r ^ { 2 } } } { \\hat { \\mathbf { r } } }$ . In the right figure, there are eight electrodes carrying static charge. The red ball also carries static charge. The controller, which is a two-layer neural network, tries to manipulate the electrodes so that the red ball follows the path of the blue ball. The bigger the electrode, the more positive charge it carries. ", + "bbox": [ + 174, + 146, + 741, + 203 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "F FIXING GRADIENTS WITH TIME OF IMPACT AND CONTINUOUS COLLISION DETECTION ", + "text_level": 1, + "bbox": [ + 174, + 238, + 728, + 272 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Here is a naive time integrator in the mass-spring system example: ", + "bbox": [ + 173, + 291, + 609, + 306 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "@ti.kernel \ndef advance(t: ti.i32): for $\\dot { 7 }$ in range(n_objects): s $=$ math.exp(-dt $\\star$ damping) new $\\underline { { \\boldsymbol { \\mathsf { U } } } } \\mathrm { ~ \\boldsymbol { \\mathsf { U } } ~ } = \\mathrm { ~ \\boldsymbol { \\mathsf { S } } ~ } \\star$ v[t - 1, i] $^ +$ dt $\\star$ gravity $\\star$ ti.Vector([0.0, 1.0]) ol $\\mathsf { I } \\_ { \\mathsf { X } } \\ = \\ \\mathsf { x } [ \\mathsf { t } \\ \\textrm { - } \\ \\mathsf { 1 }$ , i] depth $=$ old_x[1] - ground_height if depth $< ~ \\Theta$ and new_ $\\iota [ 1 ] ~ < ~ \\mathfrak { O }$ : # assuming a sticky ground (infinite coefficient of friction) new_v[0] $= \\cdot$ 0 new_v[1] $=$ 0 # Without considering time of impact, we assume the whole dt uses new_v new_x $=$ old_x + dt \\* new_v v[t, i] $=$ new_v x[t, i] $=$ new_x ", + "bbox": [ + 176, + 338, + 715, + 517 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Implementing TOI in this system is relative straightforward: ", + "bbox": [ + 173, + 536, + 568, + 551 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "@ti.kernel \ndef advance_toi(t: ti.i32): for i in range(n_objects): $\\qquad \\mathsf { s } \\quad \\mathsf { = }$ math.exp(-dt $\\star$ damping) old_v = s \\* v[t - 1, i] $^ +$ dt $\\star$ gravity $\\star$ ti.Vector([0.0, 1.0]) old_x $=$ x[t - 1, i] new_x $=$ old_x $^ +$ dt $\\star$ old_v toi $\\mathbf { \\xi } = \\mathbf { \\xi } \\odot . \\Theta$ new_v $=$ old_v if new $\\_ x [ 1 ] \\ <$ ground_height and old_v[1] < -1e-4: # The 1e-4 safe guard is important for numerical stability toi $=$ -(old_x[1] - ground_height) / old_v[1] # Compute the time of impact new_ $. { } v = { }$ ti.Vector([0.0, 0.0]) # Note that with time of impact, dt is divided into two parts, # the first part using old_v, and second part using new_v new $\\underline { { \\boldsymbol { \\mathsf { X } } } } \\ = \\ \\mathsf { o l d } _ { - } \\mathsf { { X } } \\ +$ toi $\\star$ old_v $^ +$ (dt - toi) $\\star$ new_v v[t, i] $=$ new_v x[t, i] $=$ new_x In rigid body simulation, the implementation follows the same idea yet is slightly more complex. \nPlease refer to rigid_body.py for more details. ", + "bbox": [ + 176, + 561, + 736, + 771 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 790, + 821, + 819 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "G ADDITIONAL TIPS ON GRADIENT BEHAVIORS ", + "text_level": 1, + "bbox": [ + 174, + 845, + 591, + 863 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Initialization matters: flat lands and local minima in physical processes A trivial example of objective flat land is in billiards. Without proper initialization, gradient descent will make no progress since gradients are zero (Fig. 11). Also note the local minimum near $\\left( - 5 , 0 . 0 3 \\right)$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/6abbebd93d07e719391b6547b105423faccf65fd06c9d389daa30882fcaf7f37.jpg", + "image_caption": [ + "Figure 11: Left: Scanning initial velocity in the billiard example. Middle: Most initial angles yield a flat objective (final distance between the blue ball and black destination) of 0.065, since the white ball does not collide with any other balls and imposes no effect on the pink ball via the chain reaction. Right: A zoomed-in view of the middle figure. The complex collisions lead to a lot of local minimums. [Reproduce: python3 billiards.py 1.0/0.23] " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 813, + 205 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In mass_spring and rigid_body, once the robot falls down, gradient descent will quickly become trapped. A robot on the ground will make no further progress, no matter how it changes its controller. This leads to a more non-trivial local minimum and zero gradient case. ", + "bbox": [ + 174, + 313, + 825, + 356 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Ideal physical models are only “ideal”: discontinuities and singularities Real-world macroscopic physical processes are usually continuous. However, building upon ideal physical models, even in the forward physical simulation results can contain discontinuities. For example, in a rigid body model with friction, changing the initial rotation of the box can lead to different corners hitting the ground first, and result in a discontinuity (Fig. 12). In electric and mass_spring, due to the $\\textstyle { \\frac { 1 ^ { - } } { r ^ { 2 } } }$ and $\\textstyle { \\frac { 1 } { r } }$ terms, when $r 0$ , gradients can be very inaccurate due to numerical precision issues. Note that $\\dot { d } ( 1 / r ) / d r = - 1 / r ^ { 2 }$ , and the gradient is more numerically problematic than the primal for a small $r$ . Safeguarding $r$ is critically important for gradient stability. ", + "bbox": [ + 173, + 371, + 825, + 488 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/72565f7978d7888570984925767043a72fe68b5f1e827cd467d3ff4fe1b285f9.jpg", + "image_caption": [ + "Figure 12: Friction in rigid body with collision is a common source of discontinuity. In this scene a rigid body hits the ground. Slightly rotating the rigid body changes which corner (A/B) hits the ground first, and different normal/friction impulses will be applied to the rigid body. This leads to a discontinuity in its final position $\\mathrm { l o s s } \\mathrm { = }$ final y coordinate). [Reproduce: python3 rigid_body_discontinuity.py] Please see our supplemental video for more details. 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For example, a differentiable elastic object simulator written in our language", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 346, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 152, + 358 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 152, + 346, + 174, + 356 + ], + "score": 0.88, + "content": "4 . 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 346, + 469, + 358 + ], + "score": 1.0, + "content": "shorter than the hand-engineered CUDA version yet runs as fast, and is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 356, + 469, + 369 + ], + "spans": [ + { + "bbox": [ + 142, + 357, + 167, + 367 + ], + "score": 0.87, + "content": "1 8 8 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 356, + 469, + 369 + ], + "score": 1.0, + "content": "faster than the TensorFlow implementation. Using our differentiable pro-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 469, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 469, + 380 + ], + "score": 1.0, + "content": "grams, neural network controllers are typically optimized within only tens of iter-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 379, + 171, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 379, + 171, + 389 + ], + "score": 1.0, + "content": "ations.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 247, + 470, + 389 + ] + }, + { + "type": "image", + "bbox": [ + 106, + 437, + 504, + 569 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 437, + 504, + 569 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 437, + 504, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 504, + 569 + ], + "score": 0.968, + "type": "image", + "image_path": "31d717138f86fae191183842f0f62dd34da1fdd52913d021ce76254056e3dfcd.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 106, + 437, + 504, + 481.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 106, + 481.0, + 504, + 525.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 106, + 525.0, + 504, + 569.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 577, + 505, + 622 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Figure 1: Left: Our language allows us to seamlessly integrate a neural network (NN) controller", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "and a physical simulation module, and update the weights of the controller or the initial state param-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 304, + 613 + ], + "score": 1.0, + "content": "eterization (blue). Our simulations typically have", + "type": "text" + }, + { + "bbox": [ + 305, + 600, + 355, + 610 + ], + "score": 0.88, + "content": "5 1 2 \\sim 2 0 4 8", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "time steps, and each time step has up", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 610, + 481, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 481, + 622 + ], + "score": 1.0, + "content": "to one thousand parallel operations. Right: 10 differentiable simulators built with DiffTaichi.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + } + ], + "index": 23.75 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "Differentiable physical simulators are effective components in machine learning systems. For exam-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "ple, de Avila Belbute-Peres et al. (2018a) and Hu et al. (2019b) have shown that controller optimiza-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "tion with differentiable simulators converges one to four orders of magnitude faster than model-free", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "reinforcement learning algorithms. The presence of differentiable physical simulators in the inner", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "loop of these applications makes their performance vitally important. Unfortunately, using existing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 691, + 397, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 397, + 706 + ], + "score": 1.0, + "content": "tools it is difficult to implement these simulators with high performance.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 637, + 506, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We present DiffTaichi, a new differentiable programming language for high performance physical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "score": 1.0, + "content": "simulations on both CPU and GPU. It is based on the Taichi programming language (Hu et al.,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 503, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "2019a). The DiffTaichi automatic differentiation system is designed to suit key language features", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "required by physical simulation, yet often missing in existing differentiable programming tools, as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 171, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 171, + 116 + ], + "score": 1.0, + "content": "detailed below:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "Megakernels Our language uses a “megakernel” approach, allowing the programmer to naturally", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "fuse multiple stages of computation into a single kernel, which is later differentiated using source", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "code transformations and just-in-time compilation. Compared to the linear algebra operators in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 178 + ], + "score": 1.0, + "content": "TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2017), DiffTaichi kernels have higher", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 177, + 433, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 433, + 189 + ], + "score": 1.0, + "content": "arithmetic intensity and are therefore more efficient for physical simulation tasks.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "Imperative Parallel Programming In contrast to functional array programming languages that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "are popular in modern deep learning (Bergstra et al., 2010; Abadi et al., 2016; Li et al., 2018b), most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "traditional physical simulation programs are written in imperative languages such as Fortran and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 126, + 248 + ], + "score": 0.85, + "content": "\\mathrm { C } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 237, + 505, + 249 + ], + "score": 1.0, + "content": ". DiffTaichi likewise adopts an imperative approach. The language provides parallel loops and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "control flows (such as “if” statements), which are widely used constructs in physical simulations:", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "they simplify common tasks such as handling collisions, evaluating boundary conditions, and build-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "ing iterative solvers. Using an imperative style makes it easier to port existing physical simulation", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 184, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 184, + 291 + ], + "score": 1.0, + "content": "code to DiffTaichi.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "score": 1.0, + "content": "Flexible Indexing Existing parallel differentiable programming systems provide element-wise op-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 319, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 314, + 334 + ], + "score": 1.0, + "content": "erations on arrays of the same shape, e.g. c[i, j]", + "type": "text" + }, + { + "bbox": [ + 314, + 322, + 322, + 329 + ], + "score": 0.49, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 319, + 506, + 334 + ], + "score": 1.0, + "content": "a[i, j] + b[i, j]. However, many physical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "simulation operations, such as numerical stencils and particle-grid interactions are not element-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 319, + 355 + ], + "score": 1.0, + "content": "wise. Common simulation patterns such as y[p[i]", + "type": "text" + }, + { + "bbox": [ + 319, + 343, + 334, + 352 + ], + "score": 0.59, + "content": "\\star \\ 2", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 340, + 340, + 355 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 340, + 343, + 408, + 353 + ], + "score": 0.81, + "content": "\\mathbf { j } ] \\ = \\ \\times [ \\mathbf { q } [ \\mathbf { j } + \\mathbf { j } ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 340, + 505, + 355 + ], + "score": 1.0, + "content": "can only be expressed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "with unintuitive scatter/gather operations in these existing systems, which are not only inefficient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 364, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 504, + 375 + ], + "score": 1.0, + "content": "but also hard to develop and maintain. On the other hand, in DiffTaichi, the programmer directly", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "manipulates array elements via arbitrary indexing, thus allowing partial updates of global arrays and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "making these common simulation patterns naturally expressible. The explicit indexing syntax also", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 434, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 434, + 409 + ], + "score": 1.0, + "content": "makes it easy for the compiler to perform access optimizations (Hu et al., 2019a).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "The three requirements motivated us to design a tailored two-scale automatic differentiation sys-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 504, + 436 + ], + "score": 1.0, + "content": "tem, which makes DiffTaichi especially suitable for developing complex and high-performance dif-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "ferentiable physical simulators, possibly with neural network controllers (Fig. 1, left). Using our", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "score": 1.0, + "content": "language, we are able to quickly implement and automatically differentiate 10 physical simulators1,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "covering rigid bodies, deformable objects, and fluids (Fig. 1, right). A comprehensive comparison", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 431, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 431, + 480 + ], + "score": 1.0, + "content": "between DiffTaichiand other differentiable programming tools is in Appendix A.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 499, + 414, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 415, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 415, + 514 + ], + "score": 1.0, + "content": "2 BACKGROUND: THE TAICHI PROGRAMMING LANGUAGE", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "DiffTaichi is based on the Taichi programming language (Hu et al., 2019a). Taichi is an imperative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 255, + 551 + ], + "score": 1.0, + "content": "programming language embedded in", + "type": "text" + }, + { + "bbox": [ + 255, + 538, + 285, + 549 + ], + "score": 0.86, + "content": "\\mathrm { C } { + } { + } 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 537, + 506, + 551 + ], + "score": 1.0, + "content": ". It delivers both high performance and high productiv-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "ity on modern hardware. The key design that distinguishes Taichi from other imperative program-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 205, + 572 + ], + "score": 1.0, + "content": "ming languages such as", + "type": "text" + }, + { + "bbox": [ + 206, + 560, + 256, + 570 + ], + "score": 0.75, + "content": "\\mathrm { C + + / C U D A }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "is the decoupling of computation from data structures. This", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "allows programmers to easily switch between different data layouts and access data structures with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 159, + 595 + ], + "score": 1.0, + "content": "indices (i.e.", + "type": "text" + }, + { + "bbox": [ + 159, + 583, + 204, + 593 + ], + "score": 0.72, + "content": "\\times [ \\mathfrak { i } , \\ \\mathfrak { j } , \\ \\mathsf { k } ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 581, + 505, + 595 + ], + "score": 1.0, + "content": ", as if they are normal dense arrays, regardless of the underlying layout.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "The Taichi compiler then takes both the data structure and algorithm information to apply perfor-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 504, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 504, + 616 + ], + "score": 1.0, + "content": "mance optimizations. Taichi provides “parallel-for\" loops as a first-class construct. These designs", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "score": 1.0, + "content": "make Taichi especially suitable for writing high-performance physical simulators. For more details,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 269, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 269, + 638 + ], + "score": 1.0, + "content": "readers are referred to Hu et al. 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We extend the Taichi compiler to further", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "compile and automatically differentiate the generated Taichi IR into forward and backward executa-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 685, + 128, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 128, + 699 + ], + "score": 1.0, + "content": "bles.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "1Our language, compiler, and simulator code is open-source. All the results in this work can be reproduced", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 443, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 443, + 732 + ], + "score": 1.0, + "content": "by a single Python script. Visual results in this work are presented in the supplemental video.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 503, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "2019a). The DiffTaichi automatic differentiation system is designed to suit key language features", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "required by physical simulation, yet often missing in existing differentiable programming tools, as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 171, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 171, + 116 + ], + "score": 1.0, + "content": "detailed below:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 505, + 116 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "Megakernels Our language uses a “megakernel” approach, allowing the programmer to naturally", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "fuse multiple stages of computation into a single kernel, which is later differentiated using source", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "code transformations and just-in-time compilation. Compared to the linear algebra operators in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 178 + ], + "score": 1.0, + "content": "TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al., 2017), DiffTaichi kernels have higher", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 177, + 433, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 433, + 189 + ], + "score": 1.0, + "content": "arithmetic intensity and are therefore more efficient for physical simulation tasks.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 131, + 506, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "Imperative Parallel Programming In contrast to functional array programming languages that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "are popular in modern deep learning (Bergstra et al., 2010; Abadi et al., 2016; Li et al., 2018b), most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "traditional physical simulation programs are written in imperative languages such as Fortran and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 126, + 248 + ], + "score": 0.85, + "content": "\\mathrm { C } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 237, + 505, + 249 + ], + "score": 1.0, + "content": ". DiffTaichi likewise adopts an imperative approach. The language provides parallel loops and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "control flows (such as “if” statements), which are widely used constructs in physical simulations:", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "they simplify common tasks such as handling collisions, evaluating boundary conditions, and build-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "ing iterative solvers. Using an imperative style makes it easier to port existing physical simulation", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 184, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 184, + 291 + ], + "score": 1.0, + "content": "code to DiffTaichi.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 204, + 506, + 291 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "score": 1.0, + "content": "Flexible Indexing Existing parallel differentiable programming systems provide element-wise op-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 319, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 314, + 334 + ], + "score": 1.0, + "content": "erations on arrays of the same shape, e.g. c[i, j]", + "type": "text" + }, + { + "bbox": [ + 314, + 322, + 322, + 329 + ], + "score": 0.49, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 319, + 506, + 334 + ], + "score": 1.0, + "content": "a[i, j] + b[i, j]. However, many physical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "simulation operations, such as numerical stencils and particle-grid interactions are not element-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 319, + 355 + ], + "score": 1.0, + "content": "wise. Common simulation patterns such as y[p[i]", + "type": "text" + }, + { + "bbox": [ + 319, + 343, + 334, + 352 + ], + "score": 0.59, + "content": "\\star \\ 2", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 340, + 340, + 355 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 340, + 343, + 408, + 353 + ], + "score": 0.81, + "content": "\\mathbf { j } ] \\ = \\ \\times [ \\mathbf { q } [ \\mathbf { j } + \\mathbf { j } ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 340, + 505, + 355 + ], + "score": 1.0, + "content": "can only be expressed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "with unintuitive scatter/gather operations in these existing systems, which are not only inefficient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 364, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 504, + 375 + ], + "score": 1.0, + "content": "but also hard to develop and maintain. On the other hand, in DiffTaichi, the programmer directly", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "manipulates array elements via arbitrary indexing, thus allowing partial updates of global arrays and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "making these common simulation patterns naturally expressible. The explicit indexing syntax also", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 434, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 434, + 409 + ], + "score": 1.0, + "content": "makes it easy for the compiler to perform access optimizations (Hu et al., 2019a).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 307, + 506, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "The three requirements motivated us to design a tailored two-scale automatic differentiation sys-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 504, + 436 + ], + "score": 1.0, + "content": "tem, which makes DiffTaichi especially suitable for developing complex and high-performance dif-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "ferentiable physical simulators, possibly with neural network controllers (Fig. 1, left). Using our", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "score": 1.0, + "content": "language, we are able to quickly implement and automatically differentiate 10 physical simulators1,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "covering rigid bodies, deformable objects, and fluids (Fig. 1, right). A comprehensive comparison", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 431, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 431, + 480 + ], + "score": 1.0, + "content": "between DiffTaichiand other differentiable programming tools is in Appendix A.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 412, + 506, + 480 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 499, + 414, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 415, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 415, + 514 + ], + "score": 1.0, + "content": "2 BACKGROUND: THE TAICHI PROGRAMMING LANGUAGE", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "DiffTaichi is based on the Taichi programming language (Hu et al., 2019a). Taichi is an imperative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 537, + 255, + 551 + ], + "score": 1.0, + "content": "programming language embedded in", + "type": "text" + }, + { + "bbox": [ + 255, + 538, + 285, + 549 + ], + "score": 0.86, + "content": "\\mathrm { C } { + } { + } 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 537, + 506, + 551 + ], + "score": 1.0, + "content": ". It delivers both high performance and high productiv-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "ity on modern hardware. The key design that distinguishes Taichi from other imperative program-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 205, + 572 + ], + "score": 1.0, + "content": "ming languages such as", + "type": "text" + }, + { + "bbox": [ + 206, + 560, + 256, + 570 + ], + "score": 0.75, + "content": "\\mathrm { C + + / C U D A }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "is the decoupling of computation from data structures. This", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "allows programmers to easily switch between different data layouts and access data structures with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 159, + 595 + ], + "score": 1.0, + "content": "indices (i.e.", + "type": "text" + }, + { + "bbox": [ + 159, + 583, + 204, + 593 + ], + "score": 0.72, + "content": "\\times [ \\mathfrak { i } , \\ \\mathfrak { j } , \\ \\mathsf { k } ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 581, + 505, + 595 + ], + "score": 1.0, + "content": ", as if they are normal dense arrays, regardless of the underlying layout.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "The Taichi compiler then takes both the data structure and algorithm information to apply perfor-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 504, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 504, + 616 + ], + "score": 1.0, + "content": "mance optimizations. Taichi provides “parallel-for\" loops as a first-class construct. These designs", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "score": 1.0, + "content": "make Taichi especially suitable for writing high-performance physical simulators. For more details,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 269, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 269, + 638 + ], + "score": 1.0, + "content": "readers are referred to Hu et al. 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We extend the Taichi compiler to further", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "compile and automatically differentiate the generated Taichi IR into forward and backward executa-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 685, + 128, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 128, + 699 + ], + "score": 1.0, + "content": "bles.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 643, + 506, + 699 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 454, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 455, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 455, + 95 + ], + "score": 1.0, + "content": "We demonstrate the language using a mass-spring simulator, with three springs and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 455, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 455, + 105 + ], + "score": 1.0, + "content": "three mass points, as shown right. In this section we introduce the forward simulator", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 455, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 455, + 116 + ], + "score": 1.0, + "content": "using the DiffTaichi frontend of Taichi, which is an easier-to-use wrapper of the Taichi", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 175, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 136, + 126 + ], + "score": 0.83, + "content": "\\mathrm { C } { + } { + } 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 115, + 175, + 126 + ], + "score": 1.0, + "content": "frontend.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 504, + 190 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 506, + 158 + ], + "score": 1.0, + "content": "Allocating Global Variables Firstly we allocate a set of global tensors to store the simulation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 398, + 169 + ], + "score": 1.0, + "content": "state. These tensors include a scalar loss of type float32, 2D tensors", + "type": "text" + }, + { + "bbox": [ + 399, + 159, + 418, + 168 + ], + "score": 0.3, + "content": "\\times , ~ \\lor", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 156, + 506, + 169 + ], + "score": 1.0, + "content": ", force of size steps", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 122, + 179 + ], + "score": 0.58, + "content": "\\times { \\mathsf n _ { - } }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 168, + 208, + 181 + ], + "score": 1.0, + "content": "springs and type float", + "type": "text" + }, + { + "bbox": [ + 209, + 170, + 226, + 178 + ], + "score": 0.68, + "content": "3 2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 168, + 506, + 181 + ], + "score": 1.0, + "content": ", and 1D arrays of size n_spring for spring properties: spring_anchor_a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 111, + 180, + 350, + 191 + ], + "spans": [ + { + "bbox": [ + 111, + 180, + 350, + 191 + ], + "score": 1.0, + "content": "(int32), spring_anchor_b (int32), spring_length (float32).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 104, + 199, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 406, + 216 + ], + "score": 1.0, + "content": "Defining Kernels A mass-spring system is modeled by Hooke’s law", + "type": "text" + }, + { + "bbox": [ + 406, + 201, + 505, + 214 + ], + "score": 0.9, + "content": "\\textbf { F } = \\ k ( \\| \\mathbf { x } _ { a } - \\mathbf { x } _ { b } \\| _ { 2 } \\ - ", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 214, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 157, + 230 + ], + "score": 0.92, + "content": "l _ { 0 } ) \\frac { \\mathbf { x } _ { a } - \\mathbf { x } _ { b } } { \\lVert \\mathbf { x } _ { a } - \\mathbf { x } _ { b } \\rVert _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 214, + 186, + 227 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 186, + 214, + 194, + 224 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 214, + 287, + 227 + ], + "score": 1.0, + "content": "is the spring stiffness,", + "type": "text" + }, + { + "bbox": [ + 287, + 214, + 296, + 225 + ], + "score": 0.38, + "content": "\\mathbf { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 214, + 361, + 227 + ], + "score": 1.0, + "content": "is spring force,", + "type": "text" + }, + { + "bbox": [ + 361, + 216, + 374, + 226 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 214, + 393, + 227 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 216, + 405, + 225 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "are the positions of two", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 176, + 241 + ], + "score": 1.0, + "content": "mass points, and", + "type": "text" + }, + { + "bbox": [ + 177, + 228, + 186, + 239 + ], + "score": 0.86, + "content": "l _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "is the rest length. 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\\Delta t \\alpha } { \\pmb v } _ { t - 1 , i } + } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 391, + 504, + 411 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 232, + 409 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { \\Delta t } { m _ { i } } { \\bf F } _ { t , i } , { \\bf x } _ { t , i } = { \\bf x } _ { t - 1 , i } + \\Delta t { \\bf v } _ { t , i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 391, + 263, + 411 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 264, + 397, + 315, + 407 + ], + "score": 0.9, + "content": "\\pmb { v } _ { t , i } , \\mathbf { x } _ { t , i } , m _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 391, + 498, + 411 + ], + "score": 1.0, + "content": "are the velocity, position and mass of particle", + "type": "text" + }, + { + "bbox": [ + 499, + 396, + 504, + 405 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 407, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 155, + 419 + ], + "score": 1.0, + "content": "at time step", + "type": "text" + }, + { + "bbox": [ + 155, + 409, + 160, + 417 + ], + "score": 0.66, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 408, + 216, + 419 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + }, + { + "bbox": [ + 216, + 410, + 224, + 417 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 408, + 407, + 419 + ], + "score": 1.0, + "content": "is a damping factor. 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~ { \\bf 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 462, + 217, + 474 + ], + "score": 1.0, + "content": ", i]", + "type": "text" + }, + { + "bbox": [ + 217, + 464, + 225, + 471 + ], + "score": 0.67, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 462, + 239, + 474 + ], + "score": 1.0, + "content": "dt", + "type": "text" + }, + { + "bbox": [ + 239, + 464, + 245, + 471 + ], + "score": 0.48, + "content": "\\star", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 462, + 279, + 474 + ], + "score": 1.0, + "content": "v[t, i]", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 503, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 490, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 505 + ], + "score": 1.0, + "content": "Assembling the Forward Simulator With these components, we define the forward time integra-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 128, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 128, + 515 + ], + "score": 1.0, + "content": "tion:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 108, + 518, + 223, + 552 + ], + "lines": [ + { + "bbox": [ + 108, + 516, + 171, + 529 + ], + "spans": [ + { + "bbox": [ + 108, + 516, + 171, + 529 + ], + "score": 1.0, + "content": "def forward():", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 116, + 525, + 224, + 537 + ], + "spans": [ + { + "bbox": [ + 116, + 525, + 224, + 537 + ], + "score": 1.0, + "content": "for t in range(1, steps):", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 125, + 534, + 217, + 544 + ], + "spans": [ + { + "bbox": [ + 125, + 534, + 217, + 544 + ], + "score": 1.0, + "content": "apply_spring_force(t)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 126, + 541, + 199, + 553 + ], + "spans": [ + { + "bbox": [ + 126, + 541, + 199, + 553 + ], + "score": 1.0, + "content": "time_integrate(t)", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 104, + 574, + 492, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 493, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 493, + 588 + ], + "score": 1.0, + "content": "3 AUTOMATICALLY DIFFERENTIATING PHYSICAL SIMULATORS IN TAICHI", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 105, + 598, + 503, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "The main goal of DiffTaichi’s automatic differentiation (AD) system is to generate gradient simula-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 446, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 446, + 622 + ], + "score": 1.0, + "content": "tors automatically with minimal code changes to the traditional forward simulators.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "Design Decision Source Code Transformation (SCT) (Griewank & Walther, 2008) and Trac-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "ing (Wengert, 1964) are common choices when designing AD systems. In our setting, using SCT to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "differentiate a whole simulator with thousands of time steps, results in high performance yet poor", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "flexibility and long compilation time. 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The global tensors", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 721, + 296, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 296, + 734 + ], + "score": 1.0, + "content": "are natural checkpoints for gradient evaluation.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 456, + 82, + 502, + 127 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 454, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 455, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 455, + 95 + ], + "score": 1.0, + "content": "We demonstrate the language using a mass-spring simulator, with three springs and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 455, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 455, + 105 + ], + "score": 1.0, + "content": "three mass points, as shown right. 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For instance,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 400, + 340 + ], + "score": 1.0, + "content": "in the mass-spring simulation example, we record the whole history of", + "type": "text" + }, + { + "bbox": [ + 401, + 329, + 407, + 337 + ], + "score": 0.74, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 326, + 426, + 340 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 330, + 432, + 338 + ], + "score": 0.66, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 326, + 505, + 340 + ], + "score": 1.0, + "content": ", instead of keep-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "ing only the latest values. 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We also", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "provide ti.root.lazy_grad() to automatically place the adjoint tensors following the layout of their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 471, + 142, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 142, + 483 + ], + "score": 1.0, + "content": "primals.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 495, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 496, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 496, + 512 + ], + "score": 1.0, + "content": "3.1 LOCAL AD: DIFFERENTIATING TAICHI KERNELS USING SOURCE CODE TRANSFORMS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 504, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "A typical Taichi kernel consists of multiple levels of for loops and a body block. 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We reuse some infrastructure (white boxes) from Taichi,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "while the blue boxes are our extensions for differentiable programming. Right: The tape records", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 173, + 465, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 465, + 188 + ], + "score": 1.0, + "content": "kernel launches and replays the gradient kernels in reverse order during backpropagation.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 504, + 256 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "Assumption Unlike functional programming languages where immutable output buffers are gen-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "erated, imperative programming allows programmers to freely modify global tensors. To make", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "automatic differentiation well-defined under this setting, we make the following assumption on im-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 245, + 175, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 175, + 256 + ], + "score": 1.0, + "content": "perative kernels:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 211, + 505, + 256 + ] + }, + { + "type": "title", + "bbox": [ + 112, + 264, + 226, + 275 + ], + "lines": [ + { + "bbox": [ + 112, + 263, + 227, + 276 + ], + "spans": [ + { + "bbox": [ + 112, + 263, + 227, + 276 + ], + "score": 1.0, + "content": "Global Data Access Rules:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 111, + 267, + 501, + 309 + ], + "lines": [ + { + "bbox": [ + 111, + 274, + 499, + 287 + ], + "spans": [ + { + "bbox": [ + 111, + 274, + 499, + 287 + ], + "score": 1.0, + "content": "1) If a global tensor element is written more than once, then starting from the second write, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 110, + 285, + 371, + 298 + ], + "spans": [ + { + "bbox": [ + 110, + 285, + 371, + 298 + ], + "score": 1.0, + "content": "write must come in the form of an atomic add (“accumulation”).", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 111, + 297, + 455, + 309 + ], + "spans": [ + { + "bbox": [ + 111, + 297, + 455, + 309 + ], + "score": 1.0, + "content": "2) No read accesses happen to a global tensor element, until its accumulation is done.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 110, + 274, + 499, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "In forward simulators, programmers may make subtle changes to satisfy the rules. For instance,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 400, + 340 + ], + "score": 1.0, + "content": "in the mass-spring simulation example, we record the whole history of", + "type": "text" + }, + { + "bbox": [ + 401, + 329, + 407, + 337 + ], + "score": 0.74, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 326, + 426, + 340 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 330, + 432, + 338 + ], + "score": 0.66, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 326, + 505, + 340 + ], + "score": 1.0, + "content": ", instead of keep-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "ing only the latest values. The memory consumption issues caused by this can be alleviated via", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 304, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 304, + 362 + ], + "score": 1.0, + "content": "checkpointing, as discussed later in Appendix D.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 315, + 506, + 362 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "With these assumptions, kernels will not overwrite the outputs of each other, and the goal of AD is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 221, + 390 + ], + "score": 1.0, + "content": "clear: given a primal kernel", + "type": "text" + }, + { + "bbox": [ + 222, + 378, + 229, + 389 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 376, + 306, + 390 + ], + "score": 1.0, + "content": "that takes as input", + "type": "text" + }, + { + "bbox": [ + 307, + 377, + 373, + 388 + ], + "score": 0.91, + "content": "X _ { 1 } , X _ { 2 } , \\ldots , X _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "and outputs (or accumulates to)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 107, + 388, + 168, + 399 + ], + "score": 0.89, + "content": "Y _ { 1 } , Y _ { 2 } , \\dots , Y _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 387, + 326, + 402 + ], + "score": 1.0, + "content": ", the generated gradient (adjoint) kernel", + "type": "text" + }, + { + "bbox": [ + 326, + 389, + 338, + 400 + ], + "score": 0.89, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 387, + 420, + 402 + ], + "score": 1.0, + "content": "should take as input", + "type": "text" + }, + { + "bbox": [ + 421, + 388, + 487, + 399 + ], + "score": 0.91, + "content": "X _ { 1 } , X _ { 2 } , \\ldots , X _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 396, + 507, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 120, + 411 + ], + "score": 0.53, + "content": "Y _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 399, + 173, + 411 + ], + "score": 0.86, + "content": "^ { \\prime * } _ { 1 } , Y _ { 2 } ^ { * } , \\ldots , Y _ { m } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 396, + 343, + 414 + ], + "score": 1.0, + "content": "and accumulate gradient contributions to", + "type": "text" + }, + { + "bbox": [ + 343, + 399, + 413, + 411 + ], + "score": 0.92, + "content": "X _ { 1 } ^ { * } , X _ { 2 } ^ { * } , \\ldots , X _ { m } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 396, + 466, + 414 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 467, + 400, + 481, + 411 + ], + "score": 0.89, + "content": "X _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 396, + 507, + 414 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 409, + 238, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 150, + 423 + ], + "score": 1.0, + "content": "adjoint of", + "type": "text" + }, + { + "bbox": [ + 150, + 411, + 163, + 421 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 409, + 182, + 423 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 182, + 410, + 234, + 421 + ], + "score": 0.91, + "content": "\\partial ( \\mathrm { I o s s } ) / \\partial X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 409, + 238, + 423 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 365, + 507, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "Storage Control of Adjoint Tensors Users can specify the storage of adjoint tensors using the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "Taichi data structure description language (Hu et al., 2019a), as if they are primal tensors. We also", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "provide ti.root.lazy_grad() to automatically place the adjoint tensors following the layout of their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 471, + 142, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 142, + 483 + ], + "score": 1.0, + "content": "primals.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 437, + 506, + 483 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 495, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 496, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 496, + 512 + ], + "score": 1.0, + "content": "3.1 LOCAL AD: DIFFERENTIATING TAICHI KERNELS USING SOURCE CODE TRANSFORMS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 504, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "A typical Taichi kernel consists of multiple levels of for loops and a body block. To make later AD", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 534, + 472, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 472, + 546 + ], + "score": 1.0, + "content": "easier, we introduce two basic code transforms to simplify the loop body, as detailed below.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 523, + 505, + 546 + ] + }, + { + "type": "image", + "bbox": [ + 112, + 568, + 498, + 615 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 568, + 498, + 615 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 112, + 568, + 498, + 615 + ], + "spans": [ + { + "bbox": [ + 112, + 568, + 498, + 615 + ], + "score": 0.833, + "type": "image", + "image_path": "1b4c20ad358307d39d56d6c1309b5853a1902c2d261bda590bc40d6f691560eb.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 112, + 568, + 498, + 583.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 112, + 583.6666666666666, + 498, + 599.3333333333333 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 112, + 599.3333333333333, + 498, + 614.9999999999999 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 629, + 503, + 652 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 628, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 504, + 642 + ], + "score": 1.0, + "content": "Figure 3: Simple IR preprocessing before running the AD source code transform (left to right).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 640, + 493, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 174, + 653 + ], + "score": 1.0, + "content": "Demonstrated in", + "type": "text" + }, + { + "bbox": [ + 175, + 642, + 189, + 651 + ], + "score": 0.84, + "content": "{ \\mathsf { C } } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 640, + 493, + 653 + ], + "score": 1.0, + "content": ". The actual Taichi IR is often more complex. Containing loops are ignored.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + } + ], + "index": 32.25 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "Flatten Branching In physical simulation branches are common, e.g., when implementing bound-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "ary conditions and collisions. To simplify the reverse-mode AD pass, we first replace “if” statements", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "with ternary operators select(cond, value_if_true, value_if_false), whose gradients are clearly de-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "fined (Fig. 3, middle). This is a common transformation in program vectorization (e.g. Karrenberg", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 264, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 264, + 732 + ], + "score": 1.0, + "content": "& Hack (2011); Pharr & Mark (2012)).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 677, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Eliminate Mutable Local Variables After removing branching, we end up with straight-line loop", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "bodies. To further simplify the IR and make the procedure truly single-assignment, we apply a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "series of local variable store forwarding transforms, until the mutable local variables can be fully", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 209, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 209, + 128 + ], + "score": 1.0, + "content": "eliminated (Fig. 3, right).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "After these two custom IR simplification transforms, DiffTaichi only has to differentiate the straight-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "line code without mutable variables, which it achieves with reverse-mode AD, using a standard", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "source code transformation (Griewank & Walther, 2008). More details on this transform are in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 160, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 160, + 177 + ], + "score": 1.0, + "content": "Appendix B.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 199, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "Loops Most loops in physical simulation are parallel loops, and during AD we preserve the parallel", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "score": 1.0, + "content": "loop structures. For loops that are not explicitly marked as parallel, we reverse the loop order during", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "AD transforms. We do not support loops that carry a mutating local variable since that would", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "require a complex and costly run-time stack to maintain the history of local variables. Instead, users", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 431, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 431, + 255 + ], + "score": 1.0, + "content": "are instructed to employ global variables that satisfy the global data access rules.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "Parallelism and Thread Safety For forward simulation, we inherit the “parallel-for\" construct", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "from Taichi to map each loop iteration onto CPU/GPU threads. Programmers use atomic operations", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "for thread safety. Our system can automatically differentiate these atomic operations. Gradient", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 311, + 465, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 465, + 322 + ], + "score": 1.0, + "content": "contributions in backward kernels are accumulated to the adjoint tensors via atomic adds.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 106, + 345, + 468, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 469, + 357 + ], + "score": 1.0, + "content": "3.2 GLOBAL AD: END-TO-END BACKPROPAGATION USING A LIGHT-WEIGHT TAPE", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "We construct a tape (Fig. 2, right) of the kernel execution so that gradient kernels can be replayed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "in a reversed order. The tape is very light-weight: since the intermediate results are stored in global", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "tensors, during forward simulation the tape only records kernel names and the (scalar) input param-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "eters, unlike other differentiable functional array systems where all the intermediate buffers have to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "be recorded by the tape. Whenever a DiffTaichi kernel is launched, we append the kernel function", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "score": 1.0, + "content": "pointer and parameters to the tape. When evaluating gradients, we traverse the reversed tape, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "invoke the gradient kernels with the recorded parameters. Note that DiffTaichi AD is evaluating", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 446, + 416, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 416, + 459 + ], + "score": 1.0, + "content": "gradients with respect to input global tensors instead of the input parameters.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "Learning/Optimization with Gradients Now we revisit the mass-spring example and make it", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "differentiable for optimization. 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It automatically replays", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the gradients of these kernels in reverse for backpropagation. Initially the springs have lengths", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "[0.1, 0.1, 0.14], and after optimization the rest lengths are [0.600, 0.600, 0.529]. This means the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "springs will expand the triangle according to Hooke’s law and form a larger triangle: [Reproduce:", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 211, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 211, + 734 + ], + "score": 1.0, + "content": "mass_spring_simple.py]", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 554, + 382, + 638 + ], + "lines": [ + { + "bbox": [ + 109, + 554, + 155, + 564 + ], + "spans": [ + { + "bbox": [ + 109, + 554, + 155, + 564 + ], + "score": 1.0, + "content": "@ti.kernel", + "type": "text" + } + ] + }, + { + "bbox": [ + 109, + 562, + 230, + 573 + ], + "spans": [ + { + "bbox": [ + 109, + 562, + 230, + 573 + ], + "score": 1.0, + "content": "def compute_loss(t: ti.i32):", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 570, + 218, + 582 + ], + "spans": [ + { + "bbox": [ + 118, + 572, + 145, + 579 + ], + "score": 0.28, + "content": "\\mathbf { \\nabla \\times } \\Theta \\mathbf { 1 } \\ = \\ \\mathbf { \\Omega } .", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 570, + 218, + 582 + ], + "score": 1.0, + "content": "x[t, 0] - x[t, 1]", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 578, + 218, + 590 + ], + "spans": [ + { + "bbox": [ + 118, + 580, + 158, + 588 + ], + "score": 0.32, + "content": "\\times \\odot 2 ~ = ~ \\times [ t", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 578, + 218, + 590 + ], + "score": 1.0, + "content": ", 0] - x[t, 2]", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 587, + 264, + 597 + ], + "spans": [ + { + "bbox": [ + 117, + 587, + 264, + 597 + ], + "score": 1.0, + "content": "# Triangle area from cross product", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 594, + 338, + 606 + ], + "spans": [ + { + "bbox": [ + 117, + 594, + 138, + 606 + ], + "score": 1.0, + "content": "area", + "type": "text" + }, + { + "bbox": [ + 138, + 596, + 146, + 603 + ], + "score": 0.47, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 594, + 206, + 606 + ], + "score": 1.0, + "content": "ti.abs(0.5 * (", + "type": "text" + }, + { + "bbox": [ + 206, + 596, + 220, + 604 + ], + "score": 0.32, + "content": "\\mathbf { \\boldsymbol { x } } \\Theta \\mathbf { \\boldsymbol { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 594, + 231, + 606 + ], + "score": 1.0, + "content": "[0]", + "type": "text" + }, + { + "bbox": [ + 232, + 596, + 249, + 604 + ], + "score": 0.27, + "content": "\\star \\mathsf { x } \\mathsf { 0 } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 594, + 338, + 606 + ], + "score": 1.0, + "content": "[1] - 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To further simplify the IR and make the procedure truly single-assignment, we apply a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "series of local variable store forwarding transforms, until the mutable local variables can be fully", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 209, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 209, + 128 + ], + "score": 1.0, + "content": "eliminated (Fig. 3, right).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 81, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "After these two custom IR simplification transforms, DiffTaichi only has to differentiate the straight-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "line code without mutable variables, which it achieves with reverse-mode AD, using a standard", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "source code transformation (Griewank & Walther, 2008). More details on this transform are in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 160, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 160, + 177 + ], + "score": 1.0, + "content": "Appendix B.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 132, + 505, + 177 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 199, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "Loops Most loops in physical simulation are parallel loops, and during AD we preserve the parallel", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "score": 1.0, + "content": "loop structures. For loops that are not explicitly marked as parallel, we reverse the loop order during", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "AD transforms. We do not support loops that carry a mutating local variable since that would", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "require a complex and costly run-time stack to maintain the history of local variables. Instead, users", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 431, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 431, + 255 + ], + "score": 1.0, + "content": "are instructed to employ global variables that satisfy the global data access rules.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 199, + 506, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "Parallelism and Thread Safety For forward simulation, we inherit the “parallel-for\" construct", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "from Taichi to map each loop iteration onto CPU/GPU threads. Programmers use atomic operations", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "for thread safety. Our system can automatically differentiate these atomic operations. Gradient", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 311, + 465, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 465, + 322 + ], + "score": 1.0, + "content": "contributions in backward kernels are accumulated to the adjoint tensors via atomic adds.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 276, + 506, + 322 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 345, + 468, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 469, + 357 + ], + "score": 1.0, + "content": "3.2 GLOBAL AD: END-TO-END BACKPROPAGATION USING A LIGHT-WEIGHT TAPE", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "We construct a tape (Fig. 2, right) of the kernel execution so that gradient kernels can be replayed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "in a reversed order. The tape is very light-weight: since the intermediate results are stored in global", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "tensors, during forward simulation the tape only records kernel names and the (scalar) input param-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "eters, unlike other differentiable functional array systems where all the intermediate buffers have to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "be recorded by the tape. Whenever a DiffTaichi kernel is launched, we append the kernel function", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "score": 1.0, + "content": "pointer and parameters to the tape. When evaluating gradients, we traverse the reversed tape, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "invoke the gradient kernels with the recorded parameters. 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All results can be reproduced with the provided script. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "dynamic/optimization processes are visualized in the supplemental video. In this section we focus", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 347, + 456, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 456, + 358 + ], + "score": 1.0, + "content": "our discussions on three simulators. 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ApproachForward TimeBackward TimeTotal Time# Lines of Code
TensorFlow13.20 ms35.70 ms48.90 ms (188.×)190
CUDA0.10 ms0.14 ms0.24 ms (0.92×)460
DiffTaichi0.11 ms0.15 ms0.26 ms (1.00×)110
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Apart from custom gradients, complex kernels", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 263, + 399, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 399, + 274 + ], + "score": 1.0, + "content": "can also be used to implement checkpointing, as detailed in Appendix D.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 206, + 505, + 274 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 289, + 193, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 195, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 195, + 304 + ], + "score": 1.0, + "content": "4 EVALUATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "We evaluate DiffTaichi on 10 different physical simulators covering large-scale continuum and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 325, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "small-scale rigid body simulations. All results can be reproduced with the provided script. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "dynamic/optimization processes are visualized in the supplemental video. In this section we focus", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 347, + 456, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 456, + 358 + ], + "score": 1.0, + "content": "our discussions on three simulators. 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ApproachForward TimeBackward TimeTotal Time# Lines of Code
TensorFlow13.20 ms35.70 ms48.90 ms (188.×)190
CUDA0.10 ms0.14 ms0.24 ms (0.92×)460
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ApproachForward TimeBackward TimeTotal Time#Essential LoC
PyTorch (CPU, f32)405 ms328 ms733 ms (13.8×)74
PyTorch (GPU, f32)254 ms457 ms711 ms (13.4×)74
Autograd (CPU, f64)307 ms1197 ms1504 ms (28.4×)51
JAX (GPU, f32)24 ms75 ms99 ms (1.9×)90
DiffTaichi (CPU, f32)66 ms132 ms198 ms (3.7x)75
DiffTaichi (GPU, f32)24 ms29 ms53 ms (1.0×)75
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For example, a periodic boundary condition is used so that Autograd can represent it", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "score": 1.0, + "content": "using numpy.roll, without any branching. Still, Taichi delivers higher performance than these array-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "based systems. The whole program takes 10 seconds to run in DiffTaichi on a GPU, and 2 seconds", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 291, + 327, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 327, + 303 + ], + "score": 1.0, + "content": "are spent on JIT. JAX JIT compilation takes 2 minutes.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 318, + 369, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 370, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 370, + 331 + ], + "score": 1.0, + "content": "4.3 DIFFERENTIABLE RIGID BODY SIMULATORS [rigid_body]", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "We built an impulse-based (Catto, 2009) differentiable rigid body simulator (Fig. 1, rigid_body) for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "optimizing robot controllers. This simulator supports rigid body collision and friction, spring forces,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 361, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 373 + ], + "score": 1.0, + "content": "joints, and actuation. The simulation is end-to-end differentiable except for a countable number of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "score": 1.0, + "content": "discontinuities. Interestingly, although the forward simulator works well, naively differentiating", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "it with DiffTaichi leads to completely misleading gradients, due to the rigid body collisions. We", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 393, + 307, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 307, + 406 + ], + "score": 1.0, + "content": "discuss the cause and solution of this issue below.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "Improving collision gradients Consider the rigid ball example in Fig. 4 (left), where a rigid ball", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "collides with a friction-less ground. Gravity is ignored, and due to conservation of kinetic energy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 357, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 357, + 449 + ], + "score": 1.0, + "content": "the ball keeps a constant speed even after this elastic collision.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 104, + 452, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 276, + 467 + ], + "score": 1.0, + "content": "In the forward simulation, using a small", + "type": "text" + }, + { + "bbox": [ + 276, + 454, + 290, + 464 + ], + "score": 0.81, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 452, + 505, + 467 + ], + "score": 1.0, + "content": "often leads to a reasonable result, as done in many", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "physics simulators. Lowering the initial ball height will increase the final ball height, since there is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "less distance to travel before the ball hits the ground and more after (see the loss curves in Fig.4,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 421, + 498 + ], + "score": 1.0, + "content": "middle right). However, using a naive time integrator, no matter how small", + "type": "text" + }, + { + "bbox": [ + 421, + 487, + 434, + 497 + ], + "score": 0.8, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "is, the evaluated", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 359, + 510 + ], + "score": 1.0, + "content": "gradient of final height w.r.t. initial height will be 1 instead of", + "type": "text" + }, + { + "bbox": [ + 360, + 498, + 373, + 508 + ], + "score": 0.5, + "content": "- 1", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 497, + 506, + 510 + ], + "score": 1.0, + "content": ". This counter-intuitive behavior", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "is due to the fact that time discretization itself is not differentiated by the compiler. Fig. 4 explains", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 520, + 216, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 216, + 532 + ], + "score": 1.0, + "content": "this effect in greater detail.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "image", + "bbox": [ + 110, + 540, + 499, + 611 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 540, + 499, + 611 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 540, + 499, + 611 + ], + "spans": [ + { + "bbox": [ + 110, + 540, + 499, + 611 + ], + "score": 0.959, + "type": "image", + "image_path": "8f7ca48f4fbe280ccf625c8aaa38b729d47690edbb0c833a8083c4b877937aa9.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 110, + 540, + 499, + 563.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 110, + 563.6666666666666, + 499, + 587.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 110, + 587.3333333333333, + 499, + 610.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 618, + 505, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 618, + 504, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 491, + 631 + ], + "score": 1.0, + "content": "Figure 4: How gradients can go wrong with naive time integrators. For clarity we use a large", + "type": "text" + }, + { + "bbox": [ + 491, + 619, + 504, + 628 + ], + "score": 0.77, + "content": "\\Delta t", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 365, + 642 + ], + "score": 1.0, + "content": "here. Left: Since collision detection only happens at multiples of", + "type": "text" + }, + { + "bbox": [ + 366, + 630, + 379, + 640 + ], + "score": 0.82, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 630, + 383, + 642 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 383, + 630, + 401, + 640 + ], + "score": 0.82, + "content": "2 \\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "in this case), lowering the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 637, + 507, + 655 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 507, + 655 + ], + "score": 1.0, + "content": "initial position of the ball (light blue) leads to a lowered final position. Middle Left: By improving", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 650, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 665 + ], + "score": 1.0, + "content": "the time integrator to support continuous time of impact (TOI), collisions can be detected at any", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 148, + 675 + ], + "score": 1.0, + "content": "time, e.g.", + "type": "text" + }, + { + "bbox": [ + 148, + 663, + 175, + 673 + ], + "score": 0.84, + "content": "1 . 9 \\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "(light red). Now the blue ball ends up higher than the green one. Middle Right:", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 673, + 504, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 490, + 686 + ], + "score": 1.0, + "content": "Although the two time integration techniques lead to almost identical forward results (in practice", + "type": "text" + }, + { + "bbox": [ + 491, + 673, + 504, + 684 + ], + "score": 0.82, + "content": "\\Delta t", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 684, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 697 + ], + "score": 1.0, + "content": "is small), the naive time integrator gives an incorrect gradient of 1, but adding TOI yields the correct", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "score": 1.0, + "content": "gradient. Please see our supplemental video for a better demonstration. [Reproduce: python3", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 706, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 706, + 505, + 719 + ], + "score": 1.0, + "content": "rigid_body_toi.py] Right: When zooming in, the loss of the naive integrator is decreasing, and the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 717, + 501, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 501, + 730 + ], + "score": 1.0, + "content": "saw-tooth pattern explains the positive gradients. [Reproduce: python3 rigid_body_toi.py zoom]", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + } + ], + "index": 33.25 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 133, + 508, + 224 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 125 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 455, + 92 + ], + "score": 1.0, + "content": "Table 2: smoke benchmark against Autograd, PyTorch, and JAX. We used a", + "type": "text" + }, + { + "bbox": [ + 455, + 80, + 505, + 91 + ], + "score": 0.85, + "content": "1 1 0 ~ \\times ~ 1 1 0", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "score": 1.0, + "content": "grid and 100 time steps, each with 6 Jacobi pressure projections. [Reproduce: python3", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "smoke_[autograd/pytorch/jax/taichi_cpu/taichi_gpu].py]. Note that the Autograd program uses", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 112, + 352, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 352, + 126 + ], + "score": 1.0, + "content": "float64 precision, which approximately doubles the run time.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 133, + 508, + 224 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 133, + 508, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 508, + 224 + ], + "score": 0.981, + "html": "
ApproachForward TimeBackward TimeTotal Time#Essential LoC
PyTorch (CPU, f32)405 ms328 ms733 ms (13.8×)74
PyTorch (GPU, f32)254 ms457 ms711 ms (13.4×)74
Autograd (CPU, f64)307 ms1197 ms1504 ms (28.4×)51
JAX (GPU, f32)24 ms75 ms99 ms (1.9×)90
DiffTaichi (CPU, f32)66 ms132 ms198 ms (3.7x)75
DiffTaichi (GPU, f32)24 ms29 ms53 ms (1.0×)75
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For example, a periodic boundary condition is used so that Autograd can represent it", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "score": 1.0, + "content": "using numpy.roll, without any branching. Still, Taichi delivers higher performance than these array-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "based systems. The whole program takes 10 seconds to run in DiffTaichi on a GPU, and 2 seconds", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 291, + 327, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 327, + 303 + ], + "score": 1.0, + "content": "are spent on JIT. JAX JIT compilation takes 2 minutes.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 246, + 506, + 303 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 318, + 369, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 370, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 370, + 331 + ], + "score": 1.0, + "content": "4.3 DIFFERENTIABLE RIGID BODY SIMULATORS [rigid_body]", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "We built an impulse-based (Catto, 2009) differentiable rigid body simulator (Fig. 1, rigid_body) for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "optimizing robot controllers. 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We", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 393, + 307, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 307, + 406 + ], + "score": 1.0, + "content": "discuss the cause and solution of this issue below.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 338, + 506, + 406 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "Improving collision gradients Consider the rigid ball example in Fig. 4 (left), where a rigid ball", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "collides with a friction-less ground. Gravity is ignored, and due to conservation of kinetic energy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 357, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 357, + 449 + ], + "score": 1.0, + "content": "the ball keeps a constant speed even after this elastic collision.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 414, + 505, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 104, + 452, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 276, + 467 + ], + "score": 1.0, + "content": "In the forward simulation, using a small", + "type": "text" + }, + { + "bbox": [ + 276, + 454, + 290, + 464 + ], + "score": 0.81, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 452, + 505, + 467 + ], + "score": 1.0, + "content": "often leads to a reasonable result, as done in many", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "physics simulators. Lowering the initial ball height will increase the final ball height, since there is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "less distance to travel before the ball hits the ground and more after (see the loss curves in Fig.4,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 421, + 498 + ], + "score": 1.0, + "content": "middle right). However, using a naive time integrator, no matter how small", + "type": "text" + }, + { + "bbox": [ + 421, + 487, + 434, + 497 + ], + "score": 0.8, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "is, the evaluated", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 359, + 510 + ], + "score": 1.0, + "content": "gradient of final height w.r.t. initial height will be 1 instead of", + "type": "text" + }, + { + "bbox": [ + 360, + 498, + 373, + 508 + ], + "score": 0.5, + "content": "- 1", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 497, + 506, + 510 + ], + "score": 1.0, + "content": ". This counter-intuitive behavior", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "is due to the fact that time discretization itself is not differentiated by the compiler. Fig. 4 explains", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 520, + 216, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 216, + 532 + ], + "score": 1.0, + "content": "this effect in greater detail.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 452, + 506, + 532 + ] + }, + { + "type": "image", + "bbox": [ + 110, + 540, + 499, + 611 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 540, + 499, + 611 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 540, + 499, + 611 + ], + "spans": [ + { + "bbox": [ + 110, + 540, + 499, + 611 + ], + "score": 0.959, + "type": "image", + "image_path": "8f7ca48f4fbe280ccf625c8aaa38b729d47690edbb0c833a8083c4b877937aa9.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 110, + 540, + 499, + 563.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 110, + 563.6666666666666, + 499, + 587.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 110, + 587.3333333333333, + 499, + 610.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 618, + 505, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 618, + 504, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 491, + 631 + ], + "score": 1.0, + "content": "Figure 4: How gradients can go wrong with naive time integrators. For clarity we use a large", + "type": "text" + }, + { + "bbox": [ + 491, + 619, + 504, + 628 + ], + "score": 0.77, + "content": "\\Delta t", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 365, + 642 + ], + "score": 1.0, + "content": "here. Left: Since collision detection only happens at multiples of", + "type": "text" + }, + { + "bbox": [ + 366, + 630, + 379, + 640 + ], + "score": 0.82, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 630, + 383, + 642 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 383, + 630, + 401, + 640 + ], + "score": 0.82, + "content": "2 \\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "in this case), lowering the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 637, + 507, + 655 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 507, + 655 + ], + "score": 1.0, + "content": "initial position of the ball (light blue) leads to a lowered final position. Middle Left: By improving", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 650, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 665 + ], + "score": 1.0, + "content": "the time integrator to support continuous time of impact (TOI), collisions can be detected at any", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 148, + 675 + ], + "score": 1.0, + "content": "time, e.g.", + "type": "text" + }, + { + "bbox": [ + 148, + 663, + 175, + 673 + ], + "score": 0.84, + "content": "1 . 9 \\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "(light red). Now the blue ball ends up higher than the green one. Middle Right:", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 673, + 504, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 490, + 686 + ], + "score": 1.0, + "content": "Although the two time integration techniques lead to almost identical forward results (in practice", + "type": "text" + }, + { + "bbox": [ + 491, + 673, + 504, + 684 + ], + "score": 0.82, + "content": "\\Delta t", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 684, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 697 + ], + "score": 1.0, + "content": "is small), the naive time integrator gives an incorrect gradient of 1, but adding TOI yields the correct", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "score": 1.0, + "content": "gradient. Please see our supplemental video for a better demonstration. [Reproduce: python3", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 706, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 706, + 505, + 719 + ], + "score": 1.0, + "content": "rigid_body_toi.py] Right: When zooming in, the loss of the naive integrator is decreasing, and the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 717, + 501, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 501, + 730 + ], + "score": 1.0, + "content": "saw-tooth pattern explains the positive gradients. 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Each experiment is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "repeated five times. [Reproduce: python3 [mass_spring/rigid_body.py] [1/2] plot && python3", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 195, + 171, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 171, + 209 + ], + "score": 1.0, + "content": "plot_losses.py]", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 228, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 504, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 504, + 241 + ], + "score": 1.0, + "content": "We propose a simple solution of adding continuous collision resolution (see, for example, Redon", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "score": 1.0, + "content": "et al. (2002)), which considers precise time of impact (TOI), to the forward program (Fig. 4, middle", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "left). Although it barely improves the forward simulation (Fig. 4, middle right), the gradient will be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "corrected effectively (Fig. 4, right). The details of continuous collision detection are in Appendix F.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "In real-world simulators, we find the TOI technique leads to significant improvement in gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "quality in controller optimization tasks (Fig. 5). Having TOI or not barely affects forward simula-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "tion: in the supplemental video, we show that a robot controller optimized in a simulator with TOI,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 306, + 298, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 298, + 317 + ], + "score": 1.0, + "content": "actually works well in a simulator without TOI.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "score": 1.0, + "content": "The takeaway is, differentiating physical simulators does not always yield useful gradients of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "physical system being simulated, even if the simulator does forward simulation well. In Appendix G,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 368, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 368, + 356 + ], + "score": 1.0, + "content": "we discuss some additional gradient issues we have encountered.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 366, + 211, + 379 + ], + "lines": [ + { + "bbox": [ + 104, + 365, + 213, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 213, + 381 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "Differentiable programming The recent rise of deep learning has motivated the development of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "differentiable programming libraries for deep NNs, most notably auto-differentiation frameworks", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "such as Theano (Bergstra et al., 2010), TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "2017). However, physical simulation requires complex and customizable operations due to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "intrinsic computational irregularity. Using the aforementioned frameworks, programmers have to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "score": 1.0, + "content": "compose these coarse-grained basic operations into desired complex operations. Doing so often", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 458, + 253, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 253, + 469 + ], + "score": 1.0, + "content": "leads to unsatisfactory performance.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "Earlier work on automatic differentiation focuses on transforming existing scalar code to obtain", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "derivatives (e.g. Utke et al. (2008), Hascoet & Pascual (2013), Pearlmutter & Siskind (2008)).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "A recent trend has emerged for modern programming languages to support differentiable function", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "transformations through annotation (e.g. Innes et al. (2019), Wei et al. (2019)). These frameworks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 518, + 474, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 474, + 531 + ], + "score": 1.0, + "content": "enable differentiating general programming languages, yet they provide limited parallelism.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "score": 1.0, + "content": "Differentiable array programming languages such as Halide (Ragan-Kelley et al., 2013; Li et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "2018b), Autograd (Maclaurin et al., 2015), JAX (Bradbury et al., 2018), and Enoki (Jakob, 2019)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "operate on arrays instead of scalars to utilize parallelism. Instead of operating on arrays that are im-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "mutable, DiffTaichi uses an imperative style with flexible indexing to make porting existing physical", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 224, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 224, + 591 + ], + "score": 1.0, + "content": "simulation algorithms easier.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Differentiable Physical Simulators Building differentiable simulators for robotics and machine", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "learning has recently increased in popularity. Without differentiable programming, Battaglia et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "(2016), Chang et al. (2016) and Mrowca et al. (2018) used NNs to approximate the physical pro-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "cess and used the NN gradients as the approximate simulation gradients. Degrave et al. (2016)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "and de Avila Belbute-Peres et al. (2018b) used Theano and PyTorch respectively to build differen-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "tiable rigid body simulators. Schenck & Fox (2018) differentiates position-based fluid using custom", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "CUDA kernels. Popovic et al. ´ (2000) used a differentiable rigid body simulator for manipulating", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "physically based animations. The ChainQueen differentiable elastic object simulator (Hu et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "2019b) implements forward and gradient versions of continuum mechanics in hand-written CUDA", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "kernels, leading to performance that is two orders of magnitude higher than a pure TensorFlow", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "implementation. Liang et al. (2019) built a differentiable cloth simulator for material estimation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "and motion control. 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Each experiment is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "repeated five times. [Reproduce: python3 [mass_spring/rigid_body.py] [1/2] plot && python3", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 195, + 171, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 171, + 209 + ], + "score": 1.0, + "content": "plot_losses.py]", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 228, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 504, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 504, + 241 + ], + "score": 1.0, + "content": "We propose a simple solution of adding continuous collision resolution (see, for example, Redon", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "score": 1.0, + "content": "et al. (2002)), which considers precise time of impact (TOI), to the forward program (Fig. 4, middle", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "left). Although it barely improves the forward simulation (Fig. 4, middle right), the gradient will be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "corrected effectively (Fig. 4, right). The details of continuous collision detection are in Appendix F.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "In real-world simulators, we find the TOI technique leads to significant improvement in gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "quality in controller optimization tasks (Fig. 5). 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In Appendix G,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 368, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 368, + 356 + ], + "score": 1.0, + "content": "we discuss some additional gradient issues we have encountered.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 321, + 505, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 366, + 211, + 379 + ], + "lines": [ + { + "bbox": [ + 104, + 365, + 213, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 213, + 381 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "Differentiable programming The recent rise of deep learning has motivated the development of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "differentiable programming libraries for deep NNs, most notably auto-differentiation frameworks", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "such as Theano (Bergstra et al., 2010), TensorFlow (Abadi et al., 2016) and PyTorch (Paszke et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "2017). However, physical simulation requires complex and customizable operations due to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "intrinsic computational irregularity. Using the aforementioned frameworks, programmers have to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "score": 1.0, + "content": "compose these coarse-grained basic operations into desired complex operations. Doing so often", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 458, + 253, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 253, + 469 + ], + "score": 1.0, + "content": "leads to unsatisfactory performance.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 391, + 506, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "Earlier work on automatic differentiation focuses on transforming existing scalar code to obtain", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "derivatives (e.g. Utke et al. (2008), Hascoet & Pascual (2013), Pearlmutter & Siskind (2008)).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "A recent trend has emerged for modern programming languages to support differentiable function", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "transformations through annotation (e.g. Innes et al. (2019), Wei et al. (2019)). These frameworks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 518, + 474, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 474, + 531 + ], + "score": 1.0, + "content": "enable differentiating general programming languages, yet they provide limited parallelism.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 474, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "score": 1.0, + "content": "Differentiable array programming languages such as Halide (Ragan-Kelley et al., 2013; Li et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "2018b), Autograd (Maclaurin et al., 2015), JAX (Bradbury et al., 2018), and Enoki (Jakob, 2019)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "operate on arrays instead of scalars to utilize parallelism. Instead of operating on arrays that are im-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "mutable, DiffTaichi uses an imperative style with flexible indexing to make porting existing physical", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 224, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 224, + 591 + ], + "score": 1.0, + "content": "simulation algorithms easier.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 533, + 505, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Differentiable Physical Simulators Building differentiable simulators for robotics and machine", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "learning has recently increased in popularity. 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(2018b) used Theano and PyTorch respectively to build differen-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "tiable rigid body simulators. Schenck & Fox (2018) differentiates position-based fluid using custom", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "CUDA kernels. Popovic et al. ´ (2000) used a differentiable rigid body simulator for manipulating", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "physically based animations. 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(2019) built a differentiable cloth simulator for material estimation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "and motion control. 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Imperative Scheme【【
Parallelism?
Flexible Indexing【【
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for i ∈ range(0, 16, step 1) do
%1=load x[i]
%2 = mul %1, %1
%3= sin(%2)
store y[i] = %3
end for
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for i in range(0,16, step 1) do
/ adjoint variables
%1adj = alloca 0.0
%2adj = alloca 0.0
%3adj = alloca 0.0
// original forward computation
%1=load x[] %2 = mul %1, %1
%3= sin(%2)
/ reverse accumulation
%4 = load y_adj[i]
%3adj += %4
%5 = cos(%2)
%2adj += %3adj * %5
%1adj += 2 * %1 * %2adj
atomic add x_adj[i],%1adj end for
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The actual IR has more instruc-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "tions. A following simplification pass will simplify redundant instructions generated by the AD", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 574, + 130, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 130, + 586 + ], + "score": 1.0, + "content": "pass.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 600, + 232, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 233, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 233, + 616 + ], + "score": 1.0, + "content": "C COMPLEX KERNELS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "score": 1.0, + "content": "Here we demonstrated how to use complex kernels to override the automatic differentiation system.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 298, + 649 + ], + "score": 1.0, + "content": "We use singular value decomposition (SVD) of", + "type": "text" + }, + { + "bbox": [ + 298, + 637, + 322, + 648 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 637, + 362, + 649 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 362, + 637, + 419, + 648 + ], + "score": 0.89, + "content": "\\mathbf { M } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 637, + 506, + 649 + ], + "score": 1.0, + "content": ") as an example. Fast", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "SVD solvers used in physical simulation are often iterative, yet directly evaluate the gradient of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "this iterative process is likely numerically unstable. Suppose we use McAdams et al. 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It takes straight-line code directly and operates on the hierarchical intermediate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "representation (IR) of Taichi4 . Multiple outer for loops are allowed for the primal kernel. The Taichi", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 396, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 396, + 149 + ], + "score": 1.0, + "content": "compiler will distribute these parallel iterations onto CPU/GPU threads.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 505, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "During the “make_adjoint” pass, for each SSA instruction, a local adjoint variable will be allocated", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "for gradient contribution accumulation. The compiler will traverse the statements in reverse order,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 398, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 398, + 189 + ], + "score": 1.0, + "content": "and accumulate the gradients to the corresponding adjoint local variable.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 153, + 505, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 437, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 439, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 246, + 206 + ], + "score": 1.0, + "content": "For example, a 1D array operation", + "type": "text" + }, + { + "bbox": [ + 246, + 192, + 293, + 205 + ], + "score": 0.93, + "content": "y _ { i } = \\sin x _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 192, + 439, + 206 + ], + "score": 1.0, + "content": "has its IR representation as follows:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 192, + 439, + 206 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 218, + 505, + 292 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 218, + 505, + 292 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 218, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 107, + 218, + 505, + 292 + ], + "score": 0.911, + "html": "
for i ∈ range(0, 16, step 1) do
%1=load x[i]
%2 = mul %1, %1
%3= sin(%2)
store y[i] = %3
end for
", + "type": "table", + "image_path": "b1e023b08421ea3290cc79e3ba96a4c4b0ba9a2925d5fa5c4858479a63511b3d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 107, + 218, + 505, + 242.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 107, + 242.66666666666666, + 505, + 267.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 107, + 267.3333333333333, + 505, + 292.0 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 312, + 420, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 421, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 421, + 326 + ], + "score": 1.0, + "content": "The above primal kernel will be transformed into the following adjoint kernel:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 310, + 421, + 326 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 337, + 504, + 530 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 337, + 504, + 530 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 337, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 109, + 337, + 504, + 530 + ], + "score": 0.605, + "html": "
for i in range(0,16, step 1) do
/ adjoint variables
%1adj = alloca 0.0
%2adj = alloca 0.0
%3adj = alloca 0.0
// original forward computation
%1=load x[] %2 = mul %1, %1
%3= sin(%2)
/ reverse accumulation
%4 = load y_adj[i]
%3adj += %4
%5 = cos(%2)
%2adj += %3adj * %5
%1adj += 2 * %1 * %2adj
atomic add x_adj[i],%1adj end for
", + "type": "table", + "image_path": "04e0c4cf294ba438bcb7f620510cc5e911d75f2d1521877e2505e8b9acbb5e7a.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 109, + 337, + 504, + 401.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 109, + 401.3333333333333, + 504, + 465.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 109, + 465.66666666666663, + 504, + 530.0 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 504, + 563 + ], + "score": 1.0, + "content": "Note that for clarity the transformed code is not strictly SSA here. The actual IR has more instruc-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "tions. A following simplification pass will simplify redundant instructions generated by the AD", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 574, + 130, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 130, + 586 + ], + "score": 1.0, + "content": "pass.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 550, + 505, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 600, + 232, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 233, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 233, + 616 + ], + "score": 1.0, + "content": "C COMPLEX KERNELS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "score": 1.0, + "content": "Here we demonstrated how to use complex kernels to override the automatic differentiation system.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 298, + 649 + ], + "score": 1.0, + "content": "We use singular value decomposition (SVD) of", + "type": "text" + }, + { + "bbox": [ + 298, + 637, + 322, + 648 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 637, + 362, + 649 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 362, + 637, + 419, + 648 + ], + "score": 0.89, + "content": "\\mathbf { M } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 637, + 506, + 649 + ], + "score": 1.0, + "content": ") as an example. Fast", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "SVD solvers used in physical simulation are often iterative, yet directly evaluate the gradient of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "this iterative process is likely numerically unstable. Suppose we use McAdams et al. (2011) as the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "forward SVD solver, and use the method in Jiang (2015) (Section 2.1.1.2) to evalute the gradients,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 681, + 270, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 270, + 693 + ], + "score": 1.0, + "content": "the complex kernels are used as follows:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 625, + 506, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 99, + 367, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 97, + 334, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 334, + 109 + ], + "score": 1.0, + "content": "# Do Singular Value Decomposition (SVD) on n matrices", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 108, + 106, + 154, + 116 + ], + "spans": [ + { + "bbox": [ + 108, + 106, + 154, + 116 + ], + "score": 1.0, + "content": "@ti.kernel", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 108, + 113, + 287, + 125 + ], + "spans": [ + { + "bbox": [ + 108, + 113, + 287, + 125 + ], + "score": 1.0, + "content": "def iterative_svd(num_iterations: ti.f32):", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 117, + 122, + 196, + 134 + ], + "spans": [ + { + "bbox": [ + 117, + 122, + 196, + 134 + ], + "score": 1.0, + "content": "for i in range(n):", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 126, + 131, + 207, + 141 + ], + "spans": [ + { + "bbox": [ + 126, + 131, + 150, + 141 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 150, + 132, + 158, + 139 + ], + "score": 0.75, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 131, + 207, + 141 + ], + "score": 1.0, + "content": "matrix_M[i]", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 126, + 138, + 270, + 149 + ], + "spans": [ + { + "bbox": [ + 126, + 138, + 270, + 149 + ], + "score": 1.0, + "content": "for iter in range(num_iterations):", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 136, + 146, + 353, + 157 + ], + "spans": [ + { + "bbox": [ + 136, + 146, + 353, + 157 + ], + "score": 1.0, + "content": "... iteratively solve SVD using McAdams et al. 2011", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 125, + 154, + 199, + 165 + ], + "spans": [ + { + "bbox": [ + 125, + 154, + 175, + 165 + ], + "score": 1.0, + "content": "matrix_U[i]", + "type": "text" + }, + { + "bbox": [ + 176, + 156, + 183, + 163 + ], + "score": 0.54, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 154, + 199, + 165 + ], + "score": 1.0, + "content": "...", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 124, + 162, + 200, + 173 + ], + "spans": [ + { + "bbox": [ + 124, + 162, + 192, + 173 + ], + "score": 1.0, + "content": "matrix_Sigma[i]", + "type": "text" + }, + { + "bbox": [ + 192, + 164, + 200, + 171 + ], + "score": 0.3, + "content": "=", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 125, + 169, + 200, + 181 + ], + "spans": [ + { + "bbox": [ + 125, + 169, + 175, + 181 + ], + "score": 1.0, + "content": "matrix_V[i]", + "type": "text" + }, + { + "bbox": [ + 175, + 172, + 183, + 179 + ], + "score": 0.49, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 169, + 200, + 181 + ], + "score": 1.0, + "content": "...", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 109, + 227, + 391, + 260 + ], + "lines": [ + { + "bbox": [ + 108, + 226, + 154, + 235 + ], + "spans": [ + { + "bbox": [ + 108, + 226, + 154, + 235 + ], + "score": 1.0, + "content": "@ti.kernel", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 108, + 233, + 192, + 246 + ], + "spans": [ + { + "bbox": [ + 108, + 233, + 192, + 246 + ], + "score": 1.0, + "content": "def svd_gradient():", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 117, + 242, + 194, + 253 + ], + "spans": [ + { + "bbox": [ + 117, + 242, + 194, + 253 + ], + "score": 1.0, + "content": "for i in range(n):", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 127, + 249, + 379, + 261 + ], + "spans": [ + { + "bbox": [ + 127, + 249, + 379, + 261 + ], + "score": 1.0, + "content": "... Implement, for example, section 2.1.1.2 of Jiang (2015) .", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 109, + 267, + 407, + 309 + ], + "lines": [ + { + "bbox": [ + 107, + 266, + 408, + 276 + ], + "spans": [ + { + "bbox": [ + 107, + 266, + 408, + 276 + ], + "score": 1.0, + "content": "# A complex kernel that is registered as the svd_forward complex kernel", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 108, + 273, + 262, + 284 + ], + "spans": [ + { + "bbox": [ + 108, + 273, + 262, + 284 + ], + "score": 1.0, + "content": "@ti.complex_kernel_grad(svd_forward)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 108, + 282, + 249, + 293 + ], + "spans": [ + { + "bbox": [ + 108, + 282, + 249, + 293 + ], + "score": 1.0, + "content": "def svd_backward(num_iterations):", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 290, + 200, + 300 + ], + "spans": [ + { + "bbox": [ + 116, + 290, + 200, + 300 + ], + "score": 1.0, + "content": "# differentiave SVD", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 116, + 298, + 178, + 308 + ], + "spans": [ + { + "bbox": [ + 116, + 298, + 178, + 308 + ], + "score": 1.0, + "content": "svd_gradient()", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 216, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 217, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 217, + 347 + ], + "score": 1.0, + "content": "D CHECKPOINTING", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "In this section we demonstrate how to use checkpointing via complex kernels. The goal of check-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "pointing is to use recomputation to save memory space. We demonstrate this using the diffmpm", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 378, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 504, + 390 + ], + "score": 1.0, + "content": "example, whose simulation cycle consists of particle to grid transform (p2g), grid boundary con-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 304, + 402 + ], + "score": 1.0, + "content": "ditions (grid_op), and grid to particle transform", + "type": "text" + }, + { + "bbox": [ + 304, + 390, + 324, + 401 + ], + "score": 0.41, + "content": "( \\mathtt { g } 2 \\mathsf { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 389, + 459, + 402 + ], + "score": 1.0, + "content": ". We assume the simulation has", + "type": "text" + }, + { + "bbox": [ + 460, + 389, + 482, + 401 + ], + "score": 0.91, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "time", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 399, + 133, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 133, + 415 + ], + "score": 1.0, + "content": "steps.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 424, + 296, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 298, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 298, + 437 + ], + "score": 1.0, + "content": "D.1 RECOMPUTATION WITHIN TIME STEPS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 334, + 458 + ], + "score": 1.0, + "content": "A naive implementation without checkpointing allocates", + "type": "text" + }, + { + "bbox": [ + 334, + 446, + 357, + 457 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "copied of the simulation grid, which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "can cost a lot of memory space. Actually, if we recompute the grid states during the backward", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 235, + 480 + ], + "score": 1.0, + "content": "simulation time step by redoing", + "type": "text" + }, + { + "bbox": [ + 235, + 469, + 249, + 479 + ], + "score": 0.58, + "content": "{ \\mathsf { p } } 2 { \\mathsf { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "and grid_op, we can reused the grid states and allocate only one", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 404, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 404, + 491 + ], + "score": 1.0, + "content": "copy. This checkpointing optimization is demonstrated in the code below:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "table", + "bbox": [ + 107, + 510, + 503, + 660 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 510, + 503, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 510, + 503, + 660 + ], + "spans": [ + { + "bbox": [ + 107, + 510, + 503, + 660 + ], + "score": 0.345, + "html": "
@ti.complex_kernel
def advance(s):
clear_grid()
compute_actuation(s)
p2g(s)
grid_op()
g2p(s)
@ti.complex_kernel_grad(advance)
def advance_grad(s):
clear_grid() p2g(s)
grid_op() # recompute the grid
g2p.grad(s)
grid_op.grad()
p2g.grad(s) compute_actuation.grad(s)
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"score": 1.0, + "content": "# Do Singular Value Decomposition (SVD) on n matrices", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 108, + 106, + 154, + 116 + ], + "spans": [ + { + "bbox": [ + 108, + 106, + 154, + 116 + ], + "score": 1.0, + "content": "@ti.kernel", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 108, + 113, + 287, + 125 + ], + "spans": [ + { + "bbox": [ + 108, + 113, + 287, + 125 + ], + "score": 1.0, + "content": "def iterative_svd(num_iterations: ti.f32):", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 122, + 196, + 134 + ], + "spans": [ + { + "bbox": [ + 117, + 122, + 196, + 134 + ], + "score": 1.0, + "content": "for i in range(n):", + "type": "text" + } + ], + "index": 3, + "is_list_end_line": true + }, + { + "bbox": [ + 126, + 131, + 207, + 141 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The goal of check-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "pointing is to use recomputation to save memory space. 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We assume the simulation has", + "type": "text" + }, + { + "bbox": [ + 460, + 389, + 482, + 401 + ], + "score": 0.91, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "time", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 399, + 133, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 133, + 415 + ], + "score": 1.0, + "content": "steps.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 356, + 506, + 415 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 424, + 296, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 298, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 298, + 437 + ], + "score": 1.0, + "content": "D.1 RECOMPUTATION WITHIN TIME STEPS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 334, + 458 + ], + "score": 1.0, + "content": "A naive implementation without checkpointing allocates", + "type": "text" + }, + { + "bbox": [ + 334, + 446, + 357, + 457 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "copied of the simulation grid, which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "can cost a lot of memory space. 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@ti.complex_kernel
def advance(s):
clear_grid()
compute_actuation(s)
p2g(s)
grid_op()
g2p(s)
@ti.complex_kernel_grad(advance)
def advance_grad(s):
clear_grid() p2g(s)
grid_op() # recompute the grid
g2p.grad(s)
grid_op.grad()
p2g.grad(s) compute_actuation.grad(s)
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Our liquid simulation can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 403, + 368, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 368, + 415 + ], + "score": 1.0, + "content": "two-way coupled with elastic object simulation (Figure 6, right).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 380, + 506, + 415 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 428, + 399, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 401, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 401, + 441 + ], + "score": 1.0, + "content": "E.3 DIFFERENTIABLE INCOMPRESSIBLE FLUID SIMULATOR [smoke]", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "image", + "bbox": [ + 110, + 452, + 501, + 549 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 452, + 501, + 549 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 452, + 501, + 549 + ], + "spans": [ + { + "bbox": [ + 110, + 452, + 501, + 549 + ], + "score": 0.967, + "type": "image", + "image_path": "032fd5e7cb773f9949bec038a3eaf43d815115cd2c63f7d61cc1a2a2e1ce9f67.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 110, + 452, + 501, + 484.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 110, + 484.3333333333333, + 501, + 516.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 110, + 516.6666666666666, + 501, + 549.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 552, + 505, + 585 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "Figure 7: (a): (Left to right) with an optimized initial smoke velocity field, the fluid changes its", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "pattern to a “Taichi\" symbol. [Reproduce: python3 smoke_taichi.py] (b): Unoptimized (top three)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 572, + 499, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 499, + 587 + ], + "score": 1.0, + "content": "and optimized (bottom three) waves at time step 3, 189, and 255. [Reproduce: python3 wave.py]", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "Backpropagating Through Pressure Projection We followed the baseline implementation in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "Autograd, and used 10 Jacobi iterations for pressure projection. Technically, 10 Jacobi iterations are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "not sufficient to make the velocity field fully divergence-free. However, in this example, it does a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "decent job, and we are able to successfully backpropagate through the unrolled 10 Jacobi iterations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 605, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In larger-scale simulations, 10 Jacobi iterations are likely not sufficient. Assuming the Poisson solve", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "is done by an iterative solver (e.g. multigrid preconditioned conjugate gradients, MGPCG) with 5", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "multigrid levels and 50 conjugate gradient iterations, then automatic differentiation will likely not", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "be able to provide gradients with sufficient numerical accuracy across this long iterative process.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "The accuracy is likely worse when conjugate gradients present, as they are known to numerically", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "drift as the number of iterations increases. In this case, the user can still use DiffTaichi to implement", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "the forward MGPCG solver, while implementing the backward part of the Poisson solve manually,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "likely using adjoint methods (Errico, 1997). DiffTaichi provides “complex kernels” to override the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 290, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 290, + 106 + ], + "score": 1.0, + "content": "built-in AD system, as shown in Appendix C.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 655, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "likely using adjoint methods (Errico, 1997). DiffTaichi provides “complex kernels” to override the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 290, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 290, + 106 + ], + "score": 1.0, + "content": "built-in AD system, as shown in Appendix C.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 117, + 433, + 130 + ], + "lines": [ + { + "bbox": [ + 105, + 116, + 434, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 434, + 132 + ], + "score": 1.0, + "content": "E.4 DIFFERENTIABLE HEIGHT FIELD SHALLOW WATER SIMULATOR [wave]", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 138, + 491, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 137, + 492, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 492, + 152 + ], + "score": 1.0, + "content": "We adopt the wave equation in Wang et al. (2018) to model shallow water height field evolution:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 152, + 351, + 166 + ], + "lines": [ + { + "bbox": [ + 259, + 152, + 351, + 166 + ], + "spans": [ + { + "bbox": [ + 259, + 152, + 351, + 166 + ], + "score": 0.9, + "content": "\\ddot { u } = c ^ { 2 } \\nabla ^ { 2 } u + c \\alpha \\nabla ^ { 2 } \\dot { u } ,", + "type": "interline_equation", + "image_path": "07ab8b3c6aa8fcdf7f1a2381afb5d3e5df30d0b5b0a6a5fc517f9d38e009e8f7.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 259, + 152, + 351, + 166 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 169, + 507, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 134, + 182 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 172, + 141, + 180 + ], + "score": 0.74, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 168, + 269, + 182 + ], + "score": 1.0, + "content": "is the height of shallow water,", + "type": "text" + }, + { + "bbox": [ + 269, + 172, + 275, + 180 + ], + "score": 0.76, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 168, + 393, + 182 + ], + "score": 1.0, + "content": "is the “speed of sound” and", + "type": "text" + }, + { + "bbox": [ + 393, + 172, + 401, + 180 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 168, + 505, + 182 + ], + "score": 1.0, + "content": "is a damping coefficient.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 181, + 504, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 156, + 192 + ], + "score": 1.0, + "content": "We use the", + "type": "text" + }, + { + "bbox": [ + 156, + 181, + 164, + 191 + ], + "score": 0.8, + "content": "\\dot { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 181, + 184, + 192 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 181, + 191, + 191 + ], + "score": 0.8, + "content": "\\ddot { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 181, + 447, + 192 + ], + "score": 1.0, + "content": "notations for the first and second order partial derivatives of", + "type": "text" + }, + { + "bbox": [ + 447, + 183, + 454, + 191 + ], + "score": 0.76, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 181, + 498, + 192 + ], + "score": 1.0, + "content": "w.r.t time", + "type": "text" + }, + { + "bbox": [ + 499, + 182, + 504, + 191 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 191, + 160, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 160, + 205 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 503, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "Wang et al. (2018) used the finite different time-domain (FDTD) method (Larsson & Thomée, 2008)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 220, + 299, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 299, + 233 + ], + "score": 1.0, + "content": "to discretize Eqn. 1, yielding an update scheme:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 133, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 135, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 135, + 263 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 230, + 473, + 283 + ], + "lines": [ + { + "bbox": [ + 145, + 230, + 473, + 283 + ], + "spans": [ + { + "bbox": [ + 145, + 230, + 473, + 283 + ], + "score": 0.44, + "content": "\\begin{array} { r l } & { \\mathrm { \\quad } _ { t , i , j } = 2 u _ { t - 1 , i , j } + ( c ^ { 2 } \\dot { \\Delta t ^ { 2 } } + c \\alpha \\Delta t ) ( \\nabla ^ { 2 } u ) _ { t - 1 , i , j } - p _ { t - 2 , i , j } - c \\alpha \\Delta t ( \\nabla ^ { 2 } u ) _ { t - 2 , i , j } , } \\\\ & { \\mathrm { \\quad } ( \\nabla ^ { 2 } u ) _ { t , i , j } = \\frac { - 4 u _ { t , i , j } + u _ { t , i , j + 1 } + u _ { t , i , j - 1 } + u _ { t , i + 1 , j } + u _ { t , i - 1 , j } } { { \\Delta x ^ { 2 } } } . } \\end{array}", + "type": "interline_equation", + "image_path": "f09174162393aa18666d7f419aba2cada6b04a8df2c7aa4a6f0a0ab6ef5dea48.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 138, + 230, + 473, + 247.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 138, + 247.66666666666666, + 473, + 265.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 138, + 265.3333333333333, + 473, + 283.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 288, + 504, + 311 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "We implemented this wave simulator in DiffTaichi to simulate shallow water. We used a grid of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 299, + 397, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 149, + 311 + ], + "score": 1.0, + "content": "resolution", + "type": "text" + }, + { + "bbox": [ + 149, + 300, + 192, + 310 + ], + "score": 0.9, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 299, + 397, + 311 + ], + "score": 1.0, + "content": "and 256 time steps. The loss function is defined as", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 323, + 365, + 351 + ], + "lines": [ + { + "bbox": [ + 246, + 323, + 365, + 351 + ], + "spans": [ + { + "bbox": [ + 246, + 323, + 365, + 351 + ], + "score": 0.93, + "content": "L = \\sum _ { i , j } \\Delta x ^ { 2 } ( u _ { T , i , j } - \\hat { u } _ { i , j } ) ^ { 2 }", + "type": "interline_equation", + "image_path": "284b5abfd336d90309bc4bad734d0c252d1e0b9a090341ca6973d05ef9429c3d.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 323, + 365, + 337.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 246, + 337.0, + 365, + 351.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 506, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 133, + 370 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 358, + 142, + 368 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 357, + 250, + 370 + ], + "score": 1.0, + "content": "is the final time step, and", + "type": "text" + }, + { + "bbox": [ + 250, + 358, + 258, + 368 + ], + "score": 0.82, + "content": "\\hat { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "is the target height field. 200 gradient descent iterations are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 316, + 381 + ], + "score": 1.0, + "content": "then used to optimize the initial height field. We set", + "type": "text" + }, + { + "bbox": [ + 316, + 370, + 323, + 379 + ], + "score": 0.82, + "content": "\\hat { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "to be the pattern “Taichi\", and Fig. 7b shows", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 380, + 298, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 298, + 391 + ], + "score": 1.0, + "content": "the unoptimized and optimized wave evolution.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 504, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 410 + ], + "score": 1.0, + "content": "We set the “Taichi\" symbol as the target pattern. Fig. 7b shows the unoptimized and optimized final", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 407, + 361, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 361, + 420 + ], + "score": 1.0, + "content": "wave patterns. More details on discretization is in Appendix E.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 432, + 365, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 366, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 366, + 446 + ], + "score": 1.0, + "content": "E.5 DIFFERENTIABLE MASS-SPRING SYSTEM [mass_spring]", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "We extend the mass-spring system in the main text with ground collision and a NN controller. 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We designed three mass-spring robots as shown in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 158, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 158, + 498 + ], + "score": 1.0, + "content": "Fig. 8 (left).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 354, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 355, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 355, + 523 + ], + "score": 1.0, + "content": "E.6 DIFFERENTIABLE BILLIARD SIMULATOR [billiards]", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "A differentiable rigid body simulator is built for optimizing a billiards strategy (Fig. 8, middle). 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ApproachForward TimeBackward TimeTotal Time# Lines of Code
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CUDA0.10 ms0.14 ms0.24 ms (0.92×)460
DiffTaichi0.11 ms0.15 ms0.26 ms (1.00×)110
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ApproachForward TimeBackward TimeTotal Time#Essential LoC
PyTorch (CPU, f32)405 ms328 ms733 ms (13.8×)74
PyTorch (GPU, f32)254 ms457 ms711 ms (13.4×)74
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JAX (GPU, f32)24 ms75 ms99 ms (1.9×)90
DiffTaichi (CPU, f32)66 ms132 ms198 ms (3.7x)75
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FeatureDiffTaichiPyTorchTensorFlowEnokiJAXHalideJuliaSwift
GPUMegakernels<A△<
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for i ∈ range(0, 16, step 1) do
%1=load x[i]
%2 = mul %1, %1
%3= sin(%2)
store y[i] = %3
end for
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for i in range(0,16, step 1) do
/ adjoint variables
%1adj = alloca 0.0
%2adj = alloca 0.0
%3adj = alloca 0.0
// original forward computation
%1=load x[] %2 = mul %1, %1
%3= sin(%2)
/ reverse accumulation
%4 = load y_adj[i]
%3adj += %4
%5 = cos(%2)
%2adj += %3adj * %5
%1adj += 2 * %1 * %2adj
atomic add x_adj[i],%1adj end for
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@ti.complex_kernel
def advance(s):
clear_grid()
compute_actuation(s)
p2g(s)
grid_op()
g2p(s)
@ti.complex_kernel_grad(advance)
def advance_grad(s):
clear_grid() p2g(s)
grid_op() # recompute the grid
g2p.grad(s)
grid_op.grad()
p2g.grad(s) compute_actuation.grad(s)
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Law1,2,3, Sanja Fidler1,2,3 +1 Department of Computer Science, University of Toronto +2 Vector Institute +3 NVIDIA +{jcao, law, fidler}@cs.toronto.edu + +# ABSTRACT + +Few-shot classification is the task of predicting the category of an example from few labeled examples. The number of labeled examples per category is called the number of shots (or shot number). Recent works tackle this task through metalearning, where a meta-learner extracts information from observed tasks during meta-training to quickly adapt to new tasks during meta-testing. In this formulation, the number of shots exploited during meta-training has an impact on the recognition performance at meta-test time. Generally, the shot number used in meta-training should match the one used in meta-testing to obtain the best performance. We introduce a theoretical analysis of the impact of the shot number on Prototypical Networks, a state-of-the-art few-shot classification method. From our analysis, we propose a simple method that is robust to the choice of shot number used during meta-training, which is a crucial hyperparameter. The performance of our model trained for an arbitrary meta-training shot number shows great performance for different values of meta-testing shot numbers. We experimentally demonstrate our approach on different few-shot classification benchmarks. + +# 1 INTRODUCTION + +Human cognition has the impressive ability of grasping new concepts from exposure to a handful of examples (Yger et al., 2015). In comparison, while modern deep learning methods achieve unprecedented performances with very deep neural-networks (He et al., 2016; Szegedy et al., 2015), they require extensive amounts of data to train, often ranging in the millions. Few-shot learning aims to bridge the sample-efficiency gap between deep learning and human learning in fields such as computer vision, reinforcement learning and speech recognition (Santoro et al., 2016; Ravi & Larochelle, 2017; Finn et al., 2017; Vinyals et al., 2016; Wang et al., 2019a). These methods fall under the framework of meta-learning, in which a meta-learner extracts knowledge from many related tasks (in the meta-training phase) and leverages that knowledge to quickly learn new tasks (in the meta-testing phase). In this paper, we focus on the few-shot classification problem where each task is defined as a $N$ -way classification problem with $k$ samples (shots) per class available for training. + +Many meta-learning methods use the episodic training setup in which the meta-learner iterates through episodes in the meta-training phase. In each episode, a task is drawn from some population and a limited amount of support and query data from that task is made available. The meta-learner then learns a task-specific classifier on the support data and the classifier predicts on the query data. Updates to the meta-learner is computed based on the performance of the classifier on the query set. Evaluation of the meta-learner (during a phase called meta-testing) is also carried out in episodes in a similar fashion, except that the meta-learner is no longer updated and the performance on query data across multiple episodes is aggregated. + +In the episodic setup, the selection of $k$ during meta-training time can have significant effects on the learning outcomes of the meta-learner. Intuitively, if support data is expected to be scarce, the meta-learner needs to provide strong inductive bias to the task-specific learner as the danger of overfitting is high. In contrast, if support data is expected to be abundant, then the meta-learner can provide generally more relaxed biases to the task-specific learner to achieve better fitting to the task data. Therefore it is plausible that a meta-learner trained with one $k$ value can be suboptimal at adapting to tasks with a different $k$ value and thus exhibit meta-overfitting to $k$ . In experiments, $k$ is often simply kept fixed between meta-training and meta-testing, but in real-world usage, one cannot expect to know beforehand the amount of support data from unseen tasks during deployment. + +In this paper we will focus on Prototypical networks (Snell et al., 2017), a.k.a. ProtoNet. ProtoNet is of practical interest because of its flexibility: a single trained instance of ProtoNet can be used on new tasks with any $k$ and $N$ . However, ProtoNet exhibits performance degradation when the $k$ used in training does not match the $k$ used in testing.1 First, we will undertake a theoretical investigation to elicit the connection from $k$ to a lower bound of expected performance, as well as to the intrinsic dimension of the learned embedding space. Then, we conduct experiments to empirically verify our theoretical results across various settings. Guided by our new understanding of the effects of $k$ , we propose an elegant method to tackle performance degradation in mismatched $k$ cases. Our contributions are threefold: + +• We provide performance bounds for ProtoNets given an embedding function. From which, we argue that $k$ affects learning and performance by scaling the contribution of intra-class variance. + +• Through VC-learnability theory, we connect the value of $k$ used in meta-training to the intrinsic dimension of the embedding space. + +• The most important contribution of this paper (introduced in Section 3.3) is a new method that improves upon vanilla ProtoNets by eliminating the performance degradation in cases where the $k$ is mismatched between meta-training and meta-testing. Our evaluation protocol more closely adheres to real-world scenarios where the model is exposed to different numbers of training samples. + +# 2 BACKGROUND + +# 2.1 PROBLEM SETUP + +The few-shot classification problem considered in this paper is set up as described below. Consider a space of classes $C$ with a probability distribution $\tau$ , $N$ classes $\mathbf { c } \doteq \{ c _ { 1 } , . . . , c _ { N } \}$ are sampled i.i.d. from $\tau$ to form a $N$ -way classification problem. For each class $c _ { i }$ , $k$ support data are sampled from class-conditional distribution $S _ { i } = \{ _ { s } { \bf x } _ { 1 } , . . , _ { s } { \bf x } _ { k } \} \stackrel { i i d } { \sim } P ( { \bf x } | Y ( { \bf x } ) = c _ { i } )$ , where $\mathbf { x } \in \mathbb { R } ^ { D }$ , $D$ denotes the dimension of data, and $Y ( \mathbf { x } )$ denotes the class assignment of $\mathbf { x }$ . Note that we assume that $Y ( \mathbf { x } )$ is singular (e.g. each $\mathbf { x }$ can only have 1 label), and does not depend on $N$ (e.g. a data point with a label “cat” will always have the label “cat”), in contrast to $y$ defined below. + +Additionally, the set $Q = \{ _ { q } \mathbf { x } _ { 1 } , . . . , _ { q } \mathbf { x } _ { l } \}$ containing $l$ query data is sampled from the joint distribution $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } P ( \mathbf { x } | c _ { i } ) } \end{array}$ .2 For each $\mathbf { x }$ , let $y \in \{ 1 , . . . , N \}$ denote its label in the context of the few-shot classificafor class $c _ { i }$ n task. Define , and denote t $\hat { S } _ { i } = \left\{ ( _ { s } \mathbf { x } _ { 1 } , y _ { 1 } = i ) , . . . , ( _ { s } \mathbf { x } _ { k } , y _ { k } = i ) \right\}$ $N$ as as $\textstyle S = \bigcup _ { i = 1 } ^ { N } { \hat { S } } _ { i }$ set of supports. The few-shot classification task is to predict $y$ for each $\mathbf { x }$ in $Q$ given $S$ . During meta-training, the ground truth label for $Q$ is also available to the learner. + +# 2.2 META-LEARNING SETUP + +Meta-learning approaches train on a distribution of tasks to obtain information that generalizes to unseen tasks. For few-shot classification, a task is determined by which classes are involved in the $N$ -way classification task. During meta-training, the meta-learner observes episodes of few-shot classification tasks consisting of $N$ classes, $k$ labelled samples per class, and $l$ unlabelled samples, as previously described. The collection of all classes observed during meta-training forms the metatraining split $\mathcal { D } _ { t r } = \{ { _ { t r } c _ { 1 } } , . . . , { _ { t r } c _ { R } } \}$ . Critically, we assume that every unseen class that the learner is evaluated upon (during meta-testing) is also drawn from the same distribution $\tau$ . + +# 2.3 PROTOTYPICAL NETWORKS + +ProtoNets (Snell et al., 2017) compute $E$ -dimensional embeddings for all samples in $S$ and $Q$ . The embedding function $\phi : \mathbb { R } ^ { D } \mathbb { R } ^ { \dot { E } }$ is usually a deep network.The prototype representation for each class is formed by averaging the embeddings for all supports of said class: $\begin{array} { r } { \overline { { \phi ( S _ { i } ) } } = \frac { 1 } { k } \sum _ { \mathbf { x } \in S _ { i } } \phi ( \mathbf { x } ) } \end{array}$ . Classification of any input $\mathbf { x }$ (e.g. $\mathbf { x } \in Q$ i) is performed by computing the softmax over squared Euclidean distances of the input point’s embedding to the prototypes. Let $\hat { y }$ denote the prediction of the classifier for one of the categories $j \in \{ 1 , \cdots , N \}$ : + +$$ +p _ { \phi } ( \hat { y } = j | \mathbf { x } , S ) = \frac { e ^ { - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { j } ) } } \right\| ^ { 2 } } } { \sum _ { i = 1 } ^ { N } e ^ { - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { i } ) } } ^ { 2 } \right\| } } \quad , \mathrm { w h e r e } \quad \left\| \mathbf { v } \right\| ^ { 2 } = \sum _ { d = 1 } ^ { E } v _ { d } ^ { 2 } +$$ + +The parameters of the embedding functions are learned through meta-training. Negative log-likelihood $J ( \phi ) = - \log \left( p ( \hat { y } = y | \mathbf { x } ) \right)$ of the correct class $y$ is minimized on the query points through SGD. + +As explained in (Law et al., 2019), ProtoNets can be seen as a metric learning approach optimized for the supervised hard clustering task (Law et al., 2016). The model $\phi$ is learned so that the representations of similar examples (i.e. belonging to a same category) are all grouped into the same cluster in $\mathbb { R } ^ { E }$ . We propose in this paper a subsequent metric learning step which learns a linear transformation that maximizes inter-to-intra class variance ratio. + +# 3 PROPOSED METHOD + +We first present theoretical results explaining the effect of the shot number on ProtoNets, and then introduce our method for addressing performance degradation in cases of mismatched shots. + +# 3.1 RELATING $k$ TO LOWER BOUND OF EXPECTED ACCURACY + +To better understand the role of $k$ on the performance of ProtoNets, we study how it contributes to the expected accuracy across episodes when using any kind of fixed embedding function (e.g. the embedding function obtained at the end of the meta-training phase). With $I$ denoting the indicator function, we define the expected accuracy $R$ as: + +$$ +R ( \phi ) = \mathbb { E } _ { \mathbf { c } } \mathbb { E } _ { S , \mathbf { x } , y } I [ \arg \operatorname* { m a x } _ { j } \left\{ p _ { \phi } ( \hat { y } = j | \mathbf { x } , S ) \right\} = y ] +$$ + +Definitions: Throughout this section, we will use the following symbols to denote the means and variances of embeddings under different expectations: + +$$ +{ \begin{array} { r l } & { \mathbf { \mu } _ { c } \triangleq \mathbb { E } _ { \mathbf { x } } [ \phi ( \mathbf { x } ) | \ X ( \mathbf { x } ) = c ] \qquad \quad \Sigma _ { c } \triangleq \mathbb { E } _ { \mathbf { x } } [ ( \phi ( \mathbf { x } ) - \mathbf { \mu } _ { c } ) ( \phi ( \mathbf { x } ) - \mathbf { \mu } _ { c } ) ^ { T } | \ Y ( \mathbf { x } ) = c ] } \\ & { \quad \mathbf { \mu } \triangleq \mathbb { E } _ { c } [ \mathbf { \mu } _ { c } ] \qquad \quad \Sigma \triangleq \mathbb { E } _ { c } [ ( \mathbf { \mu } _ { c } - \mathbf { \mu } ) ( \mathbf { \mu } _ { c } - \mathbf { \mu } ) ^ { T } ] } \end{array} } +$$ + +Remark. $\mu _ { c }$ is the expectation of the embedding conditioned on class $c , \mu$ is the (full) expectation of the embedding, which can be expressed as the expectation of $\mu _ { c }$ over classes. $\Sigma$ is the variance of class means in the embedding space - it can be interpreted as the signal of the input to the classifier, as larger $\Sigma$ implies larger distances between classes. $\Sigma _ { c }$ is the expected intra-class variance - it represents the noise in the above signal. + +Modelling assumptions of ProtoNets: The use of the squared Euclidean distance and softmax activation in ProtoNets implies that classification with ProtoNets is equivalent to a mixture density estimation on the support set with spherical Gaussian densities (Snell et al., 2017). Specifically, we adopt the modelling assumptions that the distribution of $\phi ( \mathbf { x } )$ given any class assignment is normally distributed $( p ( \phi ( \mathbf { x } ) \bar { | } Y ( \mathbf { x } ) \bar { = } c ) = \mathcal { N } ( \mu _ { c } , \Sigma _ { c } ) )$ , with equal covariance for all classes in the embedding space $( \forall ( c , c ^ { \prime } ) , \Sigma _ { c } = \Sigma _ { c ^ { \prime } } ) ^ { 3 }$ . + +We present the analysis for the special case of episodes with binary classification (i.e. with $N = 2$ ) for ease of presentation, but the conclusion can be generalized to arbitrary $N > 2$ (see appendix). + +Also, as noted in Section 2.1, we assume equal likelihood between the classes. We would like to emphasize that the assignment of labels can be permuted freely and the classifier’s prediction would not be affected due to symmetry. Hence, we only need to consider one case for the ground truth label without loss of generality. Let $a$ and $b$ denote any pair of classes sampled from $\tau$ . Let $\mathbf { x }$ be drawn from $a$ , and overload $a$ and $b$ to also indicate the ground truth label in the context of that episode, then equation 2 can be written as: + +$$ +R ( \phi ) = \mathbb { E } _ { a , b \sim \tau } \mathbb { E } _ { \mathbf { x } , S } I [ \hat { y } = a ] +$$ + +Additionally, noting that $p ( \hat { y } = a )$ can be expressed as a sigmoid function $\sigma$ : + +$$ +p ( \hat { y } = a | \mathbf { x } ) = \frac { e ^ { - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } } } { e ^ { - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } } + e ^ { - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } } } = \sigma ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } ) +$$ + +We can express equation 3 as a probability: + +$$ +R ( \phi ) = \operatorname* { P r } _ { a , b , \mathbf { x } , S } ( { \hat { y } } = a ) = \operatorname* { P r } _ { a , b , \mathbf { x } , S } ( \alpha > 0 ) +$$ + +We will introduce a few auxiliary results before stating the main result for this section. + +Proposition 1. From the one-sided Chebyshev’s inequality, it immediately follows that: + +$$ +R ( \phi ) = \mathrm { P r } ( \alpha > 0 ) \geq \frac { \mathbb { E } [ \alpha ] ^ { 2 } } { \mathrm { V a r } ( \alpha ) + \mathbb { E } [ \alpha ] ^ { 2 } } +$$ + +In Lemma 1 and Lemma 2, we derive the expectation and variance of $\alpha$ when conditioned on the classes sampled in a episode. Then, in Theorem 3, we compose them into the $R H S$ of Proposition 1 through law of total expectation. + +Lemma 1. Consider space of classes $C$ with sampling distribution $\tau$ $; , a , b \stackrel { i i d } { \sim } \tau .$ . Let $S = \{ S _ { a } , S _ { b } \}$ $S _ { a } = \left\{ { { \bf { \Psi } } _ { a } { \bf { x } } _ { 1 } } , . . . , { \bf { \Psi } } _ { a } { \bf { x } } _ { k } \right\}$ , $S _ { b } = \left\{ \vphantom { b } _ { b } \mathbf { x } _ { 1 } , . . . , \vphantom { b } _ { b } \mathbf { x } _ { k } \right\}$ , $k \in \mathbb N$ is the shot number, and $Y ( \mathbf { x } ) = a$ . Define ${ \overline { { \phi ( S _ { a } ) } } } \triangleq { \frac { 1 } { k } } \sum _ { \mathbf { x } \in S _ { a } } \phi ( \mathbf { x } )$ and ${ \overline { { \phi ( S _ { b } ) } } } \triangleq { \frac { 1 } { k } } \sum _ { \mathbf { x } \in S _ { b } } \phi ( \mathbf { x } )$ . Consider $\Sigma$ as defined earlier. Assume $p ( \phi ( \mathbf { x } ) | Y ( \mathbf { x } ) = c ) = N ( \mu _ { c } , \Sigma _ { c } )$ and $\Sigma _ { c } = \Sigma _ { c ^ { \prime } }$ for any choice of $c , c ^ { \prime } \in C$ , then, + +$$ +\mathbb { E } _ { \mathbf { x } , S | a , b } [ \alpha ] = \left( \mathtt { u } _ { a } - \mathtt { u } _ { b } \right) ^ { T } ( \mathtt { u } _ { a } - \mathtt { u } _ { b } ) \qquad , a n d \qquad \mathbb { E } _ { a , b , \mathbf { x } , S } [ \alpha ] = 2 \mathrm { T r } ( \Sigma ) . +$$ + +Lemma 2. Under the same notation and assumptions as Lemma $I$ , additionally invoking definition for $\Sigma _ { c } ,$ then, + +$$ +\mathbb { E } _ { a , b } [ \mathrm { V a r } ( \alpha | a , b ) ] \leq 8 ( 1 + \frac { 1 } { k } ) \mathrm { T r } \left( \Sigma _ { c } ( ( 1 + \frac { 1 } { k } ) \Sigma _ { c } + 2 \Sigma ) \right) . +$$ + +The proofs of the above lemmas are in the appendix. With the results above, we are ready to state our main theoretical result in this section. + +Theorem 3. Under the conditions where Lemma 1 and 2 hold, we have: + +$$ +R ( \phi ) \geq \frac { 4 \mathrm { T r } ( \Sigma ) ^ { 2 } } { 8 ( 1 + 1 / k ) ^ { 2 } \mathrm { T r } ( \Sigma _ { c } ^ { 2 } ) + 1 6 ( 1 + 1 / k ) \mathrm { T r } ( \Sigma \Sigma _ { c } ) + \mathbb { E } _ { a , b } [ ( \mathbf { \mu } ( \mathbf { \mu } _ { a } - \mathbf { \mu } \mathbf { \mu } _ { b } ) ^ { T } ( \mathbf { \mu } _ { a } - \mathbf { \mu } \mathbf { \mu } _ { b } ) ) ^ { 2 } ] } . +$$ + +Proof. First, we use decompose the $\mathrm { V a r } ( \alpha )$ term in Proposition 1 by Law of Total Expectation. + +$$ +\begin{array} { r l } & { \mathrm { V a r } ( \boldsymbol \alpha ) = \mathbb { E } _ { a , b , \mathbf { x } , S } [ \boldsymbol \alpha ^ { 2 } ] - \mathbb { E } _ { a , b , \mathbf { x } , S } [ \boldsymbol \alpha ] ^ { 2 } = \mathbb { E } _ { a , b } \mathbb { E } _ { \mathbf { x } , S } [ \boldsymbol \alpha ^ { 2 } | a , b ] - \mathbb { E } _ { a , b , \mathbf { x } , S } [ \boldsymbol \alpha ] ^ { 2 } } \\ & { \quad \quad \quad \quad = \mathbb { E } _ { a , b } [ \mathrm { V a r } ( \boldsymbol \alpha | a , b ) + \mathbb { E } _ { \mathbf { x } , S } [ \boldsymbol \alpha | a , b ] ^ { 2 } ] - \mathbb { E } _ { a , b , \mathbf { x } , S } [ \boldsymbol \alpha ] ^ { 2 } . } \end{array} +$$ + +Hence, Proposition 1 can also be expressed as + +$$ +R ( \phi ) \geq \frac { \mathbb { E } [ \alpha ] ^ { 2 } } { \mathbb { E } _ { a , b } [ \mathrm { V a r } ( \alpha \vert a , b ) + \mathbb { E } _ { \mathbf { x } , S } [ \alpha \vert a , b ] ^ { 2 } ] } . +$$ + +Finally, we arrive at Theorem 3 by plugging Lemma 1 and 2 into equation 10. + +Several observations can be made from Theorem 3: + +1. The shot number $k$ only appears in the first two terms of the denominator, implying that the bound saturates quickly with increasing $k$ . This is also in agreement with the empirical observation that meta-testing accuracy has diminishing improvements when more support data is added. + +2. By observing the degree of terms in equation 7 (and treating the last term of the denominator as a constant), it is clear that increasing $k$ will decrease the sensitivity (magnitude of partial derivative) of this lower bound to $\Sigma _ { c }$ , and increase its sensitivity to $\Sigma$ . + +3. If one postulates that meta-learning updates on $\phi$ are similar to gradient ascent on this accuracy lower bound, then learning with smaller $k$ emphasizes minimizing noise, while learning with higher $k$ emphasizes maximizing signal. + +In conclusion, these observations give us a plausible reason for the performance degradation observed in mismatched shots: when an embedding function has been optimized (trained) for $k _ { t r a i n } ~ >$ $k _ { t e s t }$ , the relatively high $\Sigma _ { c }$ is amplified by the now smaller $k$ , resulting in degraded performance. Conversely, an embedding function trained for $k _ { t r a i n } ~ < ~ k _ { t e s t }$ already has small $\Sigma _ { c }$ , such that increasing $k$ during testing has further diminished improvement on performance. + +# 3.2 INTERPRETATION IN TERMS OF VC DIMENSION + +In any given episode, a nearest neighbour prediction is performed from the support data (with a fixed embedding function). Therefore, a PAC learnability interpretation of the relation between the number of support data and complexity of the classifier can be made. Specifically, for binary classification, classical PAC learning theory (Vapnik et al., 1994) states that with probability of at least $1 - \delta$ , the following inequality on the difference between empirical error $e r r _ { t r a i n }$ (of the support samples) and true error $e r r _ { t r u e }$ holds for any classifier $h$ : + +$$ +e r r _ { t r u e } ( h ) - e r r _ { t r a i n } ( h ) \leq \sqrt { \frac { D ( \ln \frac { 4 k } { D } + 1 ) + \ln \frac { 4 } { \delta } } { 2 k } } +$$ + +Where $D$ is the VC dimension, and $k$ is the number of support samples per class 4. Under this binary classification setting, the predictions of prototypical network are equal to $\sigma ( \alpha )$ as shown earlier. Denoting $\mathbf { z } _ { c } = \overline { { \phi ( S _ { c } ) } }$ and $\mathbf { z } _ { c ^ { \prime } } = \overline { { \phi ( S _ { c ^ { \prime } } ) } }$ , we can manipulate $\alpha$ as follows: + +$$ +\begin{array} { r l } { \alpha = \left\| \phi ( \mathbf { x } ) - \mathbf { z } _ { c } \right\| ^ { 2 } - \left\| \phi ( \mathbf { x } ) - \mathbf { z } _ { c ^ { \prime } } \right\| ^ { 2 } } & { { } = 2 ( \mathbf { z } _ { c ^ { \prime } } - \mathbf { z } _ { c } ) ^ { T } \phi ( \mathbf { x } ) + ( \mathbf { z } _ { c } ^ { T } \mathbf { z } _ { c } - \mathbf { z } _ { c ^ { \prime } } ^ { T } \mathbf { z } _ { c ^ { \prime } } ) } \end{array} +$$ + +From equation 12, the prototypes form a linear classifier with offset in the embedding space. The VC dimension of this type of classifier is $1 + d$ where $d$ is the intrinsic dimension of the embedding space (Vapnik, 1998). In ProtoNets, the intrinsic dimension of the embedding space is not only influenced by network architecture, but more importantly determined by the parameter themselves, making it a learned property. For example, if the embedding function can be represented with a linear transformation $\phi ( \mathbf { x } ) = \Phi \cdot \mathbf { x }$ , then the intrinsic dimension of the embedding space is upper bounded by the rank of $\Phi$ (since all embeddings must lie in the column space of $\Phi$ ). Thus, the number of support samples required to learn from an episode is proportional to the intrinsic dimension of the embedding space. We hypothesize that an embedding function optimal for lower shot (e.g. one-shot) classification affords fewer intrinsic dimensions than one that is optimal for higher shot (e.g. 10-shot) classification. + +# 3.3 RECONCILING SHOT DISCREPANCY THROUGH EMBEDDING SPACE TRANSFORMATION + +Observations in Section 3.2 reveal that an ideal $\phi$ would have an output space whose intrinsic dimension $d$ is as small as possible to minimize the right-hand side of equation 11 but just large enough to allow low $e r r _ { t r a i n }$ ; the balance between the two objectives is dictated by $k$ . Similarly, observations in Section 3.1 suggest that an ideal $\phi$ would balance between minimizing $\Sigma _ { c }$ and maximizing $\Sigma$ also according to $k$ . As a result, when there is discrepancy between meta-training shots and meta-testing shots5, accuracy at meta-test time will suffer. A naive solution is to prepare many embedding functions trained for different shots, and select the embedding function according to the availability of label data at test-time. However, this solution is computationally burdensome as it requires multiple models to be trained and stored. Instead, we want to train and store a single model, and then adapt the embedding function’s variance characteristics and the embedding space dimensionality to achieve good performance for any test shot. + +Linear Discriminant Analysis (LDA) is a dimensionality reduction method suited for downstream classification (Fukunaga, 1990). Its goal is to find a maximally discriminating subspace (maximum inter-class variance and minimal intra-class variance) for a given classification task. Theoretically, performance can be maximized in each individual episode by computing the LDA transformation matrix using support samples of that episode. LDA computes the eigenvectors of the matrix $S ^ { - 1 } S _ { \mu }$ , where $S _ { \mu }$ is the covariance matrix of prototypes and $S$ is the class-conditional covariance matrix. In practice, $S _ { \mu }$ and $S$ cannot be stably estimated in few-shot episodes, preventing the direct application of LDA. + +We propose an alternative which we call Embedding Space Transformation (EST). The purpose of EST is to perform dimensionality reduction on the features, while also improving the ratio in Theorem 3. This is a different goal from LDA because Theorem 3 demonstrates that the expected performance across many episodes can be improved by maximizing $\Sigma$ and minimizing $\Sigma _ { c }$ . Similar to LDA, EST works by applying a linear transformation + +$$ +\phi ( \mathbf { x } ) \mapsto V ^ { * } ( \phi ( \mathbf { x } ) ) +$$ + +to the outputs of the embedding function. Here, $V ^ { * }$ is a linear transformation computed using $\mathcal { D } _ { t r }$ after meta-training has completed. To compute $V ^ { * }$ , we first iterate through all classes in $\mathcal { D } _ { t r }$ and compute their in-class means and covariance matrices in the embedding space. We can then find the covariance of means $\Sigma _ { \mu }$ , and the mean of covariances $\overline { { \Sigma } } _ { s }$ across $\mathcal { D } _ { t r }$ . Finally, $V ^ { * }$ is computed by taking the leading eigenvectors of $\Sigma _ { \mu } - \rho \overline { { \Sigma } } _ { s }$ - the difference between the covariance matrix of the mean and the mean covariance matrix with weight parameter $\rho$ . The exact procedure for computing $V ^ { * }$ is presented in the appendix. + +# 4 EXPERIMENTS AND RESULTS + +In this section, our first two experiments aim at supporting our theoretical results in Sections 3.1 and 3.2, while our third experiment demonstrates the improvement of EST on benchmark data sets over vanilla ProtoNets. To illustrate the applicability of our results to different embedding function architectures, all experiments are performed with both a vanilla 4-layer CNN (as in (Snell et al., 2017)) and a 7-layer Residual network (He et al., 2016). Detailed description of the architecture can be found in the appendix. Experiments are performed on three data sets: Omniglot (Lake et al., 2015), miniImageNet (Vinyals et al., 2016), and tieredImageNet (Ren et al., 2018). We followed standard data processing procedures which are detailed in the appendix. + +# 4.1 TRAINING SHOTS AFFECT VARIANCE CONTRIBUTION + +The total variance observed among embeddings of all data points can be seen as a composition of inter-class and (expected) intra-class variance based on the law of total variance $\mathrm { ( V a r } [ \mathbf { x } ] \ =$ $\mathbb { E } [ \mathrm { V a r } ( \mathbf { x } | c ) ] + \mathrm { V a r } ( \mathbb { E } [ \mathbf { x } | c ] ) = \mathbb { E } [ \Sigma _ { c } ] + \Sigma _ { \mu } )$ . Our analysis predicts that as we increase the shot number used during training, the ratio of inter-class to intra-class variance will decrease. + +To verify this hypothesis, we trained ProtoNets (vanilla and residual) with a range of shots (1 to 10 on miniImageNet and tiered imagenet, 1 to 5 on Omniglot) until convergence with 3 random initializations per group. Then, we computed the inter-class and intra-class covariance matrices across all samples in the training-set embedded by each network. To qualify the amplitude of each matrix, we take the trace of each covariance matrix. The ratio of inter-class to intra-class variance is presented in Figure 1: as we increase $k$ used during training, the inter-class to intra-class variance ratio decreases. This trend can be observed in both vanilla and residual embeddings, and across all three data sets, lending strong support to our result in Section 3.1. Another observation can be made that the ratio between inter-class and intra-class variance is significantly higher in the Omniglot data set than the other two data sets. This may indeed be reflective of the relative difficulty of each data set and the accuracy of ProtoNet on the data sets. + +![](images/654b081bf5f9f75d64263ab6ff1c8f71fc822d2b4748a3ef0ce6491dd18fc619.jpg) +Figure 1: Inter-class to Intra-class variance ratios of embedding space varies with $k$ used in training. Left: Omniglot. Right: miniImageNet and tieredImageNet. + +# 4.2 TRAINING SHOTS AFFECT INTRINSIC DIMENSION + +We consider the intrinsic dimension (id) of an embedding function with extrinsic dimension $E$ (operated on a data set) to be defined as the minimum integer $d$ where all embedded points of that data set lie within a $d$ -dimensional subspace of $\mathbb { R } ^ { E }$ (Bishop, 2006). A simple method for estimating $d$ is through principal component analysis (PCA) of the embedded data set. By eigendecomposing the covariance matrix of embeddings, we obtain the principal components expressed as the significant eigenvalues, and the principal directions expressed as the eigenvectors corresponding to those eigenvalues. The number of significant eigenvalues approximates the intrinsic dimension of the embedding space. When the subspace is linear, this approximation is exact; otherwise, it serves as an upper bound to the true intrinsic dimension (Fukunaga & Olsen, 1971). + +We determine the number of significant eigenvalues by an explained-variance over total-variance criterion. The qualifying metric is $\begin{array} { r } { r _ { d } \triangleq \sum _ { i \in [ 1 , d ] } \lambda _ { i } / \sum _ { i \in [ 1 , E ] } \lambda _ { i } } \end{array}$ . In our experiments, we set the threshold for $r _ { d }$ at 0.9. Similar to the previous experiment, we train ProtoNets with different shots to convergence. The total covariance matrix is then computed on the training set and eigendecomposition is performed. The approximate id is plotted for various values of $k$ in Figure 2. We can see a clear trend that as we increase training shot, the id of the embedding space increases. + +![](images/274fb3036767fc31f4d4c6b41ff94346ef998c29094bc159e001fc6cf33f4586.jpg) +Figure 2: Intrinsic dimension approximated by the number of significant eigenvalues varies with $k$ used in training. Left: Omniglot. Right: miniImageNet and tieredImageNet + +# 4.3 EXPERIMENTS WITH EST + +We evaluate the performance of EST on the three aforementioned data sets. The performance is compared against our implementation of vanilla ProtoNets as a baseline, as well as a variant of ProtoNets using principal components obtained from all embedding points (ProtoNet-PCA). + +All methods in this section use the same set of trained ProtoNets. As with before, networks are trained with $k \in \{ 1 , . . . , 5 \}$ on Omniglot and $k \in \{ 1 , . . . , 1 0 \}$ on miniImageNet and tieredImageNet. Additionally, we also trained a mixed-shot network for each data set. This is done by randomly selecting a value for $k$ within the specified range for each episode, and then sampling the corresponding number of support samples. Hyper-parameters for training are described in the appendix. + +(a) Omniglot-20-way, with 4 layer CNN. + +Table 1: Classification accuracies of ProtoNet variants. Best performing methods and any other runs within $9 5 \%$ confidence margin is in bold +(b) Omniglot-20-way, with 7 layer ResNet. + +
MODELTRAINING SHOTSTESTING SHOTSAVERAGE ACCURACYMODELTRAINING SHOTSTESTING SHOTSAVERAGE ACCURACY
1515
VANILLA PROTONET195.07%98.89%97.81%VANILLA PROTONET196.46%99.07%98.35%
VANILLA PROTONET593.42%98.78%97.25%VANILLA PROTONET594.42%98.99%97.75%
MIXED-k SHOT1-594.84%98.92%97.74%MIXED-k PROTONET1-596.53%99.15%98.43%
PCAPROTONET194.94%98.85%97.78%PCA PROTONET196.02%98.99%98.19%
EST PROTONET195.11%98.84%97.83 %EST PROTONET195.55%99.02%98.19%
+ +(c) miniImageNet-5-way, with 4 layer CNN. + +
MODELTRAINING SHOTSTESTING SHOTSAVERAGE ACCURACY
1510
PROTONET148.89%64.70%68.90%63.15%
PROTONET544.75%67.23%72.36%65.08%
PROTONET1039.99%66.23%72.47%63.54%
MIXED-k SHOT1-1049.36%67.96%72.27%65.83%
PCA PROTONET549.36%68.63%72.82%66.12%
EST PROTONET550.22%68.25%73.29%66.60%
+ +(d) miniImageNet-5-way, with 7 layer ResNet. + +
TRAININGTESTING SHOTSAVERAGE
MODELSHOTS1510ACCURACY
PROTONET152.65%68.27%72.29 %66.73%
PROTONET547.40%69.93%74.35%67.18%
PROTONET1042.20%68.23%74.54%65.75%
MIXED-k SHOT1-1051.74%69.09%73.63%67.41%
PCA PROTONET550.09%69.25%74.24%67.63%
EST PROTONET551.93%69.98%74.80%68.19%
+ +(e) tieredImageNet-5-way, with 4 layer CNN. + +
TRAININGTESTING SHOTSAVERAGE
MODELSHOTS1510ACCURACY
PROTONET147.37%63.70%67.99%61.85%
PROTONET542.33%66.51%72.05%64.05%
PROTONET1035.38%64.56%71.03%61.24%
MIXED-k SHOT1-1047.67%66.34%70.96%64.33%
PCA PROTONET548.34%67.07%71.65%64.96%
EST PROTONET548.85%67.24%72.09 %65.46%
+ +(f) tieredImageNet-5-way, with 7 layer ResNet. + +
TRAININGTESTING SHOTSAVERAGE
MODELSHOTS1510ACCURACY
PROTONET149.78%65.17%69.88%63.98%
PROTONET547.88%69.12%73.80%66.99 %
PROTONET1040.86%69.37%74.72%65.95%
MIXED-k SHOT1-1050.74%69.28%73.01 %66.95%
PCA PROTONET551.21%68.88%72.17 %67.14%
EST PROTONET553.05%69.30%73.63%67.91%
+ +Each model is evaluated on the test splits of the corresponding data sets (e.g. networks trained on Omniglot are only evaluated on Omniglot). Five test runs are performed per network on Omniglot to evaluate the $k$ -shot performance $( k \in [ 1 , 5 ] )$ ). Each run consists of 600 episodes formed by 1-5 support samples and 5 query sample per class. The performance is aggregated across runs for the combined performance. Similarly, on miniImageNet and tieredImageNet, 10 test runs are performed with support per sample $k \in [ 1 , 1 0 ]$ , 600 episodes per run, and 15 query samples in each episode. + +Model configuration: Vanilla ProtoNet is used as our baseline. We present the performance of multiple ProtoNets trained with different shots to illustrate the performance degradation issue. ProtoNet-PCA uses principal components of the training split embeddings in place of $V ^ { * }$ , with components other than the $d$ leading ones zeroed out. We carry out a parameter sweep on miniImageNet and set $d = 6 0$ ; the same value is used on the other two data sets. For selecting the training shot of the embedding network, we find that overall performance to be optimal using $k = 5$ . ProtoNet-EST contains three parameters that need to be determined: $\rho , d$ , and training shots of the embedding network. For our experiments, we set $\rho = 0 . 0 0 1$ and $d = 6 0$ based on performance on miniImageNet. For selecting the number of training shots, we use the same strategy as before by evaluating ProtoNetEST with all trained embedding networks and found the same trend to hold. + +As an abalation study, FC-ProtoNet adds a fully connected layer to the embedding network such that the output dimension is also 60. Results of this variant can be found in the appendix. + +EST performance results: Table 1 summarizes the performance of the evaluated methods on all data sets. Due to space constraints, only 1-shot, 5-shot, 10-shot and 1-10 shot average performance are included. Additional results are in the appendix. The best performing method in each evaluation is in bold. On Omniglot, there is no significant difference in performance between the best performing vanilla ProtoNet and any other methods. We attribute this to the already high accuracy of the baseline model. On miniImageNet and tieredImageNet, EST-ProtoNet significantly outperforms baseline methods and PCA-protonet in terms of average accuracy over test runs with different shots. + +We observe that matching the training shot to the test shot generally provides the best performance for vanilla ProtoNets. Also importantly, training with a mixture of different values of $k$ does not provide optimal performance when evaluated on the same mixture of $k$ values. Instead, the resulting performance is mediocre in all test shots. ProtoNet-EST provides minor improvements over the best-performing baseline method under most test shots settings. We hypothesize that this is due to EST aligning the embedding space to the directions with high inter-class variance and low intra-class variance. Comparison against the direct PCA approach demonstrates that the performance uplift is not entirely attributed to reducing the dimensions of the embedding space. + +In conclusion, EST improves the performance of ProtoNets on the more challenging data sets when evaluated with various test shots. It successfully tackles performance degradation when testing shots and training shots are different. This improvement is vital to the deployment of ProtoNets in real world scenarios where the number of support samples cannot be determined in advance. + +# 5 RELATED WORK + +We summarize related work on extensions of ProtoNets, on improving the few-shot classification setup, and on analyzing theoretical properties of meta-learning methods. + +Extensions of ProtoNets: Allen et al. (2019) build upon ProtoNets by allowing each class to be represented by multiple prototypes, thereby improving the representation power of ProtoNets. Oreshkin et al. (2018) use a context-conditioned embedding network to produce prototypes that are aware of the other classes. These prior works assume matched training and testing shots whereas our work focuses on setups where testing shots are not fixed (i.e. not necessarily the same as training shots). Our work is parallel to these works in that EST can be applied on the embeddings learned by these methods. + +Improvement of few-shot classification setup: Chen et al. (2019) extend the few-shot learning problem setup by considering domain adaptation in addition to learning novel classes. Specifically, they look at how well models trained on miniImageNet can perform on few-shot learning in CUB200. Importantly, they still force the number of shots to be consistent between training time and testing time. While their work deals with varying the domain of the episodes at test time, our work deals with varying shots. Concurrent to our work, Triantafillou et al. (2020) further broaden the scope of few-shot learning by introducing a benchmark composed of data from various domains; methods are tested on their ability to adapt to different domains and deal with class imbalance. We extend their work with a thorough analysis of how the number of shots affects the learning outcome, and further propose a method to overcome the negative impact of mismatched shots. + +Theoretical analysis of few-shot learning: Despite the myriad of methodological improvements, theoretical work on few-shot learning has been sparse. Wang et al. (2019b) provide a unifying formulation for few-shot learning methods, and clearly outline the key challenge in few-shot learning through a PAC argument, but do not introduce any new theoretical results. In contrast, our work introduces a novel bound for the accuracy of ProtoNets; this bound provides useful intuitions pertaining to how ProtoNets adapt to few-shot episodes. Additionally, we demonstrate theoretically and experimentally that the intrinsic dimension of the embedding function’s output space varies with the number of shots as a direct consequence of the challenges outlined in the PAC argument. To the best of our knowledge, Amit & Meir (2017) provide the only prior work to bound the error of a meta-learning agent. Specifically, they use the generalized PAC-Bayes framework to derive an error-rate bound for a MAML-style learning algorithm (where the hypothesis class is fixed). Their main result relates the performance of the learning algorithm to both the number of tasks encountered during meta-training and the number of shots given in any task. In contrast to our work, their result does not apply to non-parametric methods such as ProtoNets because the hypothesis class in ProtoNets can change from episode to episode depending on the number of ways. + +# 6 CONCLUSION AND FUTURE WORK + +We have explored how the number of support samples used during meta-training can influence the learned embedding function’s performance and intrinsic dimensions. Our proposed method transforms the embedding space to maximize inter-to-intra class variance ratio while constraining the dimensions of the space itself. In terms of applications, our method can be combined other works (Oreshkin et al., 2018; Ye et al., 2018; Rusu et al., 2019; Dong & Xing, 2018; Ren et al., 2018; Tapaswi et al., 2019) with an embedding learning component. We believe our approach is a significant step to reduce the impact of the shot number in meta-training, which is a crucial hyperparameter for few-shot classification. + +# ACKNOWLEDGMENTS + +We acknowledge partial support from NSERC COHESA NETGP485577-15 and Samsung. We thank Chaoqi Wang for discussion on the initial idea, and Clement Fuji Tsang, Mark Brophy and the ´ anonymous reviewers for helpful feedback on early versions of this paper. + +# REFERENCES + +Kelsey R. Allen, Evan Shelhamer, Hanul Shin, and Joshua B. Tenenbaum. Infinite mixture prototypes for few-shot learning. CoRR, abs/1902.04552, 2019. 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ISSN 0270-6474. doi: 10.1523/JNEUROSCI. 0607-15.2015. URL http://www.jneurosci.org/content/35/39/13351. + +# A APPENDIX + +# A.1 ALGORITHM FOR EST + +Below is the exact procedure for computing $\tau$ for embedding space transformation. + +Algorithm 1 Algorithm for computing the transformation $\tau$ . +$L _ { n }$ is the number of samples belonging to class $n$ ; $\mu _ { n }$ and $\Sigma _ { n }$ are the mean and covariance of the embeddings of that class; $\mu _ { T }$ and $\widetilde { \Sigma } _ { s }$ are the average of mean embeddings and covariances; $\Sigma _ { \mu }$ is the covariance of the mean embeddings; $V ^ { * }$ is the matrix of eigenvectors that correspoinds to the $d$ largest eigenvalues in $\Lambda$ . + +Input: Training set $\mathcal { D } _ { t r } = \{ ( \mathbf { x } _ { 1 } , y _ { 1 } ) , . . . , ( \mathbf { x } _ { M } , y _ { M } \}$ , where $y _ { i } \in \{ 1 , . . . , N \}$ , $\mathcal { D } _ { n }$ denotes the subset of $\mathcal { D } _ { t r }$ where $\forall y \in { \mathcal { D } } _ { n } , y = n$ , embedding function $\phi$ , weighting parameter $\rho$ , dimension parameter $d$ . + +$$ +\begin{array} { r } { \Sigma [ n ] = \frac { 1 } { L _ { n } } \sum _ { ( \mathbf { x } _ { i } , y _ { i } ) \in \mathcal { D } _ { n } } ^ { L _ { n } } ( \phi ( \mathbf { x } _ { i } ) - \mu _ { n } ) ( \phi ( \mathbf { x } _ { i } ) - \mu _ { n } ) ^ { T } } \end{array} +$$ + +# end for + +$$ +\begin{array} { r l } & { \mu _ { T } = \frac { 1 } { M } \sum _ { ( \mathbf { x } _ { i } , y _ { i } ) \in \mathcal { D } _ { t r } } \phi ( \mathbf { x } _ { i } ) } \\ & { \Sigma _ { \mu } = \frac { 1 } { N } \sum _ { n \in [ 1 , N ] } ( \mu [ n ] - \mu _ { T } ) ( \mu [ n ] - \mu _ { T } ) ^ { T } } \\ & { \overline { { \Sigma } } _ { s } = \frac { 1 } { N } \sum _ { n \in [ 1 , N ] } \Sigma [ n ] } \end{array} +$$ + +# A.2 NETWORK ARCHITECTURE + +The vanilla CNN has the exact same architecture as the original ProtoNet (Snell et al., 2017). It consists of four convolution layers with depth of 64; each convolution layer is followed by Relu activation, max-pooling, and batch normalization (Ioffe & Szegedy, 2015). Resnet of 7 layers is constructed with one vanilla convolution layer of depth 64 followed by three residual blocks, all joined by max-pooling layers; each residual block consists of two sets of conv-batchnorm-Relu layers, of depth 128-256-256. + +# A.3 DATA SET DESCRIPTION AND PRE-PROCESSING + +Experiments are performed on three data sets: Omniglot (Lake et al., 2015), miniImageNet (Vinyals et al., 2016), and tieredImageNet (Ren et al., 2018). For Omniglot experiments, we follow the same configuration as in the original paper where 1200 classes augmented with rotations (4800 total) are used for training, and the remaining classes are used for testing. + +For miniImageNet experiments, we use the splits proposed by (Ravi & Larochelle, 2017) where 64 classes are used for training, 16 for validation, and 20 for testing. Mirroring the original paper, we resize all miniImageNet images to $8 4 \mathrm { x } 8 4$ . No data augmentation is applied. As most state-of-art few-shot classification methods achieve saturating accuracies on Omniglot, and miniImageNet’s small number of classes make claims about generalization difficult, we also conduct experiments of tieredImageNet. + +TieredImageNet is also a subset of Imagenet1000. TieredImageNet groups classes into broader categories corresponding to higher-level nodes in the ImageNet hierarchy. It includes 34 categories, with each category containing between 10 and 30 classes. These are split into 20 training, 6 validation and 8 testing categories. In total, there are 351 classes in training, 97 in validation, and 160 in testing. Preprocessing of images follow the same steps as used for miniImageNet. + +# A.4 PROTONET TRAINING + +Training procedure of ProtoNets largely mirrors the protocol used by Snell et al. (2017). On Omniglot, we train the network to convergence after 30000 episodes. On miniImageNet and tieredImageNet, we monitor the performance of the network on the validation set and select the best performing checkpoint after training for 50000 episodes. Adam (Kingma & Ba, 2014) optimizer is used with $\alpha = 0 . 9$ , $\beta = 0 . 9 9 9$ , $\epsilon = 1 0 ^ { - 8 }$ , and an initial learning rate of 0.001 that is decayed by half every 2000 episodes. On Omniglot, we train with 60 classes and 5 query points per episode. On miniImageNet and tieredImageNet, we train with 20 classes and 15 query points per episode. + +# A.5 DERIVATION DETAILS + +Proof of Lemma 1: + +Proof. First, from the definition of $\alpha$ , we split $\mathbb { E } _ { \mathbf { x } , S \mid a , b } [ \alpha ]$ in to two parts and examine them separately: + +$$ +\mathbb { E } _ { \mathbf { x } , S | a , b } [ \alpha ] = \underbrace { \mathbb { E } [ \left. \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right. ^ { 2 } ] } _ { i } - \underbrace { \mathbb { E } [ \left. \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right. ^ { 2 } ] } _ { i i } . +$$ + +In general, for random vector $X$ , the expectation of the quadratic form is $\mathbb { E } [ \left. X \right. ^ { 2 } ] = \operatorname { T r } ( \operatorname { V a r } ( X ) ) +$ $\mathbb { E } [ { \bar { X } } ] ^ { T } \mathbb { E } [ X ]$ . Hence, + +$$ +\begin{array} { r l } & { i = \mathbb { E } _ { \mathbf { x } , S \mid a , b } [ \Big \| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \Big \| ^ { 2 } ] } \\ & { \phantom { = } = \mathrm { T r } ( \Sigma _ { \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } } ) + \mathbb { E } [ \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } ] ^ { T } \mathbb { E } [ \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } ] , } \end{array} +$$ + +where the first term inside the trace can be expanded as: + +$$ +\begin{array} { l l } { { \displaystyle \Sigma _ { \phi ( { \bf x } ) - \overline { { \phi ( S _ { b } ) } } } = \mathrm { V a r } [ \phi ( { \bf x } ) - \overline { { \phi ( S _ { b } ) } } ] } } \\ { { } } & { { = \mathbb E [ ( \phi ( { \bf x } ) - \overline { { \phi ( S _ { b } ) } } ) ( \phi ( { \bf x } ) - \overline { { \phi ( S _ { b } ) } } ) ^ { T } ] - ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ^ { T } } } \\ { { } } & { { = \Sigma _ { c } + \mathbf { \mu } _ { a } \mathbf { \mu } _ { a } ^ { T } + \displaystyle \frac { 1 } { k } \Sigma _ { c } + \mathbf { \mu } _ { b } \mathbf { \mu } _ { b } ^ { T } - \displaystyle \mathbf { \mu } _ { a } \mathbf { \mu } _ { b } ^ { T } - \displaystyle \mathbf { \mu } _ { b } \mathbf { \mu } _ { a } ^ { T } - ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ^ { T } } } \\ { { } } & { { = ( 1 + \displaystyle \frac { 1 } { k } ) \Sigma _ { c } ~ \displaystyle ( \mathrm { L a s t ~ t e r m s ~ c a n c e l ~ o u t } ) . } } \end{array} +$$ + +To go from (18) to (19), we note that $\operatorname { V a r } ( X ) = \mathbb { E } [ X X ^ { T } ] - \mathbb { E } [ X ] \mathbb { E } [ X ] ^ { T }$ and $\Sigma _ { c } \overset { \Delta } { = } \mathrm { V a r } ( \phi ( \mathbf { x } ) )$ . Hence (19) can be obtained by expanding out the first term and taking the expectation of each resulting item. + +The second term of (16) is simply: + +$$ +\mathbb { E } _ { \mathbf { x } , S | a , b } [ \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } ] = \mu _ { a } - \mu _ { b } . +$$ + +Putting them together: + +$$ +i = ( 1 + \frac { 1 } { k } ) \mathrm { T r } ( \Sigma _ { c } ) + ( \mu _ { a } - \mu _ { b } ) ^ { T } ( \mu _ { a } - \mu _ { b } ) . +$$ + +Similarly for $_ { i i }$ + +$$ +\begin{array} { l } { { \displaystyle i i = \mathbb { E } _ { { \bf x } , S \vert a , b } [ \left. \phi ( { \bf x } ) - \overline { { \phi ( S _ { a } ) } } \right. ^ { 2 } ] } \ ~ } \\ { { \displaystyle ~ = \mathrm { T r } ( \Sigma _ { \phi ( { \bf x } ) - \overline { { \phi ( S _ { a } ) } } } ) + \mathbb { E } [ \phi ( { \bf x } ) - \overline { { \phi ( S _ { a } ) } } ] ^ { T } \mathbb { E } [ \phi ( { \bf x } ) - \overline { { \phi ( S _ { a } ) } } ] } \ } \\ { { \displaystyle ~ = ( 1 + \frac { 1 } { k } ) \mathrm { T r } ( \Sigma _ { c } ) . } } \end{array} +$$ + +Putting together $i$ and $\romannumeral 2$ : + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } _ { { \mathbf { x } } , S | a , b } [ \alpha ] = ( 1 + \frac { 1 } { k } ) \mathrm { T r } ( \Sigma _ { c } ) + ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ^ { T } - ( 1 + \frac { 1 } { k } ) \mathrm { T r } ( \Sigma _ { c } ) } \\ { = ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ^ { T } ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) } \end{array} +$$ + +Then, since $\mathbb { E } _ { a , b , \mathbf { x } , S } [ \alpha ] = \mathbb { E } _ { a , b } [ \mathbb { E } _ { \mathbf { x } , S | a , b } [ \alpha ] ]$ , we have: + +$$ +\begin{array} { r l } & { \mathbb { E } _ { a , b , \mathbf { x } , S } [ \alpha ] = \mathbb { E } _ { a , b } [ ( \mu _ { a } - \mu _ { b } ) ^ { T } ( \mu _ { a } - \mu _ { b } ) ] } \\ & { \qquad = \mathbb { E } _ { a , b } [ \mu _ { a } ^ { T } \mu _ { a } + \mu _ { b } ^ { T } \mu _ { b } - \mu _ { a } ^ { T } \mu _ { b } - \mu _ { b } ^ { T } \mu _ { a } ] } \\ & { \qquad = \mathrm { T r } ( \Sigma ) + \mu ^ { T } \mu + \mathrm { T r } ( \Sigma ) + \mu ^ { T } \mu - 2 \mu ^ { T } \mu } \\ & { \qquad = 2 \mathrm { T r } ( \Sigma ) } \end{array} +$$ + +Where from the second to the third line, we note that $\mu _ { a } ^ { T } \mu _ { a }$ and $\mu _ { b } ^ { T } \mu _ { b }$ are quadratic forms while $\mu _ { a } ^ { T } \mu _ { b }$ describe a dot product between two independent randomly drawn samples which has expectation $\mu ^ { T } \mu$ . □ + +For proof of Lemma 2, we first re-state the result on quadratic forms of normally distributed random vectors by Rencher & Schaalje (2008). + +Theorem 4. Consider random vector y $\sim \cal { N } ( \{ \mathfrak { u } , \Sigma \}$ and symmetric matrix of constants $A$ , we have: + +$$ +\mathrm { V a r } ( y ^ { T } A y ) = 2 \mathrm { T r } ( ( A \Sigma ) ^ { 2 } ) + 4 { \mu } ^ { T } A \Sigma A { \mu } . +$$ + +Proof of Lemma 2: + +Proof. + +$$ +\begin{array} { r l } { \mathrm { V a r } ( \alpha | a , b ) = \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } ) } & { } \\ { = \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } ) + \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } ) } & { } \\ { - 2 \mathrm { C o v } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } , \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } ) } & { } \\ { \leq \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } ) + \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } ) } & { } \\ { + 2 \sqrt { \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } ) \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } ) } } & { } \\ { \leq 2 \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } ) + 2 \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } ) } & { } \end{array} +$$ + +From 33 to 35, we used Cauchy Schwarz inequality. From line 35 to line 37, we use the fact that $2 a b \leq a ^ { 2 } + b ^ { 2 }$ for all $a , b \in \mathcal { R } ^ { + }$ . + +By applying Theorem 4, we have: + +$$ +{ \begin{array} { r l } & { \operatorname { V a r } ( \left\| \phi ( \mathbf { x } ) - { \overline { { \phi ( S _ { b } ) } } } \right\| ^ { 2 } ) = 2 ( 1 + { \frac { 1 } { k } } ) ^ { 2 } \operatorname { T r } ( \Sigma _ { c } ^ { 2 } ) + 4 ( 1 + { \frac { 1 } { k } } ) ( \mu _ { a } - \mu _ { b } ) ^ { T } \Sigma _ { c } ( \mu _ { a } - \mu _ { b } ) } \\ & { \operatorname { V a r } ( \left\| \phi ( \mathbf { x } ) - { \overline { { \phi ( S _ { a } ) } } } \right\| ^ { 2 } ) = 2 ( 1 + { \frac { 1 } { k } } ) ^ { 2 } \operatorname { T r } ( \Sigma _ { c } ^ { 2 } ) } \end{array} } +$$ + +Finally, + +$$ +\begin{array} { r l } & { \mathbb { E } _ { a , b } [ \mathrm { V a r } ( \alpha | a , b ) ] \leq \mathbb { E } _ { a , b } [ 2 \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { b } ) } } \right\| ^ { 2 } ) + 2 \mathrm { V a r } ( \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { a } ) } } \right\| ^ { 2 } ) ] } \\ & { \qquad = \mathbb { E } _ { a , b } [ 8 ( 1 + \frac { 1 } { k } ) ^ { 2 } \mathrm { T r } ( \Sigma _ { c } ^ { 2 } ) + 8 ( 1 + \frac { 1 } { k } ) ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ^ { T } \Sigma _ { c } ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ] } \\ & { \qquad = 8 ( 1 + \frac { 1 } { k } ) \mathbb { E } _ { a , b } [ \mathrm { T r } \{ ( 1 + \frac { 1 } { k } ) \Sigma _ { c } ^ { 2 } + \Sigma _ { c } ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ( \mathbf { \mu } _ { a } - \mathbf { \mu } _ { b } ) ^ { T } \} ] } \\ & { \qquad = 8 ( 1 + \frac { 1 } { k } ) \mathrm { T r } \{ \Sigma _ { c } [ ( 1 + \frac { 1 } { k } ) \Sigma _ { c } + 2 \Sigma ] \} } \end{array} +$$ + +Extending to $N$ class: Let $\mathbf x , y$ denote the query data pair, and the set of $N$ classes be denoted as c. Let $\alpha _ { i } = \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { i } } } ) \right\| ^ { 2 } - \left\| \phi ( \mathbf { x } ) - \overline { { \phi ( S _ { y } } ) } \right\| ^ { 2 }$ . Then we have a correct prediction $\hat { y } = y$ if $\forall i \in [ 1 , N ] , i \neq y , \alpha _ { i } > 0$ . Hence: $R ( \phi ) = \operatorname* { P r } _ { \mathbf { c } , \mathbf { x } , S } ( \bigcup _ { i \neq y } ^ { N } \alpha _ { i } > 0 )$ + +By Frechet’s inequality: + +$$ +R ( \phi ) \geq \sum _ { \stackrel { i = 1 } { i \neq y } } ^ { N } \operatorname* { P r } ( \alpha _ { i } > 0 ) - ( N - 2 ) +$$ + +Noting that Theorem 3 can be applied to each term in the summation: + +$$ +\mathfrak { L } ( \phi ) \geq \sum _ { \stackrel { i = 1 } { i \neq y } } ^ { N } \frac { 4 \mathrm { T r } ( \Sigma ) ^ { 2 } } { 8 ( 1 + 1 / k ) ^ { 2 } \mathrm { T r } ( \Sigma _ { c } ^ { 2 } ) + 1 6 ( 1 + 1 / k ) \mathrm { T r } ( \Sigma \Sigma _ { c } ) + \mathbf { E } _ { i , y } [ ( ( \mu _ { y } - \mu _ { i } ) ( \mu _ { y } - \mu _ { i } ) ^ { T } ) ^ { 2 } ] } - ( N - 2 ) +$$ + +It is then clear that the observations made on the binary case also applies to the multiclass case. + +# A.6 ADDITIONAL RESULTS + +Additionally, we experimented with directly setting the output dimension of the embedding network to 60 by adding a fully connected layer to the embedding network. This variant of protonet performs worse than both the base variant and all other methods. + +Table 2: Classification Accuracy on miniImageNet-5-way, with 4 layer $\mathrm { C N N } + 1$ Fully connected layer embedding network. + +
TRAINING SHOTS2 310AVERAGE ACCURACY
45TESTING SHOTS 6789
PROTONET + FC544.77%53.75%58.04%61.06%62.26%64.60%65.19%66.63%66.65%67.52%61.05±0.28%
+ +![](images/a37c6d256f5c1f947d8f9b2c05baba36b9385f119b05e97db42f9794f3b121af.jpg) +Figure 3: Effect of hyperparameters on k-shot testing performance on miniImageNet. + +Table 3: Classification Accuracy on miniImageNet-5-way, with 4 layer CNN embedding network. + +
MODELTRAININGTESTING SHOTSAVERAGE
SHOTS12345678910ACCURACY
VANILLA PROTONET148.89%56.54%60.31%63.12%64.70%66.02%66.62%67.99%68.41%68.90%63.15±0.21%
EST PROTONET149.07%56.49%60.62%62.67%64.83%66.23%66.84%67.90%67.68%68.73%63.11±0.22%
PCA PROTONET149.01%56.35%60.07%62.42%64.38%65.28%66.56%67.81%67.59%68.26%62.77 ±0.22%
VANILLA PROTONET544.75%56.61%61.52%65.32%67.23%69.04%70.66%71.47%71.84%72.36%65.08±0.23%
EST PROTONET550.22%59.04 %64.14%66.61%68.25%69.46%70.80%71.60%72.61%73.29%66.60±0.23%
PCA PROTONET48.72%58.43%63.17%66.07%68.63%69.56%70.55%71.21%72.09%72.82%66.12±0.24%
VANILLA PROTONET1039.99%52.73%59.71%63.41%66.23%68.27%69.86%71.03%71.72%72.47%63.54±0.25%
EST PROTONET1048.98%57.83%63.13%66.39%68.12%69.82%70.63%71.85%72.79%73.22%66.28±0.23%
PCA PROTONET1048.04%57.05%62.46%64.63%67.61%68.98%69.71%71.78%71.75%72.42%65.44±0.24%
VANILLA PROTONET1-1049.36%58.67%62.77 %65.76%67.96%69.20%70.05%70.90%71.34%72.27%65.83±0.21%
+ +![](images/0a8ddca89f408f726d27585899c5c5c84a7e9af46d7419e3f80969f00e34ec2d.jpg) +Figure 4: Comparison between estimated accuracy lower bound and empirical accuracy for various training and test shots. Experiment is 2-way few-shot classification performed on miniImageNet. + +Table 4: Classification Accuracy on tieredImageNet-5-way, with 4 layer CNN embedding network. + +
MODELTRAINING SHOTS245TESTING SHOTS678910AVERAGE ACCURACY
1361.85±0.29%
VANILLA PROTONET EST PROTONET1 147.37% 47.71%55.72% 55.05%59.27% 59.31%61.50% 61.97%63.70% 63.48%64.28% 64.65%65.01% 65.35%66.43% 66.33%67.20% 66.42%67.99% 67.44%61.77±0.31%
PCA PROTONET147.72%54.60%58.69%60.73%62.78%64.92%65.88%65.88%66.11%66.87%61.42±0.32%
VANILLA PROTONET
EST PROTONET5542.33% 48.85%55.06%61.32%64.57%66.51%67.86%69.33%70.15%71.32% 70.87%72.05% 72.09%64.05±0.33% 65.46±0.32%
PCA PROTONET548.34%58.38% 57.44%62.75% 0.79%65.16% 64.96%67.24% 67.07%68.39% 67.93%69.89% 69.08%70.99% 70.40%70.38%71.65%64.96±0.31%
VANILLA PROTONET
EST PROTONET1035.38%49.40%56.76%61.25%64.56%66.56%68.05%69.31%70.12%71.03%61.24±0.36% 65.26±0.32%
PCA PROTONET10 1047.33% 46.55%56.75% 56.61%62.80% 60.81%65.78% 64.01%66.84% 66.53%69.07% 67.70%69.98% 69.18%70.82% 69.83%71.99% 70.62%71.20% 71.22%64.31 ±0.33%
VANILLA PROTONET1-1047.65%56.23%62.12 %63.93%66.34%67.94%68.44%68.93%70.80%70.96%64.33±0.30%
+ +Table 5: Classification Accuracy on Omniglot-20-way, with 4 layer CNN embedding network. + +
MODELTRAINING SHOTS 1TESTING SHOTS
2345AVERAGE ACCURACY
VANILLA PROTONET195.07 ±0.17 %97.89 ±0.09 %98.45 ±0.08 %98.75 ±0.06 %98.89 ±0.06%97.81±0.06 %
VANILLA PROTONET294.59 ±0.18 %97.69 ±0.09 %98.44 ±0.07 %98.69 ±0.06 %98.89 ±0.06 %97.66 ±0.07 %
VANILLA PROTONET394.19 ±0.18 %97.57 ±0.09 %98.30±0.07 %98.63±0.07 %98.79 ±0.06 %97.50 ±0.07 %
VANILLA PROTONET493.79 ±0.18 %97.41 ±0.10 %98.19 ±0.08 %98.54 ±0.07 %98.75±0.06%97.34 ±0.07 %
VANILLA PROTONET593.42 ±0.18 %97.34±0.10 %98.18 ±0.07 %98.53 ±0.07 %98.78 ±0.05 %97.25 ±0.07 %
MIXED-k SHOT1-594.84±0.17 %97.81 ±0.09 %98.45±0.07 %98.70 ±0.06 %98.92 ±0.54 %97.74 ±0.06 %
PCA PROTONET194.94 ±0.16 %97.81 ±0.09 %98.53±0.07 %98.79 ±0.06 %98.85±0.06%97.78 ±0.06 %
EST PROTONET195.11 ±0.17 %97.95 ±0.09 %98.46±0.07 %98.77 ±0.06 %98.84±0.06%97.83 ±0.06 %
+ +Table 6: Classification Accuracy on miniImageNet-5-way, with 7 layer ResNet embedding network. + +
MODELTRAINING SHOTS2TESTING SHOTS 3 4AVERAGEACCURACY
5678
VANILLA PROTONET152.65%60.46%64.18%66.83%68.27%69.23%70.19%71.37%71.83%72.29%66.73±0.20%
EST PROTONET152.56%60.63%64.50%66.53%68.33%69.36%70.04%71.21%71.37%71.60%66.61±0.22%
PCA PROTONET152.78%60.35%64.05%66.24%67.51%69.64%69.85%71.13%71.88%71.79%66.52±0.22%
VANILLA PROTONET547.40%58.23%64.60%67.52%69.93%71.05%72.27%72.90%73.55%74.35%67.18±0.23%
EST PROTONET551.93%60.70%65.33%68.06%69.98%71.26%72.33%73.46%74.03%74.80%68.19±0.23 %
PCA PROTONET550.90%60.38%64.66%67.23%69.25%71.30%71.72%72.75%73.84%74.24%67.63±0.23%
VANILLA PROTONET68.23%
EST PROTONET10 1042.20% 51.24%55.98% 60.46%61.67%66.08%70.06%69.95% 71.07%72.20% 72.55%72.56% 73.39%74.04% 73.85%74.54 % 74.69%65.75±0.25% 68.00±0.23%
PCAPROTONET1050.52%59.20%65.11% 64.44%67.63% 66.86%69.36%71.00%71.73%73.19%73.73%73.82%67.38±0.23%
VANILLA PROTONET1-1051.74%60.10%65.13%67.12%69.09%70.53%71.57%72.22%72.99%73.63%67.41±0.21%
+ +Table 7: Classification Accuracy on tieredImageNet-5-way, with 7 layer ResNet embedding network. + +
MODELTRAININGTESTING SHOTSAVERAGE
SHOTS12345678910ACCURACY
VANILLA PROTONET149.78%57.67%61.37%64.88%65.17%66.73%67.53%68.58%68.21%69.88%63.98±0.29%
EST PROTONET151.19%57.61%61.85%64.43%65.48%67.64%67.69%67.76%68.51%68.20%64.04±0.30%
PCA PROTONET151.38%57.89%61.68%64.32%65.96%66.15%66.83%68.00%68.58%68.77%63.95±0.30%
VANILLA PROTONET547.88%58.70%63.95%66.58%69.12%71.69%71.95%73.15%73.11%73.80%66.99 ±0.31%
EST PROTONET553.05%61.13 %65.67%68.07%69.30%71.47%71.59 %72.42%72.80%73.63%67.91 ±0.31%
PCA PROTONET51.21%59.86%63.91%66.92%68.88%70.38%71.48%72.77%73.81%72.17%67.14±0.32%
VANILLA PROTONET1040.84%56.24%62.10%66.28%69.37%71.43%72.09%72.91%73.56%74.72%65.95±0.35%
EST PROTONET1050.45%60.41%65.19%68.46%69.55%70.87%72.07%72.66%73.78%74.79%67.82 ±0.32%
PCA PROTONET1050.18%59.59%64.24%67.43%69.52%70.65%71.78%72.10%72.78%73.65%67.19±0.31%
VANILLA PROTONET1-1050.74%60.03%64.17 %67.26%69.28%69.56%71.07%72.41%71.99%73.01%66.95 ±0.29 %
+ +Table 8: Classification Accuracy on Omniglot-20-way, with 7 layer ResNet embedding network. + +
TESTING SHOTS
MODELTRAINING SHOTS12345AVERAGE ACCURACY
VANILLA PROTONET196.46%98.39 %98.82%99.01 %99.07%98.35 ±0.05 %
VANILLA PROTONET295.85%98.32%98.80%98.95%99.07%98.20±0.05 %
VANILLA PROTONET395.35 %98.15%98.73 %98.91 %99.03%98.03 ±0.06%
VANILLA PROTONET495.00%98.05%98.62 %98.90%98.99%97.91 ±0.06 %
VANILLA PROTONET594.42 %97.98 %98.60%98.77 %98.99%97.75 ±0.06 %
VANILLA PROTONET1-596.53%98.53%98.90%99.06%99.15%98.43 ±0.05 %
EST PROTONET196.18%98.23 %98.68%98.87%98.99%98.19 ±0.05 %
PCA PROTONET196.02%98.22 %98.76%98.93%99.02 %98.19 ±0.05 %
\ No newline at end of file diff --git a/parse/train/HkgB2TNYPS/HkgB2TNYPS_content_list.json b/parse/train/HkgB2TNYPS/HkgB2TNYPS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..880b5c0c9d0e6fe4e2b03205e682bb38e3583366 --- /dev/null +++ b/parse/train/HkgB2TNYPS/HkgB2TNYPS_content_list.json @@ -0,0 +1,2423 @@ +[ + { + "type": "text", + "text": "A THEORETICAL ANALYSIS OF THE NUMBER OF SHOTS IN FEW-SHOT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 99, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tianshi $\\mathbf { C a o ^ { 1 , 2 } }$ , Marc T. Law1,2,3, Sanja Fidler1,2,3 \n1 Department of Computer Science, University of Toronto \n2 Vector Institute \n3 NVIDIA \n{jcao, law, fidler}@cs.toronto.edu ", + "bbox": [ + 184, + 167, + 565, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 279, + 544, + 294 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Few-shot classification is the task of predicting the category of an example from few labeled examples. The number of labeled examples per category is called the number of shots (or shot number). Recent works tackle this task through metalearning, where a meta-learner extracts information from observed tasks during meta-training to quickly adapt to new tasks during meta-testing. In this formulation, the number of shots exploited during meta-training has an impact on the recognition performance at meta-test time. Generally, the shot number used in meta-training should match the one used in meta-testing to obtain the best performance. We introduce a theoretical analysis of the impact of the shot number on Prototypical Networks, a state-of-the-art few-shot classification method. From our analysis, we propose a simple method that is robust to the choice of shot number used during meta-training, which is a crucial hyperparameter. The performance of our model trained for an arbitrary meta-training shot number shows great performance for different values of meta-testing shot numbers. We experimentally demonstrate our approach on different few-shot classification benchmarks. ", + "bbox": [ + 233, + 309, + 766, + 517 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 544, + 336, + 560 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Human cognition has the impressive ability of grasping new concepts from exposure to a handful of examples (Yger et al., 2015). In comparison, while modern deep learning methods achieve unprecedented performances with very deep neural-networks (He et al., 2016; Szegedy et al., 2015), they require extensive amounts of data to train, often ranging in the millions. Few-shot learning aims to bridge the sample-efficiency gap between deep learning and human learning in fields such as computer vision, reinforcement learning and speech recognition (Santoro et al., 2016; Ravi & Larochelle, 2017; Finn et al., 2017; Vinyals et al., 2016; Wang et al., 2019a). These methods fall under the framework of meta-learning, in which a meta-learner extracts knowledge from many related tasks (in the meta-training phase) and leverages that knowledge to quickly learn new tasks (in the meta-testing phase). In this paper, we focus on the few-shot classification problem where each task is defined as a $N$ -way classification problem with $k$ samples (shots) per class available for training. ", + "bbox": [ + 174, + 575, + 825, + 728 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many meta-learning methods use the episodic training setup in which the meta-learner iterates through episodes in the meta-training phase. In each episode, a task is drawn from some population and a limited amount of support and query data from that task is made available. The meta-learner then learns a task-specific classifier on the support data and the classifier predicts on the query data. Updates to the meta-learner is computed based on the performance of the classifier on the query set. Evaluation of the meta-learner (during a phase called meta-testing) is also carried out in episodes in a similar fashion, except that the meta-learner is no longer updated and the performance on query data across multiple episodes is aggregated. ", + "bbox": [ + 174, + 736, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In the episodic setup, the selection of $k$ during meta-training time can have significant effects on the learning outcomes of the meta-learner. Intuitively, if support data is expected to be scarce, the meta-learner needs to provide strong inductive bias to the task-specific learner as the danger of overfitting is high. In contrast, if support data is expected to be abundant, then the meta-learner can provide generally more relaxed biases to the task-specific learner to achieve better fitting to the task data. Therefore it is plausible that a meta-learner trained with one $k$ value can be suboptimal at adapting to tasks with a different $k$ value and thus exhibit meta-overfitting to $k$ . In experiments, $k$ is often simply kept fixed between meta-training and meta-testing, but in real-world usage, one cannot expect to know beforehand the amount of support data from unseen tasks during deployment. ", + "bbox": [ + 176, + 854, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper we will focus on Prototypical networks (Snell et al., 2017), a.k.a. ProtoNet. ProtoNet is of practical interest because of its flexibility: a single trained instance of ProtoNet can be used on new tasks with any $k$ and $N$ . However, ProtoNet exhibits performance degradation when the $k$ used in training does not match the $k$ used in testing.1 First, we will undertake a theoretical investigation to elicit the connection from $k$ to a lower bound of expected performance, as well as to the intrinsic dimension of the learned embedding space. Then, we conduct experiments to empirically verify our theoretical results across various settings. Guided by our new understanding of the effects of $k$ , we propose an elegant method to tackle performance degradation in mismatched $k$ cases. Our contributions are threefold: ", + "bbox": [ + 173, + 166, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We provide performance bounds for ProtoNets given an embedding function. From which, we argue that $k$ affects learning and performance by scaling the contribution of intra-class variance. ", + "bbox": [ + 173, + 299, + 823, + 327 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Through VC-learnability theory, we connect the value of $k$ used in meta-training to the intrinsic dimension of the embedding space. ", + "bbox": [ + 173, + 334, + 823, + 362 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• The most important contribution of this paper (introduced in Section 3.3) is a new method that improves upon vanilla ProtoNets by eliminating the performance degradation in cases where the $k$ is mismatched between meta-training and meta-testing. Our evaluation protocol more closely adheres to real-world scenarios where the model is exposed to different numbers of training samples. ", + "bbox": [ + 174, + 369, + 825, + 424 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 450, + 326, + 467 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 PROBLEM SETUP ", + "text_level": 1, + "bbox": [ + 176, + 484, + 331, + 500 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The few-shot classification problem considered in this paper is set up as described below. Consider a space of classes $C$ with a probability distribution $\\tau$ , $N$ classes $\\mathbf { c } \\doteq \\{ c _ { 1 } , . . . , c _ { N } \\}$ are sampled i.i.d. from $\\tau$ to form a $N$ -way classification problem. For each class $c _ { i }$ , $k$ support data are sampled from class-conditional distribution $S _ { i } = \\{ _ { s } { \\bf x } _ { 1 } , . . , _ { s } { \\bf x } _ { k } \\} \\stackrel { i i d } { \\sim } P ( { \\bf x } | Y ( { \\bf x } ) = c _ { i } )$ , where $\\mathbf { x } \\in \\mathbb { R } ^ { D }$ , $D$ denotes the dimension of data, and $Y ( \\mathbf { x } )$ denotes the class assignment of $\\mathbf { x }$ . Note that we assume that $Y ( \\mathbf { x } )$ is singular (e.g. each $\\mathbf { x }$ can only have 1 label), and does not depend on $N$ (e.g. a data point with a label “cat” will always have the label “cat”), in contrast to $y$ defined below. ", + "bbox": [ + 173, + 513, + 825, + 616 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Additionally, the set $Q = \\{ _ { q } \\mathbf { x } _ { 1 } , . . . , _ { q } \\mathbf { x } _ { l } \\}$ containing $l$ query data is sampled from the joint distribution $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } P ( \\mathbf { x } | c _ { i } ) } \\end{array}$ .2 For each $\\mathbf { x }$ , let $y \\in \\{ 1 , . . . , N \\}$ denote its label in the context of the few-shot classificafor class $c _ { i }$ n task. Define , and denote t $\\hat { S } _ { i } = \\left\\{ ( _ { s } \\mathbf { x } _ { 1 } , y _ { 1 } = i ) , . . . , ( _ { s } \\mathbf { x } _ { k } , y _ { k } = i ) \\right\\}$ $N$ as as $\\textstyle S = \\bigcup _ { i = 1 } ^ { N } { \\hat { S } } _ { i }$ set of supports. The few-shot classification task is to predict $y$ for each $\\mathbf { x }$ in $Q$ given $S$ . During meta-training, the ground truth label for $Q$ is also available to the learner. ", + "bbox": [ + 174, + 622, + 825, + 717 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 META-LEARNING SETUP ", + "text_level": 1, + "bbox": [ + 174, + 738, + 383, + 752 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Meta-learning approaches train on a distribution of tasks to obtain information that generalizes to unseen tasks. For few-shot classification, a task is determined by which classes are involved in the $N$ -way classification task. During meta-training, the meta-learner observes episodes of few-shot classification tasks consisting of $N$ classes, $k$ labelled samples per class, and $l$ unlabelled samples, as previously described. The collection of all classes observed during meta-training forms the metatraining split $\\mathcal { D } _ { t r } = \\{ { _ { t r } c _ { 1 } } , . . . , { _ { t r } c _ { R } } \\}$ . Critically, we assume that every unseen class that the learner is evaluated upon (during meta-testing) is also drawn from the same distribution $\\tau$ . ", + "bbox": [ + 174, + 766, + 825, + 864 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.3 PROTOTYPICAL NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 408, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "ProtoNets (Snell et al., 2017) compute $E$ -dimensional embeddings for all samples in $S$ and $Q$ . The embedding function $\\phi : \\mathbb { R } ^ { D } \\mathbb { R } ^ { \\dot { E } }$ is usually a deep network.The prototype representation for each class is formed by averaging the embeddings for all supports of said class: $\\begin{array} { r } { \\overline { { \\phi ( S _ { i } ) } } = \\frac { 1 } { k } \\sum _ { \\mathbf { x } \\in S _ { i } } \\phi ( \\mathbf { x } ) } \\end{array}$ . Classification of any input $\\mathbf { x }$ (e.g. $\\mathbf { x } \\in Q$ i) is performed by computing the softmax over squared Euclidean distances of the input point’s embedding to the prototypes. Let $\\hat { y }$ denote the prediction of the classifier for one of the categories $j \\in \\{ 1 , \\cdots , N \\}$ : ", + "bbox": [ + 174, + 128, + 826, + 217 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1e23287a42926213e58784a2d097b09aa63c866b260c117aee88c31902df9168.jpg", + "text": "$$\np _ { \\phi } ( \\hat { y } = j | \\mathbf { x } , S ) = \\frac { e ^ { - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { j } ) } } \\right\\| ^ { 2 } } } { \\sum _ { i = 1 } ^ { N } e ^ { - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { i } ) } } ^ { 2 } \\right\\| } } \\quad , \\mathrm { w h e r e } \\quad \\left\\| \\mathbf { v } \\right\\| ^ { 2 } = \\sum _ { d = 1 } ^ { E } v _ { d } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 267, + 223, + 730, + 271 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The parameters of the embedding functions are learned through meta-training. Negative log-likelihood $J ( \\phi ) = - \\log \\left( p ( \\hat { y } = y | \\mathbf { x } ) \\right)$ of the correct class $y$ is minimized on the query points through SGD. ", + "bbox": [ + 169, + 276, + 825, + 306 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As explained in (Law et al., 2019), ProtoNets can be seen as a metric learning approach optimized for the supervised hard clustering task (Law et al., 2016). The model $\\phi$ is learned so that the representations of similar examples (i.e. belonging to a same category) are all grouped into the same cluster in $\\mathbb { R } ^ { E }$ . We propose in this paper a subsequent metric learning step which learns a linear transformation that maximizes inter-to-intra class variance ratio. ", + "bbox": [ + 174, + 311, + 825, + 382 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PROPOSED METHOD ", + "text_level": 1, + "bbox": [ + 176, + 402, + 372, + 419 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We first present theoretical results explaining the effect of the shot number on ProtoNets, and then introduce our method for addressing performance degradation in cases of mismatched shots. ", + "bbox": [ + 173, + 434, + 823, + 463 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 RELATING $k$ TO LOWER BOUND OF EXPECTED ACCURACY ", + "text_level": 1, + "bbox": [ + 173, + 481, + 611, + 494 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To better understand the role of $k$ on the performance of ProtoNets, we study how it contributes to the expected accuracy across episodes when using any kind of fixed embedding function (e.g. the embedding function obtained at the end of the meta-training phase). With $I$ denoting the indicator function, we define the expected accuracy $R$ as: ", + "bbox": [ + 173, + 506, + 825, + 563 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9747916e79419c3ebebc17611bb6c22a3fb5eecdd42b1a51422cca94136c14bf.jpg", + "text": "$$\nR ( \\phi ) = \\mathbb { E } _ { \\mathbf { c } } \\mathbb { E } _ { S , \\mathbf { x } , y } I [ \\arg \\operatorname* { m a x } _ { j } \\left\\{ p _ { \\phi } ( \\hat { y } = j | \\mathbf { x } , S ) \\right\\} = y ]\n$$", + "text_format": "latex", + "bbox": [ + 325, + 569, + 671, + 597 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definitions: Throughout this section, we will use the following symbols to denote the means and variances of embeddings under different expectations: ", + "bbox": [ + 171, + 603, + 823, + 632 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e03a432aceb7cbd5a4a462b4ed1202faeecf1b5a7fc033c974dfa7c8e2590d1a.jpg", + "text": "$$\n{ \\begin{array} { r l } & { \\mathbf { \\mu } _ { c } \\triangleq \\mathbb { E } _ { \\mathbf { x } } [ \\phi ( \\mathbf { x } ) | \\ X ( \\mathbf { x } ) = c ] \\qquad \\quad \\Sigma _ { c } \\triangleq \\mathbb { E } _ { \\mathbf { x } } [ ( \\phi ( \\mathbf { x } ) - \\mathbf { \\mu } _ { c } ) ( \\phi ( \\mathbf { x } ) - \\mathbf { \\mu } _ { c } ) ^ { T } | \\ Y ( \\mathbf { x } ) = c ] } \\\\ & { \\quad \\mathbf { \\mu } \\triangleq \\mathbb { E } _ { c } [ \\mathbf { \\mu } _ { c } ] \\qquad \\quad \\Sigma \\triangleq \\mathbb { E } _ { c } [ ( \\mathbf { \\mu } _ { c } - \\mathbf { \\mu } ) ( \\mathbf { \\mu } _ { c } - \\mathbf { \\mu } ) ^ { T } ] } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 225, + 637, + 772, + 681 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Remark. $\\mu _ { c }$ is the expectation of the embedding conditioned on class $c , \\mu$ is the (full) expectation of the embedding, which can be expressed as the expectation of $\\mu _ { c }$ over classes. $\\Sigma$ is the variance of class means in the embedding space - it can be interpreted as the signal of the input to the classifier, as larger $\\Sigma$ implies larger distances between classes. $\\Sigma _ { c }$ is the expected intra-class variance - it represents the noise in the above signal. ", + "bbox": [ + 173, + 685, + 826, + 756 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Modelling assumptions of ProtoNets: The use of the squared Euclidean distance and softmax activation in ProtoNets implies that classification with ProtoNets is equivalent to a mixture density estimation on the support set with spherical Gaussian densities (Snell et al., 2017). Specifically, we adopt the modelling assumptions that the distribution of $\\phi ( \\mathbf { x } )$ given any class assignment is normally distributed $( p ( \\phi ( \\mathbf { x } ) \\bar { | } Y ( \\mathbf { x } ) \\bar { = } c ) = \\mathcal { N } ( \\mu _ { c } , \\Sigma _ { c } ) )$ , with equal covariance for all classes in the embedding space $( \\forall ( c , c ^ { \\prime } ) , \\Sigma _ { c } = \\Sigma _ { c ^ { \\prime } } ) ^ { 3 }$ . ", + "bbox": [ + 173, + 767, + 825, + 852 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We present the analysis for the special case of episodes with binary classification (i.e. with $N = 2$ ) for ease of presentation, but the conclusion can be generalized to arbitrary $N > 2$ (see appendix). ", + "bbox": [ + 174, + 858, + 823, + 887 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Also, as noted in Section 2.1, we assume equal likelihood between the classes. We would like to emphasize that the assignment of labels can be permuted freely and the classifier’s prediction would not be affected due to symmetry. Hence, we only need to consider one case for the ground truth label without loss of generality. Let $a$ and $b$ denote any pair of classes sampled from $\\tau$ . Let $\\mathbf { x }$ be drawn from $a$ , and overload $a$ and $b$ to also indicate the ground truth label in the context of that episode, then equation 2 can be written as: ", + "bbox": [ + 173, + 102, + 826, + 188 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/277183cd7a4d862b4ee733cc0eb884d9ad8dff14e359507182f93b667e8d5861.jpg", + "text": "$$\nR ( \\phi ) = \\mathbb { E } _ { a , b \\sim \\tau } \\mathbb { E } _ { \\mathbf { x } , S } I [ \\hat { y } = a ]\n$$", + "text_format": "latex", + "bbox": [ + 401, + 194, + 598, + 212 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Additionally, noting that $p ( \\hat { y } = a )$ can be expressed as a sigmoid function $\\sigma$ : ", + "bbox": [ + 173, + 218, + 676, + 234 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1eccdd90d25fd59d264fc9e60f72466abf54d4a0e074444d09f544bfc3b342aa.jpg", + "text": "$$\np ( \\hat { y } = a | \\mathbf { x } ) = \\frac { e ^ { - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } } } { e ^ { - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } } + e ^ { - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } } } = \\sigma ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 239, + 820, + 316 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can express equation 3 as a probability: ", + "bbox": [ + 174, + 319, + 457, + 334 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8f4d5c75acd4b722477f1a3366d2fe7a59a27084057deb17ccc36167796e9d9c.jpg", + "text": "$$\nR ( \\phi ) = \\operatorname* { P r } _ { a , b , \\mathbf { x } , S } ( { \\hat { y } } = a ) = \\operatorname* { P r } _ { a , b , \\mathbf { x } , S } ( \\alpha > 0 )\n$$", + "text_format": "latex", + "bbox": [ + 362, + 340, + 635, + 366 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We will introduce a few auxiliary results before stating the main result for this section. ", + "bbox": [ + 176, + 372, + 738, + 387 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proposition 1. From the one-sided Chebyshev’s inequality, it immediately follows that: ", + "bbox": [ + 173, + 390, + 741, + 406 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9827575e08a973ba03f91df009dfe2724fc620186cc631ba6adf39662f12d9b5.jpg", + "text": "$$\nR ( \\phi ) = \\mathrm { P r } ( \\alpha > 0 ) \\geq \\frac { \\mathbb { E } [ \\alpha ] ^ { 2 } } { \\mathrm { V a r } ( \\alpha ) + \\mathbb { E } [ \\alpha ] ^ { 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 411, + 630, + 446 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In Lemma 1 and Lemma 2, we derive the expectation and variance of $\\alpha$ when conditioned on the classes sampled in a episode. Then, in Theorem 3, we compose them into the $R H S$ of Proposition 1 through law of total expectation. ", + "bbox": [ + 173, + 460, + 825, + 503 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 1. Consider space of classes $C$ with sampling distribution $\\tau$ $; , a , b \\stackrel { i i d } { \\sim } \\tau .$ . Let $S = \\{ S _ { a } , S _ { b } \\}$ $S _ { a } = \\left\\{ { { \\bf { \\Psi } } _ { a } { \\bf { x } } _ { 1 } } , . . . , { \\bf { \\Psi } } _ { a } { \\bf { x } } _ { k } \\right\\}$ , $S _ { b } = \\left\\{ \\vphantom { b } _ { b } \\mathbf { x } _ { 1 } , . . . , \\vphantom { b } _ { b } \\mathbf { x } _ { k } \\right\\}$ , $k \\in \\mathbb N$ is the shot number, and $Y ( \\mathbf { x } ) = a$ . Define ${ \\overline { { \\phi ( S _ { a } ) } } } \\triangleq { \\frac { 1 } { k } } \\sum _ { \\mathbf { x } \\in S _ { a } } \\phi ( \\mathbf { x } )$ and ${ \\overline { { \\phi ( S _ { b } ) } } } \\triangleq { \\frac { 1 } { k } } \\sum _ { \\mathbf { x } \\in S _ { b } } \\phi ( \\mathbf { x } )$ . Consider $\\Sigma$ as defined earlier. Assume $p ( \\phi ( \\mathbf { x } ) | Y ( \\mathbf { x } ) = c ) = N ( \\mu _ { c } , \\Sigma _ { c } )$ and $\\Sigma _ { c } = \\Sigma _ { c ^ { \\prime } }$ for any choice of $c , c ^ { \\prime } \\in C$ , then, ", + "bbox": [ + 173, + 508, + 825, + 574 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/718eff581d07c6cd93d0f9ba77b1574bd3455ee971c0114c479635fcb835491e.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathbf { x } , S | a , b } [ \\alpha ] = \\left( \\mathtt { u } _ { a } - \\mathtt { u } _ { b } \\right) ^ { T } ( \\mathtt { u } _ { a } - \\mathtt { u } _ { b } ) \\qquad , a n d \\qquad \\mathbb { E } _ { a , b , \\mathbf { x } , S } [ \\alpha ] = 2 \\mathrm { T r } ( \\Sigma ) .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 579, + 767, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 2. Under the same notation and assumptions as Lemma $I$ , additionally invoking definition for $\\Sigma _ { c } ,$ then, ", + "bbox": [ + 173, + 604, + 823, + 633 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/492a6df51ab57cea59d936abcdc5b21f16b4026855cb0af4051d71595bddafb9.jpg", + "text": "$$\n\\mathbb { E } _ { a , b } [ \\mathrm { V a r } ( \\alpha | a , b ) ] \\leq 8 ( 1 + \\frac { 1 } { k } ) \\mathrm { T r } \\left( \\Sigma _ { c } ( ( 1 + \\frac { 1 } { k } ) \\Sigma _ { c } + 2 \\Sigma ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 300, + 632, + 696, + 666 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The proofs of the above lemmas are in the appendix. With the results above, we are ready to state our main theoretical result in this section. ", + "bbox": [ + 173, + 676, + 825, + 704 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 3. Under the conditions where Lemma 1 and 2 hold, we have: ", + "bbox": [ + 174, + 708, + 650, + 724 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8ba188cbc949bf95ec557e24d05365d2ff150624a4ee79c1aa4a1529e33288bb.jpg", + "text": "$$\nR ( \\phi ) \\geq \\frac { 4 \\mathrm { T r } ( \\Sigma ) ^ { 2 } } { 8 ( 1 + 1 / k ) ^ { 2 } \\mathrm { T r } ( \\Sigma _ { c } ^ { 2 } ) + 1 6 ( 1 + 1 / k ) \\mathrm { T r } ( \\Sigma \\Sigma _ { c } ) + \\mathbb { E } _ { a , b } [ ( \\mathbf { \\mu } ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } \\mathbf { \\mu } _ { b } ) ^ { T } ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } \\mathbf { \\mu } _ { b } ) ) ^ { 2 } ] } .\n$$", + "text_format": "latex", + "bbox": [ + 196, + 728, + 781, + 765 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proof. First, we use decompose the $\\mathrm { V a r } ( \\alpha )$ term in Proposition 1 by Law of Total Expectation. ", + "bbox": [ + 178, + 779, + 794, + 795 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f817e8c941fe801dfbdd23dbc95a4e1cfa94f14bc32a26c29fec5e68e7c2d596.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V a r } ( \\boldsymbol \\alpha ) = \\mathbb { E } _ { a , b , \\mathbf { x } , S } [ \\boldsymbol \\alpha ^ { 2 } ] - \\mathbb { E } _ { a , b , \\mathbf { x } , S } [ \\boldsymbol \\alpha ] ^ { 2 } = \\mathbb { E } _ { a , b } \\mathbb { E } _ { \\mathbf { x } , S } [ \\boldsymbol \\alpha ^ { 2 } | a , b ] - \\mathbb { E } _ { a , b , \\mathbf { x } , S } [ \\boldsymbol \\alpha ] ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\quad = \\mathbb { E } _ { a , b } [ \\mathrm { V a r } ( \\boldsymbol \\alpha | a , b ) + \\mathbb { E } _ { \\mathbf { x } , S } [ \\boldsymbol \\alpha | a , b ] ^ { 2 } ] - \\mathbb { E } _ { a , b , \\mathbf { x } , S } [ \\boldsymbol \\alpha ] ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 800, + 743, + 843 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Hence, Proposition 1 can also be expressed as ", + "bbox": [ + 174, + 847, + 477, + 861 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/857deaed4cc8bb7f40d1984fb20b0e1bcf2d7afbc1f56d5baaa530f22556890c.jpg", + "text": "$$\nR ( \\phi ) \\geq \\frac { \\mathbb { E } [ \\alpha ] ^ { 2 } } { \\mathbb { E } _ { a , b } [ \\mathrm { V a r } ( \\alpha \\vert a , b ) + \\mathbb { E } _ { \\mathbf { x } , S } [ \\alpha \\vert a , b ] ^ { 2 } ] } .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 867, + 643, + 904 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Finally, we arrive at Theorem 3 by plugging Lemma 1 and 2 into equation 10. ", + "bbox": [ + 171, + 909, + 683, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Several observations can be made from Theorem 3: ", + "bbox": [ + 176, + 103, + 509, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. The shot number $k$ only appears in the first two terms of the denominator, implying that the bound saturates quickly with increasing $k$ . This is also in agreement with the empirical observation that meta-testing accuracy has diminishing improvements when more support data is added. ", + "bbox": [ + 174, + 125, + 825, + 167 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2. By observing the degree of terms in equation 7 (and treating the last term of the denominator as a constant), it is clear that increasing $k$ will decrease the sensitivity (magnitude of partial derivative) of this lower bound to $\\Sigma _ { c }$ , and increase its sensitivity to $\\Sigma$ . ", + "bbox": [ + 174, + 174, + 825, + 215 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3. If one postulates that meta-learning updates on $\\phi$ are similar to gradient ascent on this accuracy lower bound, then learning with smaller $k$ emphasizes minimizing noise, while learning with higher $k$ emphasizes maximizing signal. ", + "bbox": [ + 174, + 223, + 825, + 265 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In conclusion, these observations give us a plausible reason for the performance degradation observed in mismatched shots: when an embedding function has been optimized (trained) for $k _ { t r a i n } ~ >$ $k _ { t e s t }$ , the relatively high $\\Sigma _ { c }$ is amplified by the now smaller $k$ , resulting in degraded performance. Conversely, an embedding function trained for $k _ { t r a i n } ~ < ~ k _ { t e s t }$ already has small $\\Sigma _ { c }$ , such that increasing $k$ during testing has further diminished improvement on performance. ", + "bbox": [ + 174, + 272, + 825, + 342 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 INTERPRETATION IN TERMS OF VC DIMENSION ", + "text_level": 1, + "bbox": [ + 174, + 359, + 542, + 373 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In any given episode, a nearest neighbour prediction is performed from the support data (with a fixed embedding function). Therefore, a PAC learnability interpretation of the relation between the number of support data and complexity of the classifier can be made. Specifically, for binary classification, classical PAC learning theory (Vapnik et al., 1994) states that with probability of at least $1 - \\delta$ , the following inequality on the difference between empirical error $e r r _ { t r a i n }$ (of the support samples) and true error $e r r _ { t r u e }$ holds for any classifier $h$ : ", + "bbox": [ + 174, + 385, + 825, + 469 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fc80273b26d8be5b236798c7c241bcc8a6a984a2b3148b298183df2a68252da1.jpg", + "text": "$$\ne r r _ { t r u e } ( h ) - e r r _ { t r a i n } ( h ) \\leq \\sqrt { \\frac { D ( \\ln \\frac { 4 k } { D } + 1 ) + \\ln \\frac { 4 } { \\delta } } { 2 k } }\n$$", + "text_format": "latex", + "bbox": [ + 325, + 477, + 674, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Where $D$ is the VC dimension, and $k$ is the number of support samples per class 4. Under this binary classification setting, the predictions of prototypical network are equal to $\\sigma ( \\alpha )$ as shown earlier. Denoting $\\mathbf { z } _ { c } = \\overline { { \\phi ( S _ { c } ) } }$ and $\\mathbf { z } _ { c ^ { \\prime } } = \\overline { { \\phi ( S _ { c ^ { \\prime } } ) } }$ , we can manipulate $\\alpha$ as follows: ", + "bbox": [ + 173, + 526, + 825, + 571 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0c844513f93640e48dfd8eb7783294ca596834b97530042e47f7c6a4cddac101.jpg", + "text": "$$\n\\begin{array} { r l } { \\alpha = \\left\\| \\phi ( \\mathbf { x } ) - \\mathbf { z } _ { c } \\right\\| ^ { 2 } - \\left\\| \\phi ( \\mathbf { x } ) - \\mathbf { z } _ { c ^ { \\prime } } \\right\\| ^ { 2 } } & { { } = 2 ( \\mathbf { z } _ { c ^ { \\prime } } - \\mathbf { z } _ { c } ) ^ { T } \\phi ( \\mathbf { x } ) + ( \\mathbf { z } _ { c } ^ { T } \\mathbf { z } _ { c } - \\mathbf { z } _ { c ^ { \\prime } } ^ { T } \\mathbf { z } _ { c ^ { \\prime } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 579, + 759, + 599 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "From equation 12, the prototypes form a linear classifier with offset in the embedding space. The VC dimension of this type of classifier is $1 + d$ where $d$ is the intrinsic dimension of the embedding space (Vapnik, 1998). In ProtoNets, the intrinsic dimension of the embedding space is not only influenced by network architecture, but more importantly determined by the parameter themselves, making it a learned property. For example, if the embedding function can be represented with a linear transformation $\\phi ( \\mathbf { x } ) = \\Phi \\cdot \\mathbf { x }$ , then the intrinsic dimension of the embedding space is upper bounded by the rank of $\\Phi$ (since all embeddings must lie in the column space of $\\Phi$ ). Thus, the number of support samples required to learn from an episode is proportional to the intrinsic dimension of the embedding space. We hypothesize that an embedding function optimal for lower shot (e.g. one-shot) classification affords fewer intrinsic dimensions than one that is optimal for higher shot (e.g. 10-shot) classification. ", + "bbox": [ + 173, + 604, + 826, + 757 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 RECONCILING SHOT DISCREPANCY THROUGH EMBEDDING SPACE TRANSFORMATION ", + "text_level": 1, + "bbox": [ + 171, + 776, + 800, + 790 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Observations in Section 3.2 reveal that an ideal $\\phi$ would have an output space whose intrinsic dimension $d$ is as small as possible to minimize the right-hand side of equation 11 but just large enough to allow low $e r r _ { t r a i n }$ ; the balance between the two objectives is dictated by $k$ . Similarly, observations in Section 3.1 suggest that an ideal $\\phi$ would balance between minimizing $\\Sigma _ { c }$ and maximizing $\\Sigma$ also according to $k$ . As a result, when there is discrepancy between meta-training shots and meta-testing shots5, accuracy at meta-test time will suffer. A naive solution is to prepare many embedding functions trained for different shots, and select the embedding function according to the availability of label data at test-time. However, this solution is computationally burdensome as it requires multiple models to be trained and stored. Instead, we want to train and store a single model, and then adapt the embedding function’s variance characteristics and the embedding space dimensionality to achieve good performance for any test shot. ", + "bbox": [ + 176, + 801, + 825, + 886 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Linear Discriminant Analysis (LDA) is a dimensionality reduction method suited for downstream classification (Fukunaga, 1990). Its goal is to find a maximally discriminating subspace (maximum inter-class variance and minimal intra-class variance) for a given classification task. Theoretically, performance can be maximized in each individual episode by computing the LDA transformation matrix using support samples of that episode. LDA computes the eigenvectors of the matrix $S ^ { - 1 } S _ { \\mu }$ , where $S _ { \\mu }$ is the covariance matrix of prototypes and $S$ is the class-conditional covariance matrix. In practice, $S _ { \\mu }$ and $S$ cannot be stably estimated in few-shot episodes, preventing the direct application of LDA. ", + "bbox": [ + 174, + 180, + 825, + 291 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We propose an alternative which we call Embedding Space Transformation (EST). The purpose of EST is to perform dimensionality reduction on the features, while also improving the ratio in Theorem 3. This is a different goal from LDA because Theorem 3 demonstrates that the expected performance across many episodes can be improved by maximizing $\\Sigma$ and minimizing $\\Sigma _ { c }$ . Similar to LDA, EST works by applying a linear transformation ", + "bbox": [ + 173, + 297, + 825, + 368 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/b8221aa1f4e79a1494cf91246de64aaeb2b9961bef9ab0e1e96091e15aae3864.jpg", + "text": "$$\n\\phi ( \\mathbf { x } ) \\mapsto V ^ { * } ( \\phi ( \\mathbf { x } ) )\n$$", + "text_format": "latex", + "bbox": [ + 436, + 376, + 562, + 393 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "to the outputs of the embedding function. Here, $V ^ { * }$ is a linear transformation computed using $\\mathcal { D } _ { t r }$ after meta-training has completed. To compute $V ^ { * }$ , we first iterate through all classes in $\\mathcal { D } _ { t r }$ and compute their in-class means and covariance matrices in the embedding space. We can then find the covariance of means $\\Sigma _ { \\mu }$ , and the mean of covariances $\\overline { { \\Sigma } } _ { s }$ across $\\mathcal { D } _ { t r }$ . Finally, $V ^ { * }$ is computed by taking the leading eigenvectors of $\\Sigma _ { \\mu } - \\rho \\overline { { \\Sigma } } _ { s }$ - the difference between the covariance matrix of the mean and the mean covariance matrix with weight parameter $\\rho$ . The exact procedure for computing $V ^ { * }$ is presented in the appendix. ", + "bbox": [ + 174, + 400, + 825, + 501 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS AND RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 522, + 446, + 536 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, our first two experiments aim at supporting our theoretical results in Sections 3.1 and 3.2, while our third experiment demonstrates the improvement of EST on benchmark data sets over vanilla ProtoNets. To illustrate the applicability of our results to different embedding function architectures, all experiments are performed with both a vanilla 4-layer CNN (as in (Snell et al., 2017)) and a 7-layer Residual network (He et al., 2016). Detailed description of the architecture can be found in the appendix. Experiments are performed on three data sets: Omniglot (Lake et al., 2015), miniImageNet (Vinyals et al., 2016), and tieredImageNet (Ren et al., 2018). We followed standard data processing procedures which are detailed in the appendix. ", + "bbox": [ + 174, + 553, + 825, + 664 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 TRAINING SHOTS AFFECT VARIANCE CONTRIBUTION ", + "text_level": 1, + "bbox": [ + 174, + 683, + 578, + 695 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The total variance observed among embeddings of all data points can be seen as a composition of inter-class and (expected) intra-class variance based on the law of total variance $\\mathrm { ( V a r } [ \\mathbf { x } ] \\ =$ $\\mathbb { E } [ \\mathrm { V a r } ( \\mathbf { x } | c ) ] + \\mathrm { V a r } ( \\mathbb { E } [ \\mathbf { x } | c ] ) = \\mathbb { E } [ \\Sigma _ { c } ] + \\Sigma _ { \\mu } )$ . Our analysis predicts that as we increase the shot number used during training, the ratio of inter-class to intra-class variance will decrease. ", + "bbox": [ + 174, + 708, + 825, + 763 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To verify this hypothesis, we trained ProtoNets (vanilla and residual) with a range of shots (1 to 10 on miniImageNet and tiered imagenet, 1 to 5 on Omniglot) until convergence with 3 random initializations per group. Then, we computed the inter-class and intra-class covariance matrices across all samples in the training-set embedded by each network. To qualify the amplitude of each matrix, we take the trace of each covariance matrix. The ratio of inter-class to intra-class variance is presented in Figure 1: as we increase $k$ used during training, the inter-class to intra-class variance ratio decreases. This trend can be observed in both vanilla and residual embeddings, and across all three data sets, lending strong support to our result in Section 3.1. Another observation can be made that the ratio between inter-class and intra-class variance is significantly higher in the Omniglot data set than the other two data sets. This may indeed be reflective of the relative difficulty of each data set and the accuracy of ProtoNet on the data sets. ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/654b081bf5f9f75d64263ab6ff1c8f71fc822d2b4748a3ef0ce6491dd18fc619.jpg", + "image_caption": [ + "Figure 1: Inter-class to Intra-class variance ratios of embedding space varies with $k$ used in training. Left: Omniglot. Right: miniImageNet and tieredImageNet. " + ], + "image_footnote": [], + "bbox": [ + 209, + 104, + 776, + 256 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 TRAINING SHOTS AFFECT INTRINSIC DIMENSION ", + "text_level": 1, + "bbox": [ + 174, + 324, + 552, + 338 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We consider the intrinsic dimension (id) of an embedding function with extrinsic dimension $E$ (operated on a data set) to be defined as the minimum integer $d$ where all embedded points of that data set lie within a $d$ -dimensional subspace of $\\mathbb { R } ^ { E }$ (Bishop, 2006). A simple method for estimating $d$ is through principal component analysis (PCA) of the embedded data set. By eigendecomposing the covariance matrix of embeddings, we obtain the principal components expressed as the significant eigenvalues, and the principal directions expressed as the eigenvectors corresponding to those eigenvalues. The number of significant eigenvalues approximates the intrinsic dimension of the embedding space. When the subspace is linear, this approximation is exact; otherwise, it serves as an upper bound to the true intrinsic dimension (Fukunaga & Olsen, 1971). ", + "bbox": [ + 173, + 349, + 825, + 474 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We determine the number of significant eigenvalues by an explained-variance over total-variance criterion. The qualifying metric is $\\begin{array} { r } { r _ { d } \\triangleq \\sum _ { i \\in [ 1 , d ] } \\lambda _ { i } / \\sum _ { i \\in [ 1 , E ] } \\lambda _ { i } } \\end{array}$ . In our experiments, we set the threshold for $r _ { d }$ at 0.9. Similar to the previous experiment, we train ProtoNets with different shots to convergence. The total covariance matrix is then computed on the training set and eigendecomposition is performed. The approximate id is plotted for various values of $k$ in Figure 2. We can see a clear trend that as we increase training shot, the id of the embedding space increases. ", + "bbox": [ + 174, + 482, + 825, + 569 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/274fb3036767fc31f4d4c6b41ff94346ef998c29094bc159e001fc6cf33f4586.jpg", + "image_caption": [ + "Figure 2: Intrinsic dimension approximated by the number of significant eigenvalues varies with $k$ used in training. Left: Omniglot. Right: miniImageNet and tieredImageNet " + ], + "image_footnote": [], + "bbox": [ + 207, + 584, + 772, + 734 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 EXPERIMENTS WITH EST ", + "text_level": 1, + "bbox": [ + 174, + 779, + 392, + 792 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate the performance of EST on the three aforementioned data sets. The performance is compared against our implementation of vanilla ProtoNets as a baseline, as well as a variant of ProtoNets using principal components obtained from all embedding points (ProtoNet-PCA). ", + "bbox": [ + 176, + 805, + 821, + 847 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "All methods in this section use the same set of trained ProtoNets. As with before, networks are trained with $k \\in \\{ 1 , . . . , 5 \\}$ on Omniglot and $k \\in \\{ 1 , . . . , 1 0 \\}$ on miniImageNet and tieredImageNet. Additionally, we also trained a mixed-shot network for each data set. This is done by randomly selecting a value for $k$ within the specified range for each episode, and then sampling the corresponding number of support samples. Hyper-parameters for training are described in the appendix. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(a) Omniglot-20-way, with 4 layer CNN. ", + "bbox": [ + 218, + 136, + 457, + 150 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/559ea61fc7b0fd9e504b39f9586fae3de580ee7b82349eba9aec2e213e04aae4.jpg", + "table_caption": [ + "Table 1: Classification accuracies of ProtoNet variants. Best performing methods and any other runs within $9 5 \\%$ confidence margin is in bold ", + "(b) Omniglot-20-way, with 7 layer ResNet. " + ], + "table_footnote": [], + "table_body": "
MODELTRAINING SHOTSTESTING SHOTSAVERAGE ACCURACYMODELTRAINING SHOTSTESTING SHOTSAVERAGE ACCURACY
1515
VANILLA PROTONET195.07%98.89%97.81%VANILLA PROTONET196.46%99.07%98.35%
VANILLA PROTONET593.42%98.78%97.25%VANILLA PROTONET594.42%98.99%97.75%
MIXED-k SHOT1-594.84%98.92%97.74%MIXED-k PROTONET1-596.53%99.15%98.43%
PCAPROTONET194.94%98.85%97.78%PCA PROTONET196.02%98.99%98.19%
EST PROTONET195.11%98.84%97.83 %EST PROTONET195.55%99.02%98.19%
", + "bbox": [ + 178, + 155, + 821, + 231 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/bbd39650d5af468eeb5dfdbd1ce7aa958ff0e3fda0b8358517ae1cb792c52fa6.jpg", + "table_caption": [ + "(c) miniImageNet-5-way, with 4 layer CNN. " + ], + "table_footnote": [], + "table_body": "
MODELTRAINING SHOTSTESTING SHOTSAVERAGE ACCURACY
1510
PROTONET148.89%64.70%68.90%63.15%
PROTONET544.75%67.23%72.36%65.08%
PROTONET1039.99%66.23%72.47%63.54%
MIXED-k SHOT1-1049.36%67.96%72.27%65.83%
PCA PROTONET549.36%68.63%72.82%66.12%
EST PROTONET550.22%68.25%73.29%66.60%
", + "bbox": [ + 179, + 261, + 496, + 335 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/ce19f8932d3494c6e6a6205d6094df52311c11ec23341615ac9ee3d59ef14bcb.jpg", + "table_caption": [ + "(d) miniImageNet-5-way, with 7 layer ResNet. " + ], + "table_footnote": [], + "table_body": "
TRAININGTESTING SHOTSAVERAGE
MODELSHOTS1510ACCURACY
PROTONET152.65%68.27%72.29 %66.73%
PROTONET547.40%69.93%74.35%67.18%
PROTONET1042.20%68.23%74.54%65.75%
MIXED-k SHOT1-1051.74%69.09%73.63%67.41%
PCA PROTONET550.09%69.25%74.24%67.63%
EST PROTONET551.93%69.98%74.80%68.19%
", + "bbox": [ + 501, + 261, + 818, + 335 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/cd7c17112aeb2171ab1a0071cbe68aa1fc56141836197c743ebf66a884cff874.jpg", + "table_caption": [ + "(e) tieredImageNet-5-way, with 4 layer CNN. " + ], + "table_footnote": [], + "table_body": "
TRAININGTESTING SHOTSAVERAGE
MODELSHOTS1510ACCURACY
PROTONET147.37%63.70%67.99%61.85%
PROTONET542.33%66.51%72.05%64.05%
PROTONET1035.38%64.56%71.03%61.24%
MIXED-k SHOT1-1047.67%66.34%70.96%64.33%
PCA PROTONET548.34%67.07%71.65%64.96%
EST PROTONET548.85%67.24%72.09 %65.46%
", + "bbox": [ + 179, + 366, + 496, + 440 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/688f5543513089f97511637495f868740b77922d18838cc4efbabfdd7164d759.jpg", + "table_caption": [ + "(f) tieredImageNet-5-way, with 7 layer ResNet. " + ], + "table_footnote": [], + "table_body": "
TRAININGTESTING SHOTSAVERAGE
MODELSHOTS1510ACCURACY
PROTONET149.78%65.17%69.88%63.98%
PROTONET547.88%69.12%73.80%66.99 %
PROTONET1040.86%69.37%74.72%65.95%
MIXED-k SHOT1-1050.74%69.28%73.01 %66.95%
PCA PROTONET551.21%68.88%72.17 %67.14%
EST PROTONET553.05%69.30%73.63%67.91%
", + "bbox": [ + 501, + 366, + 818, + 440 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Each model is evaluated on the test splits of the corresponding data sets (e.g. networks trained on Omniglot are only evaluated on Omniglot). Five test runs are performed per network on Omniglot to evaluate the $k$ -shot performance $( k \\in [ 1 , 5 ] )$ ). Each run consists of 600 episodes formed by 1-5 support samples and 5 query sample per class. The performance is aggregated across runs for the combined performance. Similarly, on miniImageNet and tieredImageNet, 10 test runs are performed with support per sample $k \\in [ 1 , 1 0 ]$ , 600 episodes per run, and 15 query samples in each episode. ", + "bbox": [ + 173, + 474, + 825, + 559 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Model configuration: Vanilla ProtoNet is used as our baseline. We present the performance of multiple ProtoNets trained with different shots to illustrate the performance degradation issue. ProtoNet-PCA uses principal components of the training split embeddings in place of $V ^ { * }$ , with components other than the $d$ leading ones zeroed out. We carry out a parameter sweep on miniImageNet and set $d = 6 0$ ; the same value is used on the other two data sets. For selecting the training shot of the embedding network, we find that overall performance to be optimal using $k = 5$ . ProtoNet-EST contains three parameters that need to be determined: $\\rho , d$ , and training shots of the embedding network. For our experiments, we set $\\rho = 0 . 0 0 1$ and $d = 6 0$ based on performance on miniImageNet. For selecting the number of training shots, we use the same strategy as before by evaluating ProtoNetEST with all trained embedding networks and found the same trend to hold. ", + "bbox": [ + 173, + 565, + 826, + 705 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As an abalation study, FC-ProtoNet adds a fully connected layer to the embedding network such that the output dimension is also 60. Results of this variant can be found in the appendix. ", + "bbox": [ + 176, + 704, + 823, + 732 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "EST performance results: Table 1 summarizes the performance of the evaluated methods on all data sets. Due to space constraints, only 1-shot, 5-shot, 10-shot and 1-10 shot average performance are included. Additional results are in the appendix. The best performing method in each evaluation is in bold. On Omniglot, there is no significant difference in performance between the best performing vanilla ProtoNet and any other methods. We attribute this to the already high accuracy of the baseline model. On miniImageNet and tieredImageNet, EST-ProtoNet significantly outperforms baseline methods and PCA-protonet in terms of average accuracy over test runs with different shots. ", + "bbox": [ + 174, + 736, + 825, + 833 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We observe that matching the training shot to the test shot generally provides the best performance for vanilla ProtoNets. Also importantly, training with a mixture of different values of $k$ does not provide optimal performance when evaluated on the same mixture of $k$ values. Instead, the resulting performance is mediocre in all test shots. ProtoNet-EST provides minor improvements over the best-performing baseline method under most test shots settings. We hypothesize that this is due to EST aligning the embedding space to the directions with high inter-class variance and low intra-class variance. Comparison against the direct PCA approach demonstrates that the performance uplift is not entirely attributed to reducing the dimensions of the embedding space. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In conclusion, EST improves the performance of ProtoNets on the more challenging data sets when evaluated with various test shots. It successfully tackles performance degradation when testing shots and training shots are different. This improvement is vital to the deployment of ProtoNets in real world scenarios where the number of support samples cannot be determined in advance. ", + "bbox": [ + 174, + 138, + 823, + 195 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 214, + 341, + 229 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We summarize related work on extensions of ProtoNets, on improving the few-shot classification setup, and on analyzing theoretical properties of meta-learning methods. ", + "bbox": [ + 176, + 244, + 821, + 273 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Extensions of ProtoNets: Allen et al. (2019) build upon ProtoNets by allowing each class to be represented by multiple prototypes, thereby improving the representation power of ProtoNets. Oreshkin et al. (2018) use a context-conditioned embedding network to produce prototypes that are aware of the other classes. These prior works assume matched training and testing shots whereas our work focuses on setups where testing shots are not fixed (i.e. not necessarily the same as training shots). Our work is parallel to these works in that EST can be applied on the embeddings learned by these methods. ", + "bbox": [ + 174, + 287, + 825, + 385 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Improvement of few-shot classification setup: Chen et al. (2019) extend the few-shot learning problem setup by considering domain adaptation in addition to learning novel classes. Specifically, they look at how well models trained on miniImageNet can perform on few-shot learning in CUB200. Importantly, they still force the number of shots to be consistent between training time and testing time. While their work deals with varying the domain of the episodes at test time, our work deals with varying shots. Concurrent to our work, Triantafillou et al. (2020) further broaden the scope of few-shot learning by introducing a benchmark composed of data from various domains; methods are tested on their ability to adapt to different domains and deal with class imbalance. We extend their work with a thorough analysis of how the number of shots affects the learning outcome, and further propose a method to overcome the negative impact of mismatched shots. ", + "bbox": [ + 174, + 400, + 825, + 540 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Theoretical analysis of few-shot learning: Despite the myriad of methodological improvements, theoretical work on few-shot learning has been sparse. Wang et al. (2019b) provide a unifying formulation for few-shot learning methods, and clearly outline the key challenge in few-shot learning through a PAC argument, but do not introduce any new theoretical results. In contrast, our work introduces a novel bound for the accuracy of ProtoNets; this bound provides useful intuitions pertaining to how ProtoNets adapt to few-shot episodes. Additionally, we demonstrate theoretically and experimentally that the intrinsic dimension of the embedding function’s output space varies with the number of shots as a direct consequence of the challenges outlined in the PAC argument. To the best of our knowledge, Amit & Meir (2017) provide the only prior work to bound the error of a meta-learning agent. Specifically, they use the generalized PAC-Bayes framework to derive an error-rate bound for a MAML-style learning algorithm (where the hypothesis class is fixed). Their main result relates the performance of the learning algorithm to both the number of tasks encountered during meta-training and the number of shots given in any task. In contrast to our work, their result does not apply to non-parametric methods such as ProtoNets because the hypothesis class in ProtoNets can change from episode to episode depending on the number of ways. ", + "bbox": [ + 174, + 555, + 825, + 762 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 781, + 486, + 797 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have explored how the number of support samples used during meta-training can influence the learned embedding function’s performance and intrinsic dimensions. Our proposed method transforms the embedding space to maximize inter-to-intra class variance ratio while constraining the dimensions of the space itself. In terms of applications, our method can be combined other works (Oreshkin et al., 2018; Ye et al., 2018; Rusu et al., 2019; Dong & Xing, 2018; Ren et al., 2018; Tapaswi et al., 2019) with an embedding learning component. We believe our approach is a significant step to reduce the impact of the shot number in meta-training, which is a crucial hyperparameter for few-shot classification. 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In Proceedings of the 30th International Conference on Neural Information Processing Systems (NIPS), NIPS’16, pp. 3637–3645, USA, 2016. Curran Associates Inc. ISBN 978-1-5108-3881-9. URL http://dl.acm.org/citation.cfm? id=3157382.3157504. ", + "bbox": [ + 174, + 679, + 826, + 750 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Jixuan Wang, Kuan-Chieh Wang, Marc T. Law, Frank Rudzicz, and Michael Brudno. Centroid-based deep metric learning for speaker recognition. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3652–3656. IEEE, 2019a. ", + "bbox": [ + 173, + 757, + 825, + 801 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Yaqing Wang, Quanming Yao, James T. Kwok, and Lionel M. Ni. Generalizing from a few examples: A survey on few-shot learning. 2019b. ", + "bbox": [ + 174, + 808, + 823, + 837 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Han-Jia Ye, Hexiang Hu, De-Chuan Zhan, and Fei Sha. Learning embedding adaptation for few-shot learning. CoRR, abs/1812.03664, 2018. URL http://arxiv.org/abs/1812.03664. ", + "bbox": [ + 173, + 844, + 825, + 875 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Pierre Yger, Marcel Stimberg, and Romain Brette. Fast learning with weak synaptic plasticity. Journal of Neuroscience, 35(39):13351–13362, 2015. ISSN 0270-6474. doi: 10.1523/JNEUROSCI. 0607-15.2015. URL http://www.jneurosci.org/content/35/39/13351. ", + "bbox": [ + 174, + 882, + 826, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 ALGORITHM FOR EST ", + "text_level": 1, + "bbox": [ + 176, + 135, + 374, + 150 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Below is the exact procedure for computing $\\tau$ for embedding space transformation. ", + "bbox": [ + 174, + 162, + 722, + 178 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1 Algorithm for computing the transformation $\\tau$ . \n$L _ { n }$ is the number of samples belonging to class $n$ ; $\\mu _ { n }$ and $\\Sigma _ { n }$ are the mean and covariance of the embeddings of that class; $\\mu _ { T }$ and $\\widetilde { \\Sigma } _ { s }$ are the average of mean embeddings and covariances; $\\Sigma _ { \\mu }$ is the covariance of the mean embeddings; $V ^ { * }$ is the matrix of eigenvectors that correspoinds to the $d$ largest eigenvalues in $\\Lambda$ . ", + "bbox": [ + 173, + 195, + 826, + 265 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Input: Training set $\\mathcal { D } _ { t r } = \\{ ( \\mathbf { x } _ { 1 } , y _ { 1 } ) , . . . , ( \\mathbf { x } _ { M } , y _ { M } \\}$ , where $y _ { i } \\in \\{ 1 , . . . , N \\}$ , $\\mathcal { D } _ { n }$ denotes the subset of $\\mathcal { D } _ { t r }$ where $\\forall y \\in { \\mathcal { D } } _ { n } , y = n$ , embedding function $\\phi$ , weighting parameter $\\rho$ , dimension parameter $d$ . ", + "bbox": [ + 187, + 270, + 825, + 313 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/05dc6ccf9a72711f4067b799c63491d7fcbccf27ab45834bd78251221e837592.jpg", + "text": "$$\n\\begin{array} { r } { \\Sigma [ n ] = \\frac { 1 } { L _ { n } } \\sum _ { ( \\mathbf { x } _ { i } , y _ { i } ) \\in \\mathcal { D } _ { n } } ^ { L _ { n } } ( \\phi ( \\mathbf { x } _ { i } ) - \\mu _ { n } ) ( \\phi ( \\mathbf { x } _ { i } ) - \\mu _ { n } ) ^ { T } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 366, + 557, + 390 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "end for ", + "text_level": 1, + "bbox": [ + 191, + 388, + 243, + 401 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/05fb79a737c7787be2b7a7e3776c5a4d15724a72c2915c23bdc8b86424941ce2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mu _ { T } = \\frac { 1 } { M } \\sum _ { ( \\mathbf { x } _ { i } , y _ { i } ) \\in \\mathcal { D } _ { t r } } \\phi ( \\mathbf { x } _ { i } ) } \\\\ & { \\Sigma _ { \\mu } = \\frac { 1 } { N } \\sum _ { n \\in [ 1 , N ] } ( \\mu [ n ] - \\mu _ { T } ) ( \\mu [ n ] - \\mu _ { T } ) ^ { T } } \\\\ & { \\overline { { \\Sigma } } _ { s } = \\frac { 1 } { N } \\sum _ { n \\in [ 1 , N ] } \\Sigma [ n ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 398, + 490, + 455 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 NETWORK ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 542, + 403, + 556 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The vanilla CNN has the exact same architecture as the original ProtoNet (Snell et al., 2017). It consists of four convolution layers with depth of 64; each convolution layer is followed by Relu activation, max-pooling, and batch normalization (Ioffe & Szegedy, 2015). Resnet of 7 layers is constructed with one vanilla convolution layer of depth 64 followed by three residual blocks, all joined by max-pooling layers; each residual block consists of two sets of conv-batchnorm-Relu layers, of depth 128-256-256. ", + "bbox": [ + 173, + 569, + 826, + 652 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.3 DATA SET DESCRIPTION AND PRE-PROCESSING ", + "text_level": 1, + "bbox": [ + 174, + 674, + 544, + 688 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Experiments are performed on three data sets: Omniglot (Lake et al., 2015), miniImageNet (Vinyals et al., 2016), and tieredImageNet (Ren et al., 2018). For Omniglot experiments, we follow the same configuration as in the original paper where 1200 classes augmented with rotations (4800 total) are used for training, and the remaining classes are used for testing. ", + "bbox": [ + 174, + 700, + 825, + 756 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For miniImageNet experiments, we use the splits proposed by (Ravi & Larochelle, 2017) where 64 classes are used for training, 16 for validation, and 20 for testing. Mirroring the original paper, we resize all miniImageNet images to $8 4 \\mathrm { x } 8 4$ . No data augmentation is applied. As most state-of-art few-shot classification methods achieve saturating accuracies on Omniglot, and miniImageNet’s small number of classes make claims about generalization difficult, we also conduct experiments of tieredImageNet. ", + "bbox": [ + 174, + 763, + 825, + 847 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "TieredImageNet is also a subset of Imagenet1000. TieredImageNet groups classes into broader categories corresponding to higher-level nodes in the ImageNet hierarchy. It includes 34 categories, with each category containing between 10 and 30 classes. These are split into 20 training, 6 validation and 8 testing categories. In total, there are 351 classes in training, 97 in validation, and 160 in testing. Preprocessing of images follow the same steps as used for miniImageNet. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.4 PROTONET TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 369, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Training procedure of ProtoNets largely mirrors the protocol used by Snell et al. (2017). On Omniglot, we train the network to convergence after 30000 episodes. On miniImageNet and tieredImageNet, we monitor the performance of the network on the validation set and select the best performing checkpoint after training for 50000 episodes. Adam (Kingma & Ba, 2014) optimizer is used with $\\alpha = 0 . 9$ , $\\beta = 0 . 9 9 9$ , $\\epsilon = 1 0 ^ { - 8 }$ , and an initial learning rate of 0.001 that is decayed by half every 2000 episodes. On Omniglot, we train with 60 classes and 5 query points per episode. On miniImageNet and tieredImageNet, we train with 20 classes and 15 query points per episode. ", + "bbox": [ + 173, + 128, + 826, + 228 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.5 DERIVATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 243, + 370, + 258 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof of Lemma 1: ", + "bbox": [ + 174, + 270, + 300, + 284 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. First, from the definition of $\\alpha$ , we split $\\mathbb { E } _ { \\mathbf { x } , S \\mid a , b } [ \\alpha ]$ in to two parts and examine them separately: ", + "bbox": [ + 171, + 299, + 825, + 316 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/32891f2773296e05b4f76f5e9a37baa3545054d0cd37c62e744be67cce668768.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathbf { x } , S | a , b } [ \\alpha ] = \\underbrace { \\mathbb { E } [ \\left. \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right. ^ { 2 } ] } _ { i } - \\underbrace { \\mathbb { E } [ \\left. \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right. ^ { 2 } ] } _ { i i } .\n$$", + "text_format": "latex", + "bbox": [ + 299, + 321, + 699, + 368 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In general, for random vector $X$ , the expectation of the quadratic form is $\\mathbb { E } [ \\left. X \\right. ^ { 2 } ] = \\operatorname { T r } ( \\operatorname { V a r } ( X ) ) +$ $\\mathbb { E } [ { \\bar { X } } ] ^ { T } \\mathbb { E } [ X ]$ . Hence, ", + "bbox": [ + 173, + 375, + 826, + 406 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1031053d6c192d05784058cbe7c5f70fd0e95ed95d2709b3f4036b67f1a40736.jpg", + "text": "$$\n\\begin{array} { r l } & { i = \\mathbb { E } _ { \\mathbf { x } , S \\mid a , b } [ \\Big \\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\Big \\| ^ { 2 } ] } \\\\ & { \\phantom { = } = \\mathrm { T r } ( \\Sigma _ { \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } } ) + \\mathbb { E } [ \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } ] ^ { T } \\mathbb { E } [ \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 410, + 699, + 465 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where the first term inside the trace can be expanded as: ", + "bbox": [ + 173, + 468, + 540, + 483 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/27ff8705f35a31976e912dd8d81d47fb74526ac7f3b78031c5d9ad3531314d20.jpg", + "text": "$$\n\\begin{array} { l l } { { \\displaystyle \\Sigma _ { \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { b } ) } } } = \\mathrm { V a r } [ \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { b } ) } } ] } } \\\\ { { } } & { { = \\mathbb E [ ( \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { b } ) } } ) ( \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { b } ) } } ) ^ { T } ] - ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ^ { T } } } \\\\ { { } } & { { = \\Sigma _ { c } + \\mathbf { \\mu } _ { a } \\mathbf { \\mu } _ { a } ^ { T } + \\displaystyle \\frac { 1 } { k } \\Sigma _ { c } + \\mathbf { \\mu } _ { b } \\mathbf { \\mu } _ { b } ^ { T } - \\displaystyle \\mathbf { \\mu } _ { a } \\mathbf { \\mu } _ { b } ^ { T } - \\displaystyle \\mathbf { \\mu } _ { b } \\mathbf { \\mu } _ { a } ^ { T } - ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ^ { T } } } \\\\ { { } } & { { = ( 1 + \\displaystyle \\frac { 1 } { k } ) \\Sigma _ { c } ~ \\displaystyle ( \\mathrm { L a s t ~ t e r m s ~ c a n c e l ~ o u t } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 488, + 774, + 594 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To go from (18) to (19), we note that $\\operatorname { V a r } ( X ) = \\mathbb { E } [ X X ^ { T } ] - \\mathbb { E } [ X ] \\mathbb { E } [ X ] ^ { T }$ and $\\Sigma _ { c } \\overset { \\Delta } { = } \\mathrm { V a r } ( \\phi ( \\mathbf { x } ) )$ . Hence (19) can be obtained by expanding out the first term and taking the expectation of each resulting item. ", + "bbox": [ + 173, + 602, + 826, + 646 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The second term of (16) is simply: ", + "bbox": [ + 176, + 646, + 400, + 660 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/e5ae6f9d9a744584437c2351b8ec0503280ac814962fe35947397f5bda6c09f8.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathbf { x } , S | a , b } [ \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } ] = \\mu _ { a } - \\mu _ { b } .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 665, + 619, + 685 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Putting them together: ", + "bbox": [ + 174, + 690, + 321, + 704 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/0d56b66b3258db2aeaf6833be001dc6f8b97c6a2901dbe16c765c885bb3531b0.jpg", + "text": "$$\ni = ( 1 + \\frac { 1 } { k } ) \\mathrm { T r } ( \\Sigma _ { c } ) + ( \\mu _ { a } - \\mu _ { b } ) ^ { T } ( \\mu _ { a } - \\mu _ { b } ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 709, + 650, + 739 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Similarly for $_ { i i }$ ", + "bbox": [ + 174, + 744, + 279, + 758 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/5a8f4e09e3a1b0df84794c5b142f0711ae4ad20990f7e531a5ddf4a1f3595551.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle i i = \\mathbb { E } _ { { \\bf x } , S \\vert a , b } [ \\left. \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right. ^ { 2 } ] } \\ ~ } \\\\ { { \\displaystyle ~ = \\mathrm { T r } ( \\Sigma _ { \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { a } ) } } } ) + \\mathbb { E } [ \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { a } ) } } ] ^ { T } \\mathbb { E } [ \\phi ( { \\bf x } ) - \\overline { { \\phi ( S _ { a } ) } } ] } \\ } \\\\ { { \\displaystyle ~ = ( 1 + \\frac { 1 } { k } ) \\mathrm { T r } ( \\Sigma _ { c } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 763, + 700, + 849 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Putting together $i$ and $\\romannumeral 2$ : ", + "bbox": [ + 173, + 852, + 338, + 867 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/0b7cad37faf9fa1ed9e95ded8b47eab55ee430a248b27b9abbd80196dfd749c7.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { { \\mathbf { x } } , S | a , b } [ \\alpha ] = ( 1 + \\frac { 1 } { k } ) \\mathrm { T r } ( \\Sigma _ { c } ) + ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ^ { T } - ( 1 + \\frac { 1 } { k } ) \\mathrm { T r } ( \\Sigma _ { c } ) } \\\\ { = ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ^ { T } ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 872, + 741, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Then, since $\\mathbb { E } _ { a , b , \\mathbf { x } , S } [ \\alpha ] = \\mathbb { E } _ { a , b } [ \\mathbb { E } _ { \\mathbf { x } , S | a , b } [ \\alpha ] ]$ , we have: ", + "bbox": [ + 173, + 102, + 524, + 119 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/09e9e11632c89540ab0aa272e4b595f9af8088fb04c4d50c660b5b4581fb92d7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { a , b , \\mathbf { x } , S } [ \\alpha ] = \\mathbb { E } _ { a , b } [ ( \\mu _ { a } - \\mu _ { b } ) ^ { T } ( \\mu _ { a } - \\mu _ { b } ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { a , b } [ \\mu _ { a } ^ { T } \\mu _ { a } + \\mu _ { b } ^ { T } \\mu _ { b } - \\mu _ { a } ^ { T } \\mu _ { b } - \\mu _ { b } ^ { T } \\mu _ { a } ] } \\\\ & { \\qquad = \\mathrm { T r } ( \\Sigma ) + \\mu ^ { T } \\mu + \\mathrm { T r } ( \\Sigma ) + \\mu ^ { T } \\mu - 2 \\mu ^ { T } \\mu } \\\\ & { \\qquad = 2 \\mathrm { T r } ( \\Sigma ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 316, + 123, + 679, + 203 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Where from the second to the third line, we note that $\\mu _ { a } ^ { T } \\mu _ { a }$ and $\\mu _ { b } ^ { T } \\mu _ { b }$ are quadratic forms while $\\mu _ { a } ^ { T } \\mu _ { b }$ describe a dot product between two independent randomly drawn samples which has expectation $\\mu ^ { T } \\mu$ . □ ", + "bbox": [ + 176, + 205, + 820, + 248 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For proof of Lemma 2, we first re-state the result on quadratic forms of normally distributed random vectors by Rencher & Schaalje (2008). ", + "bbox": [ + 174, + 263, + 823, + 292 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Theorem 4. Consider random vector y $\\sim \\cal { N } ( \\{ \\mathfrak { u } , \\Sigma \\}$ and symmetric matrix of constants $A$ , we have: ", + "bbox": [ + 176, + 295, + 821, + 311 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d031d9f529fc6cdde6bab3d032d5cd8c162ef84c082290a9660834c0ef913bcb.jpg", + "text": "$$\n\\mathrm { V a r } ( y ^ { T } A y ) = 2 \\mathrm { T r } ( ( A \\Sigma ) ^ { 2 } ) + 4 { \\mu } ^ { T } A \\Sigma A { \\mu } .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 314, + 640, + 333 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof of Lemma 2: ", + "bbox": [ + 174, + 344, + 300, + 358 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 173, + 375, + 217, + 388 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/88ca5802903b72748f6fc45fb8c4324a8c97ea4df367a8e4ebf5fc326abc32b2.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathrm { V a r } ( \\alpha | a , b ) = \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } ) } & { } \\\\ { = \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } ) + \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } ) } & { } \\\\ { - 2 \\mathrm { C o v } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } , \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } ) } & { } \\\\ { \\leq \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } ) + \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } ) } & { } \\\\ { + 2 \\sqrt { \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } ) \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } ) } } & { } \\\\ { \\leq 2 \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } ) + 2 \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } ) } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 392, + 727, + 579 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "From 33 to 35, we used Cauchy Schwarz inequality. From line 35 to line 37, we use the fact that $2 a b \\leq a ^ { 2 } + b ^ { 2 }$ for all $a , b \\in \\mathcal { R } ^ { + }$ . ", + "bbox": [ + 176, + 582, + 823, + 609 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By applying Theorem 4, we have: ", + "bbox": [ + 174, + 609, + 398, + 623 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c8cc60d906796619b3fb9696396e8daed98e0ecb20ac2c406d22a6a41f1fa267.jpg", + "text": "$$\n{ \\begin{array} { r l } & { \\operatorname { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - { \\overline { { \\phi ( S _ { b } ) } } } \\right\\| ^ { 2 } ) = 2 ( 1 + { \\frac { 1 } { k } } ) ^ { 2 } \\operatorname { T r } ( \\Sigma _ { c } ^ { 2 } ) + 4 ( 1 + { \\frac { 1 } { k } } ) ( \\mu _ { a } - \\mu _ { b } ) ^ { T } \\Sigma _ { c } ( \\mu _ { a } - \\mu _ { b } ) } \\\\ & { \\operatorname { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - { \\overline { { \\phi ( S _ { a } ) } } } \\right\\| ^ { 2 } ) = 2 ( 1 + { \\frac { 1 } { k } } ) ^ { 2 } \\operatorname { T r } ( \\Sigma _ { c } ^ { 2 } ) } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 225, + 626, + 772, + 690 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Finally, ", + "bbox": [ + 173, + 690, + 225, + 705 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e5cd0b6a6f14250eddacbc8c3af8226d98aa3a0791a9d9ccbcaa44f7474a6b65.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { a , b } [ \\mathrm { V a r } ( \\alpha | a , b ) ] \\leq \\mathbb { E } _ { a , b } [ 2 \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { b } ) } } \\right\\| ^ { 2 } ) + 2 \\mathrm { V a r } ( \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { a } ) } } \\right\\| ^ { 2 } ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { a , b } [ 8 ( 1 + \\frac { 1 } { k } ) ^ { 2 } \\mathrm { T r } ( \\Sigma _ { c } ^ { 2 } ) + 8 ( 1 + \\frac { 1 } { k } ) ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ^ { T } \\Sigma _ { c } ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ] } \\\\ & { \\qquad = 8 ( 1 + \\frac { 1 } { k } ) \\mathbb { E } _ { a , b } [ \\mathrm { T r } \\{ ( 1 + \\frac { 1 } { k } ) \\Sigma _ { c } ^ { 2 } + \\Sigma _ { c } ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ( \\mathbf { \\mu } _ { a } - \\mathbf { \\mu } _ { b } ) ^ { T } \\} ] } \\\\ & { \\qquad = 8 ( 1 + \\frac { 1 } { k } ) \\mathrm { T r } \\{ \\Sigma _ { c } [ ( 1 + \\frac { 1 } { k } ) \\Sigma _ { c } + 2 \\Sigma ] \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 227, + 708, + 771, + 832 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Extending to $N$ class: Let $\\mathbf x , y$ denote the query data pair, and the set of $N$ classes be denoted as c. Let $\\alpha _ { i } = \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { i } } } ) \\right\\| ^ { 2 } - \\left\\| \\phi ( \\mathbf { x } ) - \\overline { { \\phi ( S _ { y } } ) } \\right\\| ^ { 2 }$ . Then we have a correct prediction $\\hat { y } = y$ if $\\forall i \\in [ 1 , N ] , i \\neq y , \\alpha _ { i } > 0$ . Hence: $R ( \\phi ) = \\operatorname* { P r } _ { \\mathbf { c } , \\mathbf { x } , S } ( \\bigcup _ { i \\neq y } ^ { N } \\alpha _ { i } > 0 )$ ", + "bbox": [ + 173, + 861, + 826, + 928 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By Frechet’s inequality: ", + "bbox": [ + 174, + 103, + 333, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d5fe8ba6d5f35c41d809b99a09f44b00b4e4939436310cb4d4a71af71ee18092.jpg", + "text": "$$\nR ( \\phi ) \\geq \\sum _ { \\stackrel { i = 1 } { i \\neq y } } ^ { N } \\operatorname* { P r } ( \\alpha _ { i } > 0 ) - ( N - 2 )\n$$", + "text_format": "latex", + "bbox": [ + 379, + 116, + 619, + 169 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Noting that Theorem 3 can be applied to each term in the summation: ", + "bbox": [ + 173, + 171, + 629, + 185 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/e1eac0ab13873d697f57b7ff2505a17ffc52df34b409a4ececf89cca636ac064.jpg", + "text": "$$\n\\mathfrak { L } ( \\phi ) \\geq \\sum _ { \\stackrel { i = 1 } { i \\neq y } } ^ { N } \\frac { 4 \\mathrm { T r } ( \\Sigma ) ^ { 2 } } { 8 ( 1 + 1 / k ) ^ { 2 } \\mathrm { T r } ( \\Sigma _ { c } ^ { 2 } ) + 1 6 ( 1 + 1 / k ) \\mathrm { T r } ( \\Sigma \\Sigma _ { c } ) + \\mathbf { E } _ { i , y } [ ( ( \\mu _ { y } - \\mu _ { i } ) ( \\mu _ { y } - \\mu _ { i } ) ^ { T } ) ^ { 2 } ] } - ( N - 2 )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 193, + 838, + 242 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "It is then clear that the observations made on the binary case also applies to the multiclass case. ", + "bbox": [ + 169, + 250, + 795, + 265 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.6 ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 281, + 377, + 296 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Additionally, we experimented with directly setting the output dimension of the embedding network to 60 by adding a fully connected layer to the embedding network. This variant of protonet performs worse than both the base variant and all other methods. ", + "bbox": [ + 174, + 308, + 825, + 349 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/ca15b304f8445dae325c0315a841394d2ab0c88fd239ff017b702f60d1d9a41a.jpg", + "table_caption": [ + "Table 2: Classification Accuracy on miniImageNet-5-way, with 4 layer $\\mathrm { C N N } + 1$ Fully connected layer embedding network. " + ], + "table_footnote": [], + "table_body": "
TRAINING SHOTS2 310AVERAGE ACCURACY
45TESTING SHOTS 6789
PROTONET + FC544.77%53.75%58.04%61.06%62.26%64.60%65.19%66.63%66.65%67.52%61.05±0.28%
", + "bbox": [ + 174, + 409, + 823, + 443 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/a37c6d256f5c1f947d8f9b2c05baba36b9385f119b05e97db42f9794f3b121af.jpg", + "image_caption": [ + "Figure 3: Effect of hyperparameters on k-shot testing performance on miniImageNet. " + ], + "image_footnote": [], + "bbox": [ + 207, + 478, + 802, + 616 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/6bb9582b1b8cf5c08addcb1d0dd24394e9e58be79e552478beb71111765d4f0c.jpg", + "table_caption": [ + "Table 3: Classification Accuracy on miniImageNet-5-way, with 4 layer CNN embedding network. " + ], + "table_footnote": [], + "table_body": "
MODELTRAININGTESTING SHOTSAVERAGE
SHOTS12345678910ACCURACY
VANILLA PROTONET148.89%56.54%60.31%63.12%64.70%66.02%66.62%67.99%68.41%68.90%63.15±0.21%
EST PROTONET149.07%56.49%60.62%62.67%64.83%66.23%66.84%67.90%67.68%68.73%63.11±0.22%
PCA PROTONET149.01%56.35%60.07%62.42%64.38%65.28%66.56%67.81%67.59%68.26%62.77 ±0.22%
VANILLA PROTONET544.75%56.61%61.52%65.32%67.23%69.04%70.66%71.47%71.84%72.36%65.08±0.23%
EST PROTONET550.22%59.04 %64.14%66.61%68.25%69.46%70.80%71.60%72.61%73.29%66.60±0.23%
PCA PROTONET48.72%58.43%63.17%66.07%68.63%69.56%70.55%71.21%72.09%72.82%66.12±0.24%
VANILLA PROTONET1039.99%52.73%59.71%63.41%66.23%68.27%69.86%71.03%71.72%72.47%63.54±0.25%
EST PROTONET1048.98%57.83%63.13%66.39%68.12%69.82%70.63%71.85%72.79%73.22%66.28±0.23%
PCA PROTONET1048.04%57.05%62.46%64.63%67.61%68.98%69.71%71.78%71.75%72.42%65.44±0.24%
VANILLA PROTONET1-1049.36%58.67%62.77 %65.76%67.96%69.20%70.05%70.90%71.34%72.27%65.83±0.21%
", + "bbox": [ + 173, + 696, + 825, + 805 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/0a8ddca89f408f726d27585899c5c5c84a7e9af46d7419e3f80969f00e34ec2d.jpg", + "image_caption": [ + "Figure 4: Comparison between estimated accuracy lower bound and empirical accuracy for various training and test shots. Experiment is 2-way few-shot classification performed on miniImageNet. " + ], + "image_footnote": [], + "bbox": [ + 256, + 146, + 761, + 328 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/f2b6a0cbbd52320aca1bd455570ea4ba1863b6b9be9b531f0b71ca9c078770bb.jpg", + "table_caption": [ + "Table 4: Classification Accuracy on tieredImageNet-5-way, with 4 layer CNN embedding network. " + ], + "table_footnote": [], + "table_body": "
MODELTRAINING SHOTS245TESTING SHOTS678910AVERAGE ACCURACY
1361.85±0.29%
VANILLA PROTONET EST PROTONET1 147.37% 47.71%55.72% 55.05%59.27% 59.31%61.50% 61.97%63.70% 63.48%64.28% 64.65%65.01% 65.35%66.43% 66.33%67.20% 66.42%67.99% 67.44%61.77±0.31%
PCA PROTONET147.72%54.60%58.69%60.73%62.78%64.92%65.88%65.88%66.11%66.87%61.42±0.32%
VANILLA PROTONET
EST PROTONET5542.33% 48.85%55.06%61.32%64.57%66.51%67.86%69.33%70.15%71.32% 70.87%72.05% 72.09%64.05±0.33% 65.46±0.32%
PCA PROTONET548.34%58.38% 57.44%62.75% 0.79%65.16% 64.96%67.24% 67.07%68.39% 67.93%69.89% 69.08%70.99% 70.40%70.38%71.65%64.96±0.31%
VANILLA PROTONET
EST PROTONET1035.38%49.40%56.76%61.25%64.56%66.56%68.05%69.31%70.12%71.03%61.24±0.36% 65.26±0.32%
PCA PROTONET10 1047.33% 46.55%56.75% 56.61%62.80% 60.81%65.78% 64.01%66.84% 66.53%69.07% 67.70%69.98% 69.18%70.82% 69.83%71.99% 70.62%71.20% 71.22%64.31 ±0.33%
VANILLA PROTONET1-1047.65%56.23%62.12 %63.93%66.34%67.94%68.44%68.93%70.80%70.96%64.33±0.30%
", + "bbox": [ + 173, + 458, + 823, + 566 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/aca3931cf520734720dd6e8343fa6c7d5f9a8bc06d60ce57e0249a23d3e61dc8.jpg", + "table_caption": [ + "Table 5: Classification Accuracy on Omniglot-20-way, with 4 layer CNN embedding network. " + ], + "table_footnote": [], + "table_body": "
MODELTRAINING SHOTS 1TESTING SHOTS
2345AVERAGE ACCURACY
VANILLA PROTONET195.07 ±0.17 %97.89 ±0.09 %98.45 ±0.08 %98.75 ±0.06 %98.89 ±0.06%97.81±0.06 %
VANILLA PROTONET294.59 ±0.18 %97.69 ±0.09 %98.44 ±0.07 %98.69 ±0.06 %98.89 ±0.06 %97.66 ±0.07 %
VANILLA PROTONET394.19 ±0.18 %97.57 ±0.09 %98.30±0.07 %98.63±0.07 %98.79 ±0.06 %97.50 ±0.07 %
VANILLA PROTONET493.79 ±0.18 %97.41 ±0.10 %98.19 ±0.08 %98.54 ±0.07 %98.75±0.06%97.34 ±0.07 %
VANILLA PROTONET593.42 ±0.18 %97.34±0.10 %98.18 ±0.07 %98.53 ±0.07 %98.78 ±0.05 %97.25 ±0.07 %
MIXED-k SHOT1-594.84±0.17 %97.81 ±0.09 %98.45±0.07 %98.70 ±0.06 %98.92 ±0.54 %97.74 ±0.06 %
PCA PROTONET194.94 ±0.16 %97.81 ±0.09 %98.53±0.07 %98.79 ±0.06 %98.85±0.06%97.78 ±0.06 %
EST PROTONET195.11 ±0.17 %97.95 ±0.09 %98.46±0.07 %98.77 ±0.06 %98.84±0.06%97.83 ±0.06 %
", + "bbox": [ + 174, + 637, + 823, + 727 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/7c1b116977a4efe894c71e5d3ffa53a75a5887e1bb3a1cefec5e1fcc6181a6bd.jpg", + "table_caption": [ + "Table 6: Classification Accuracy on miniImageNet-5-way, with 7 layer ResNet embedding network. " + ], + "table_footnote": [], + "table_body": "
MODELTRAINING SHOTS2TESTING SHOTS 3 4AVERAGEACCURACY
5678
VANILLA PROTONET152.65%60.46%64.18%66.83%68.27%69.23%70.19%71.37%71.83%72.29%66.73±0.20%
EST PROTONET152.56%60.63%64.50%66.53%68.33%69.36%70.04%71.21%71.37%71.60%66.61±0.22%
PCA PROTONET152.78%60.35%64.05%66.24%67.51%69.64%69.85%71.13%71.88%71.79%66.52±0.22%
VANILLA PROTONET547.40%58.23%64.60%67.52%69.93%71.05%72.27%72.90%73.55%74.35%67.18±0.23%
EST PROTONET551.93%60.70%65.33%68.06%69.98%71.26%72.33%73.46%74.03%74.80%68.19±0.23 %
PCA PROTONET550.90%60.38%64.66%67.23%69.25%71.30%71.72%72.75%73.84%74.24%67.63±0.23%
VANILLA PROTONET68.23%
EST PROTONET10 1042.20% 51.24%55.98% 60.46%61.67%66.08%70.06%69.95% 71.07%72.20% 72.55%72.56% 73.39%74.04% 73.85%74.54 % 74.69%65.75±0.25% 68.00±0.23%
PCAPROTONET1050.52%59.20%65.11% 64.44%67.63% 66.86%69.36%71.00%71.73%73.19%73.73%73.82%67.38±0.23%
VANILLA PROTONET1-1051.74%60.10%65.13%67.12%69.09%70.53%71.57%72.22%72.99%73.63%67.41±0.21%
", + "bbox": [ + 173, + 799, + 825, + 909 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/75d35454e1d3c09bdeb52bc01f70dca9e91d574e5c01f3674294bb0281498677.jpg", + "table_caption": [ + "Table 7: Classification Accuracy on tieredImageNet-5-way, with 7 layer ResNet embedding network. " + ], + "table_footnote": [], + "table_body": "
MODELTRAININGTESTING SHOTSAVERAGE
SHOTS12345678910ACCURACY
VANILLA PROTONET149.78%57.67%61.37%64.88%65.17%66.73%67.53%68.58%68.21%69.88%63.98±0.29%
EST PROTONET151.19%57.61%61.85%64.43%65.48%67.64%67.69%67.76%68.51%68.20%64.04±0.30%
PCA PROTONET151.38%57.89%61.68%64.32%65.96%66.15%66.83%68.00%68.58%68.77%63.95±0.30%
VANILLA PROTONET547.88%58.70%63.95%66.58%69.12%71.69%71.95%73.15%73.11%73.80%66.99 ±0.31%
EST PROTONET553.05%61.13 %65.67%68.07%69.30%71.47%71.59 %72.42%72.80%73.63%67.91 ±0.31%
PCA PROTONET51.21%59.86%63.91%66.92%68.88%70.38%71.48%72.77%73.81%72.17%67.14±0.32%
VANILLA PROTONET1040.84%56.24%62.10%66.28%69.37%71.43%72.09%72.91%73.56%74.72%65.95±0.35%
EST PROTONET1050.45%60.41%65.19%68.46%69.55%70.87%72.07%72.66%73.78%74.79%67.82 ±0.32%
PCA PROTONET1050.18%59.59%64.24%67.43%69.52%70.65%71.78%72.10%72.78%73.65%67.19±0.31%
VANILLA PROTONET1-1050.74%60.03%64.17 %67.26%69.28%69.56%71.07%72.41%71.99%73.01%66.95 ±0.29 %
", + "bbox": [ + 173, + 266, + 825, + 375 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/dde202f76915f056c670a6300cbe14707dc3ac8651f013905333d38c608e9427.jpg", + "table_caption": [ + "Table 8: Classification Accuracy on Omniglot-20-way, with 7 layer ResNet embedding network. " + ], + "table_footnote": [], + "table_body": "
TESTING SHOTS
MODELTRAINING SHOTS12345AVERAGE ACCURACY
VANILLA PROTONET196.46%98.39 %98.82%99.01 %99.07%98.35 ±0.05 %
VANILLA PROTONET295.85%98.32%98.80%98.95%99.07%98.20±0.05 %
VANILLA PROTONET395.35 %98.15%98.73 %98.91 %99.03%98.03 ±0.06%
VANILLA PROTONET495.00%98.05%98.62 %98.90%98.99%97.91 ±0.06 %
VANILLA PROTONET594.42 %97.98 %98.60%98.77 %98.99%97.75 ±0.06 %
VANILLA PROTONET1-596.53%98.53%98.90%99.06%99.15%98.43 ±0.05 %
EST PROTONET196.18%98.23 %98.68%98.87%98.99%98.19 ±0.05 %
PCA PROTONET196.02%98.22 %98.76%98.93%99.02 %98.19 ±0.05 %
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MODELTRAININGTESTING SHOTSAVERAGE
SHOTS12345678910ACCURACY
VANILLA PROTONET148.89%56.54%60.31%63.12%64.70%66.02%66.62%67.99%68.41%68.90%63.15±0.21%
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EST PROTONET1048.98%57.83%63.13%66.39%68.12%69.82%70.63%71.85%72.79%73.22%66.28±0.23%
PCA PROTONET1048.04%57.05%62.46%64.63%67.61%68.98%69.71%71.78%71.75%72.42%65.44±0.24%
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TRAINING SHOTS2 310AVERAGE ACCURACY
45TESTING SHOTS 6789
PROTONET + FC544.77%53.75%58.04%61.06%62.26%64.60%65.19%66.63%66.65%67.52%61.05±0.28%
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MODELTRAINING SHOTS245TESTING SHOTS678910AVERAGE ACCURACY
1361.85±0.29%
VANILLA PROTONET EST PROTONET1 147.37% 47.71%55.72% 55.05%59.27% 59.31%61.50% 61.97%63.70% 63.48%64.28% 64.65%65.01% 65.35%66.43% 66.33%67.20% 66.42%67.99% 67.44%61.77±0.31%
PCA PROTONET147.72%54.60%58.69%60.73%62.78%64.92%65.88%65.88%66.11%66.87%61.42±0.32%
VANILLA PROTONET
EST PROTONET5542.33% 48.85%55.06%61.32%64.57%66.51%67.86%69.33%70.15%71.32% 70.87%72.05% 72.09%64.05±0.33% 65.46±0.32%
PCA PROTONET548.34%58.38% 57.44%62.75% 0.79%65.16% 64.96%67.24% 67.07%68.39% 67.93%69.89% 69.08%70.99% 70.40%70.38%71.65%64.96±0.31%
VANILLA PROTONET
EST PROTONET1035.38%49.40%56.76%61.25%64.56%66.56%68.05%69.31%70.12%71.03%61.24±0.36% 65.26±0.32%
PCA PROTONET10 1047.33% 46.55%56.75% 56.61%62.80% 60.81%65.78% 64.01%66.84% 66.53%69.07% 67.70%69.98% 69.18%70.82% 69.83%71.99% 70.62%71.20% 71.22%64.31 ±0.33%
VANILLA PROTONET1-1047.65%56.23%62.12 %63.93%66.34%67.94%68.44%68.93%70.80%70.96%64.33±0.30%
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MODELTRAINING SHOTS2TESTING SHOTS 3 4AVERAGEACCURACY
5678
VANILLA PROTONET152.65%60.46%64.18%66.83%68.27%69.23%70.19%71.37%71.83%72.29%66.73±0.20%
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VANILLA PROTONET547.40%58.23%64.60%67.52%69.93%71.05%72.27%72.90%73.55%74.35%67.18±0.23%
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PCA PROTONET550.90%60.38%64.66%67.23%69.25%71.30%71.72%72.75%73.84%74.24%67.63±0.23%
VANILLA PROTONET68.23%
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PCAPROTONET1050.52%59.20%65.11% 64.44%67.63% 66.86%69.36%71.00%71.73%73.19%73.73%73.82%67.38±0.23%
VANILLA PROTONET1-1051.74%60.10%65.13%67.12%69.09%70.53%71.57%72.22%72.99%73.63%67.41±0.21%
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MODELTRAINING SHOTS 1TESTING SHOTS
2345AVERAGE ACCURACY
VANILLA PROTONET195.07 ±0.17 %97.89 ±0.09 %98.45 ±0.08 %98.75 ±0.06 %98.89 ±0.06%97.81±0.06 %
VANILLA PROTONET294.59 ±0.18 %97.69 ±0.09 %98.44 ±0.07 %98.69 ±0.06 %98.89 ±0.06 %97.66 ±0.07 %
VANILLA PROTONET394.19 ±0.18 %97.57 ±0.09 %98.30±0.07 %98.63±0.07 %98.79 ±0.06 %97.50 ±0.07 %
VANILLA PROTONET493.79 ±0.18 %97.41 ±0.10 %98.19 ±0.08 %98.54 ±0.07 %98.75±0.06%97.34 ±0.07 %
VANILLA PROTONET593.42 ±0.18 %97.34±0.10 %98.18 ±0.07 %98.53 ±0.07 %98.78 ±0.05 %97.25 ±0.07 %
MIXED-k SHOT1-594.84±0.17 %97.81 ±0.09 %98.45±0.07 %98.70 ±0.06 %98.92 ±0.54 %97.74 ±0.06 %
PCA PROTONET194.94 ±0.16 %97.81 ±0.09 %98.53±0.07 %98.79 ±0.06 %98.85±0.06%97.78 ±0.06 %
EST PROTONET195.11 ±0.17 %97.95 ±0.09 %98.46±0.07 %98.77 ±0.06 %98.84±0.06%97.83 ±0.06 %
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TESTING SHOTS
MODELTRAINING SHOTS12345AVERAGE ACCURACY
VANILLA PROTONET196.46%98.39 %98.82%99.01 %99.07%98.35 ±0.05 %
VANILLA PROTONET295.85%98.32%98.80%98.95%99.07%98.20±0.05 %
VANILLA PROTONET395.35 %98.15%98.73 %98.91 %99.03%98.03 ±0.06%
VANILLA PROTONET495.00%98.05%98.62 %98.90%98.99%97.91 ±0.06 %
VANILLA PROTONET594.42 %97.98 %98.60%98.77 %98.99%97.75 ±0.06 %
VANILLA PROTONET1-596.53%98.53%98.90%99.06%99.15%98.43 ±0.05 %
EST PROTONET196.18%98.23 %98.68%98.87%98.99%98.19 ±0.05 %
PCA PROTONET196.02%98.22 %98.76%98.93%99.02 %98.19 ±0.05 %
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MODELTRAININGTESTING SHOTSAVERAGE
SHOTS12345678910ACCURACY
VANILLA PROTONET149.78%57.67%61.37%64.88%65.17%66.73%67.53%68.58%68.21%69.88%63.98±0.29%
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PCA PROTONET151.38%57.89%61.68%64.32%65.96%66.15%66.83%68.00%68.58%68.77%63.95±0.30%
VANILLA PROTONET547.88%58.70%63.95%66.58%69.12%71.69%71.95%73.15%73.11%73.80%66.99 ±0.31%
EST PROTONET553.05%61.13 %65.67%68.07%69.30%71.47%71.59 %72.42%72.80%73.63%67.91 ±0.31%
PCA PROTONET51.21%59.86%63.91%66.92%68.88%70.38%71.48%72.77%73.81%72.17%67.14±0.32%
VANILLA PROTONET1040.84%56.24%62.10%66.28%69.37%71.43%72.09%72.91%73.56%74.72%65.95±0.35%
EST PROTONET1050.45%60.41%65.19%68.46%69.55%70.87%72.07%72.66%73.78%74.79%67.82 ±0.32%
PCA PROTONET1050.18%59.59%64.24%67.43%69.52%70.65%71.78%72.10%72.78%73.65%67.19±0.31%
VANILLA PROTONET1-1050.74%60.03%64.17 %67.26%69.28%69.56%71.07%72.41%71.99%73.01%66.95 ±0.29 %
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b/parse/train/L7Irrt5sMQa/L7Irrt5sMQa.md new file mode 100644 index 0000000000000000000000000000000000000000..fa0d14ab828d8d94ad66c0789fbc87f5ac4282ec --- /dev/null +++ b/parse/train/L7Irrt5sMQa/L7Irrt5sMQa.md @@ -0,0 +1,537 @@ +# THE SURPRISING POWER OF GRAPH NEURAL NETWORKS WITH RANDOM NODE INITIALIZATION + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Graph neural networks (GNNs) are effective models for representation learning on graph-structured data. However, standard GNNs are limited in their expressive power, as they cannot distinguish graphs beyond the capability of the WeisfeilerLeman (1-WL) graph isomorphism heuristic. This limitation motivated a large body of work, including higher-order GNNs, which are provably more powerful models. To date, higher-order invariant and equivariant networks are the only models with known universality results, but these results are practically hindered by prohibitive computational complexity. Thus, despite their limitations, standard GNNs are commonly used, due to their strong practical performance. In practice, GNNs have shown a promising performance when enhanced with random node initialization (RNI), where the idea is to train and run the models with randomized initial node features. In this paper, we analyze the expressive power of GNNs with RNI, and pose the following question: are GNNs with RNI more expressive than GNNs? We prove that this is indeed the case, by showing that GNNs with RNI are universal, a first such result for GNNs not relying on computationally demanding higher-order properties. We then empirically analyze the effect of RNI on GNNs, based on carefully constructed datasets. Our empirical findings support the superior performance of GNNs with RNI over standard GNNs. In fact, we demonstrate that the performance of GNNs with RNI is often comparable with or better than that of higher-order GNNs, while keeping the much lower memory requirements of standard GNNs. However, this improvement typically comes at the cost of slower model convergence. Somewhat surprisingly, we found that the convergence rate and the accuracy of the models can be improved by using only a partial random initialization regime. + +# 1 INTRODUCTION + +Graph neural networks (GNNs) (Scarselli et al., 2009; Gori et al., 2005) are neural architectures designed for learning functions over graph-structured data, and naturally encode desirable properties such as permutation invariance (resp., equivariance) relative to graph nodes, and node-level computation based on message passing between these nodes. These properties provide GNNs with a strong inductive bias, enabling them to effectively learn and combine both local and global graph features (Battaglia et al., 2018). As a result, GNNs have been applied to a multitude of tasks, ranging from protein classification (Gilmer et al., 2017) and synthesis (You et al., 2018), protein-protein interaction (Fout et al., 2017), and social network analysis (Hamilton et al., 2017), to recommender systems (Ying et al., 2018) and combinatorial optimization (Bengio et al., 2018; Selsam et al., 2019). + +However, popular GNN architectures, primarily based on message passing (MPNNs), are limited in their expressive power. In particular, MPNNs are at most as powerful as the Weisfeiler-Leman (1-WL) graph isomorphism heuristic (Morris et al., 2019; Xu et al., 2019), and thus cannot discern between several families of non-isomorphic graphs, e.g., sets of regular graphs (Cai et al., 1992). To address this limitation, alternative GNN architectures with provably higher expressive power than MPNNs have been proposed. These models, which we refer to as higher-order GNNs, are inspired by the more powerful generalization of 1-WL to $k$ −tuples of nodes, known as $k$ -WL (Grohe, 2017). These models are the only GNNs with an established universality result, but these models are computationally very demanding. As a result, MPNNs, despite their limited expressiveness, remain the standard GNN model for graph learning applications. + +In a parallel development, MPNNs have recently achieved significant empirical improvements using random node initialization (RNI), through which initial graph node embeddings are randomly set. Indeed, RNI has enabled MPNNs to distinguish instances that 1-WL cannot distinguish, and is proven to enable better approximation of a class of combinatorial problems (Sato et al., 2020). However, the effect of RNI on the expressive power of GNNs has not yet been comprehensively studied, and its impact on the inductive capacity and learning ability of GNNs remains unclear. + +In this paper, we thoroughly study the impact of RNI on MPNNs. First, we prove that MPNNs enhanced with RNI are universal, in the sense that they can approximate every function defined on graphs of any fixed order. This follows from a logical characterisation of the expressiveness of MPNNs (Barcelo et al., 2020) combined with an argument on order-invariant definability. Our ´ result strongly contrasts with existing 1-WL limitations for deterministic MPNNs, and provides a foundation for developing very expressive and memory-efficient MPNN models. + +To empirically verify our theoretical findings, we carry out a careful empirical study to quantify the practical impact of RNI. To this end, we design EXP, a synthetic dataset requiring 2-WL expressive power for models to achieve above-random performance, and run MPNNs with RNI on it, to observe how well and how easily this model can learn and generalize based on this dataset. Then, we propose CEXP, a modification of EXP with partially 1-WL distinguishable data, and evaluate the same questions in this more variable setting. Overall, the contributions of this paper are as follows: + +- We prove that MPNNs with RNI are universal, a significant improvement over the 1-WL limit of standard MPNNs and, to our knowledge, a first universality result for memory-efficient GNNs. +- We introduce two carefully designed datasets, EXP and CEXP, based on graph pairs only distinguishable by 2-WL or higher, to rigorously evaluate the impact of RNI. +- Using these datasets, we thoroughly analyze the effects of RNI on MPNN, and observe that (i) MPNNs with RNI can closely match the performance of higher-order GNNs, (ii) the improved performance of MPNNs with RNI comes at the cost of slower convergence (compared to higherorder GNNs), and (iii) using a partial random initialization regime over node features typically improves convergence rate and the accuracy of the models. +- We additionally perform the same experiments with analog, sparser datasets, with longer training, and observe similar behavior, but more volatility. + +# 2 GRAPH NEURAL NETWORKS + +Graph neural networks (GNNs) (Gori et al., 2005; Scarselli et al., 2009) are neural architectures dedicated to learning functions over graph-structured data. In a GNN, nodes in the input graph are assigned vector representations, which are updated iteratively through series of invariant or equivariant computational layers. We recall message passing neural networks (MPNNs) (Gilmer et al., 2017), a popular family of GNN models, and its expressive power in relation to the WeisfeilerLeman graph isomorphism heuristic. We discuss alternative GNN models in Section 3; for a broader coverage, we refer the reader to the literature (Hamilton, 2020). + +In MPNNs, node representations aggregate messages from their neighboring nodes, and use this information to iteratively update their representations. Formally, given a node $x$ , its vector representation $v _ { x , t }$ at time $t$ , and its neighborhood $N ( x )$ , a message passing update can be written as: + +$$ +v _ { x , t + 1 } = c o m b i n e { \Big ( } v _ { x , t } , a g g r e g a t e { \big ( } \{ v _ { y , t } | y \in N ( x ) \} { \big ) } { \Big ) } , +$$ + +where combine and aggregate are functions, and aggregate is typically permutation-invariant. Once message passing is complete, the final node representations are then used to compute target outputs. Prominent message passing GNN architectures include graph convolutional networks (GCNs) (Kipf & Welling, 2017) and gated graph neural networks (GGNNs) (Li et al., 2016). + +It is well-known that standard MPNNs have the same power as the 1-dimensional Weisfeiler-Leman algorithm (1-WL) (Xu et al., 2019; Morris et al., 2019). This entails that two nodes in a graph cannot be distinguished if 1-WL does not distinguish them, and neither can two graphs be distinguished if 1-WL cannot distinguish them. + +Consider the graphs $G$ and $H$ shown in Figure 1. 1-WL cannot distinguish any two nodes of the graph $G$ . Thus, for example, the invariant function $f : V ( G ) \to \mathbb { R }$ that maps all nodes in the 4-cycle of $G$ to 1 and all nodes in the triangle to 0 is not expressible (or approximable) by a GNN. Moreover, 1-WL cannot distinguish the graphs $G , H$ , even though they are obviously non-isomorphic, so no classifier based on the node embeddings computed by an MPNN can distinguish the two graphs. + +![](images/aa77d9397242894aa279436cb0ba0d6163929d466561f5dcda8966db0ef5a676.jpg) +Figure 1: $G$ and $H$ are indistinguishable by 1-WL and hence by (1-WL) GNNs. + +A somewhat trivial limitation in the expressiveness of MPNNs is that information is only propagated along edges, and hence can never be shared between distinct connected components of a graph (Barcelo et al., 2020; Xu et al., 2019). An easy way to overcome this limitation is by adding ´ global readouts, that is, permutation-invariant functions that aggregate the current states of all nodes1. Throughout the paper, we therefore focus on MPNNs with global readouts (also called aggregatecombine GNNs with global readout, i.e., ACR-GNNs (Barcelo et al., 2020)). ´ + +# 3 RELATED WORK & MOTIVATION + +Developing more expressive GNNs is an active research area due to the prominence of GNNs for relational learning (Hamilton et al., 2017) and combinatorial optimization (Bengio et al., 2018). As mentioned earlier, standard GNN models are at most as expressive as 1-WL (Morris et al., 2019; Xu et al., 2019), and thus cannot distinguish between non-isomorphic input instances. In this section, we describe theoretical results quantifying the expressive power of existing GNNs. + +Higher-order GNNs. We recall the following families of higher-order GNN models: + +- Invariant (resp., equivariant) graph networks: Invariant (resp., equivariant) graph networks (Maron et al., 2019b) represent graphs as a tensor where node adjacency is directly encoded, and implicitly pass information between nodes through invariant (resp., equivariant) computational blocks. Hence, these models are themselves invariant (resp., equivariant), a desirable property for computations on graphs. Following intermediate blocks, higher-order tensors are typically returned, and the order of these tensors correlates directly with the expressive power of the overall model. Indeed, invariant networks (Maron et al., 2019c), and later equivariant networks (Keriven & Peyre, 2019), are shown to be universal, but with tensor orders of ´ $\dot { O } ( | V | ^ { 2 } )$ , where $| V |$ denotes the number of graph nodes. Furthermore, invariant (resp., equivariant) networks with intermediate tensor order $k$ are shown to be equivalent in power to $\left( k - 1 \right)$ -WL (Maron et al., 2019a), which is strictly more expressive as $k$ increases (Cai et al., 1992). Therefore, such universal higher-order models require intractably-sized intermediate tensors in practice. + +- Higher-order MPNNs: The $k { \mathrm { - } } \mathbf { W } \mathbf { L }$ hierarchy has been directly emulated in GNNs, such that these models learn embeddings for tuples of nodes, and perform messaging passing between them, as opposed to individual nodes. This approach has yielded models such as $k$ -GNNs (Morris et al., 2019). $k$ -GNNs have $\left( k - 1 \right)$ -WL expressive power,2 but need $O ( | V | ^ { k } )$ memory to run, leading to excessive memory requirements. + +- Provably powerful graph networks (PPGNs): PPGN is an invariant GNN (Maron et al., 2019a), based on “blocks” of multilayer perceptrons (MLPs) and matrix multiplication, which theoretically has 2-WL expressive power, and only requires memory $O ( | V | ^ { 2 } )$ (compared to $O ( | V | ^ { 3 } )$ for 3-GNNs). However, PPGN theoretically requires exponentially many samples in the number of graph nodes to learn necessary functions for 2-WL expressiveness (Puny et al., 2020). + +GNNs with random node initialization. MPNNs have been enhanced with random node initialization (Sato et al., 2020), such that the model trains and runs with partially randomized initial node features. These models, denoted rGNNs, are shown to near-optimally approximate solutions to specific combinatorial optimization problems, and are able to distinguish between 1-WL indistinguishable graph pairs. rGNNs can also detect characteristic sub-graphs in an input graph with high probability. Nonetheless, it remains open as to how much expressive power is exactly gained through RNI, and, in general, whether a GNN model that is universal, scalable, and structure-preserving, can be developed. Our work strongly improves the theoretical result of Sato et al. (2020), as it shows universality of MPNNs with RNI, and thus that arbitrary real-valued functions over graphs can be learned by MPNNs with the help of RNI. On the empirical side, we highlight the power of RNI in a significantly more challenging setting than rGNN, using a target function (SAT) beyond their theoretical scope. Indeed, for SAT, approximation is known to be hard, and fixed local structures are not useful for prediction. + +Similar work to RNI has also been conducted in terms of randomly adding features from a predetermined set of colors (Dasoulas et al., 2020) to disambiguate between nodes. This model, known as CLIP, is similar in spirit to RNI, in that it introduces randomness to node representations, but explicitly makes graphs distinguishable by construction. By contrast, we study random features produced by RNI, which (i) are not designed a priori to distinguish nodes, (ii) do not explicitly introduce a fixed underlying structure, and (iii) yield potentially infinitely many representations for a single graph. In this more general setting, we nonetheless show that RNI adds expressive power to distinguish between nodes with high probability, leads to a universality result, and performs strongly in challenging problem settings. + +# 4 RANDOM NODE INITIALIZATION MAKES GNNS UNIVERSAL + +We present the main result of the paper, showing that random node initialization significantly increases the expressiveness and makes MPNNs universal, in a natural sense. Our work is a first positive result for the universality of MPNNs. This result is not based on a new model, but rather on random initialization of node features, which is widely used in practice, and in this respect, it also serves as a theoretical justification for models that are successfully employed in practice. + +It may appear somewhat surprising, and even counter-intuitive, that randomly initializing node features, on its own, would deliver such a gain in expressiveness. In fact, on the surface, random initialization no longer preserves the invariance of MPNNs, since the result of the computation of an MPNN with RNI not only depends on the structure (i.e., the isomorphism type) of the input graph, but also on the random initialization. The broader picture is, however, rather subtle, as we can view such a model as computing a random variable (or as generating an output distribution), and this random variable would still be invariant. This means that the outcome of the computation of an MPNN with RNI does still not depend on the specific representation of the input graph, which fundamentally maintains invariance. Indeed, random features vary around a mean which, in expectation, will inform GNN predictions, and is identical across all nodes as randomization is i.i.d. However, the variability between different samples, and the variability of a random sample relative to this mean, enable graph discrimination and improve expressiveness. Hence, in expectation, all samples over training and evaluation fluctuate around a unique value, preserving invariance, whereas single-sample variance achieves the improved expressiveness. + +Formally, let ${ \mathcal { G } } _ { n }$ be the class of all $n$ -vertex graphs, i.e., graphs that consist of at most $n$ vertices, and let $f : { \mathcal { G } } _ { n } \to \mathbb { R }$ . We say that $f$ is invariant if for isomorphic graphs $G , H \ \in \ G _ { n }$ it holds that $f ( G ) = f ( H )$ . We say that a randomized function $\mathcal { X }$ that associates with every graph $G \in \mathcal G _ { n }$ a random variable $\mathcal { X } ( G )$ is an $( \epsilon , \delta )$ -approximation of $f$ if for all $G \in \mathcal G _ { n }$ it holds that $\operatorname* { P r } \left( | f ( G ) - \mathcal { X } ( G ) | \leq \epsilon \right) \geq 1 - \delta$ . Note that MPNNs with RNI compute functions $\mathcal { X }$ of this type. If $\mathcal { X }$ is computed by an MPNN $\mathcal { N }$ with RNI, we say that $\mathcal { N } \left( \epsilon , \delta \right)$ -approximates $f$ . + +Theorem 4.1 (Universal approximation). Let $n \geq 1$ , and let $f : { \mathcal { G } } _ { n } \to \mathbb { R }$ be invariant. Then, for all $\epsilon , \delta > 0$ , there is a MPNN with RNI that $( \epsilon , \delta )$ -approximates $f$ . + +For ease of presentation, we state the theorem only for real-valued functions, but it can be extended to equivariant functions defined on the nodes of a graph (that is, functions $f$ mapping graphs $G$ to vectors $\pmb { x } \in \mathbb { R } ^ { V ( G ) }$ with the property that for every permutation $\pi$ of $V ( G )$ it holds that $f ( G ^ { \pi } ) = f ( G ) ^ { \pi } )$ and even higher-order equivariant functions. The result can also be extended to weighted graphs, but then the function $f$ that we approximate needs to be continuous. + +To prove Theorem 4.1, we first show that MPNNs with RNI can capture arbitrary Boolean functions, by building on the result of Barcelo et al. (2020), which states that any logical sentence in ´ $\mathsf { C } ^ { 2 }$ can be captured by an MPNN with global readout. The logic $\complement$ is the extension of first-order predicate logic using counting quantifiers of the form $\exists ^ { \geq k } x$ for $k \geq 0$ , where $\exists ^ { \geq k } x \varphi ( x )$ means that there are at least $k$ elements $x$ satisfying $\varphi$ , and $\mathsf { C } ^ { 2 }$ is the two-variable fragment of $\mathrm { c }$ (see appendix for further details). We establish that any graph with identifying node features, which we call individualized graphs, can be represented by a sentence in $\mathsf { C } ^ { 2 }$ . Then, we extend this result to sets of individualized graphs, and thus to Boolean functions mapping these sets to True, by showing that these functions are represented by a $\mathsf { C } ^ { 2 }$ sentence, namely the disjunction of all constituent graph sentences. Following this, we provide a construction with node embeddings based on random node initialization, and show that, with high probability, RNI makes the input graph individualized. Thus, with high probability, RNI makes that MPNNs learn a Boolean function over individualized graphs. Since all such functions can be captured by a sentence in $\mathsf { C } ^ { 2 }$ , and an MPNN with a global readout can capture any Boolean function by Barcelo et al. (2020), we conclude that MPNNs with RNI can capture arbitrary ´ Boolean functions. Finally, the result is extended to real-valued functions via a natural mapping, yielding universality. + +The concrete implications of Theorem 4.1 can be summarized as follows: First, MPNNs enhanced with RNI are able to distinguish individual graphs, already with an embedding dimensionality polynomial in the inverse of desired confidence $\delta$ , namely $O ( n ^ { 2 } \delta ^ { - 1 } )$ , where $n$ is the number of graph nodes. Second, our universality results holds also with partial RNI. More specifically, it already holds with only one randomized dimension. Third, although Theorem 4.1 can potentially result in very large constructions, we note that it is very adaptive and tightly linked to the descriptive complexity of the function that we want to approximate. That is, for a more restricted class of functions, there may be more efficient constructions, and our proof does not rely on a particular construction. This deserves a more thorough investigation, which we leave for future work. Finally, our construction provides a logical characterization of the power of MPNNs with RNI, and substantiates the means through which randomization yields expressiveness improvements. This construction therefore also serves as a basis for a more logically grounded theoretical study of randomized MPNN models, based on particular architectural or parametric choices. + +Similarly to other universality results, Theorem 4.1 can potentially result in very large constructions. This is a simple consequence of the general nature of such universality results: Theorem 4.1 applies to families of functions, describing problems of arbitrary computational complexity, including problems that are computationally hard (even to approximate). Thus, a practically more relevant aspect is to empirically verify the formal statement, and test the capacity of MPNNs with RNI, in comparison to higher-order GNNs. Higher-order GNNs typically suffer from prohibitive space requirements, which is not the case for MPNNs with RNI, and this already makes them more viable in practice. As we discuss later in detail, our experiments demonstrate that MPNNs with RNI indeed combine expressiveness with efficiency in practice. + +# 5 DATASETS FOR EVALUATING THE EXPRESSIVE POWER OF GNNS + +GNN models are evaluated on prominent real-world datasets, such as IMDB, TU, and Proteins (Kersting et al., 2016). These datasets are not tailored for evaluating the expressive power of GNNs, as they do not contain instances or edge cases requiring expressiveness beyond 1-WL. In fact, higherorder models only marginally outperform MPNNs on these datasets (Maron et al., 2019a; Dwivedi et al., 2020), which further highlights their unsuitability for expressiveness evaluation. + +We develop the datasets EXP and CEXP. EXP is designed to explicitly evaluate the expressiveness of GNN models, and consists of a set of graph instances $\left\{ { G _ { 1 } \dots { \bar { G } } _ { n } , \dot { H } _ { 1 } \dots { \cal H } _ { n } } \right\}$ , such that each instance is a graph encoding of a propositional formula. The classification task is to determine whether the formula is satisfiable (SAT). Each pair $( G _ { i } , H _ { i } )$ respects the following properties: (i) $G _ { i }$ and $H _ { i }$ are non-isomorphic, (ii) $G _ { i }$ and $H _ { i }$ have different SAT outcomes, that is, $G _ { i }$ encodes a satisfiable formula, while $H _ { i }$ encodes an unsatisfiable formula, (iii) $G _ { i }$ and $H _ { i }$ are 1-WL indistinguishable, so are guaranteed to be classified in the same way by standard MPNNs, and (iv) $G _ { i }$ and $H _ { i }$ are 2-WL distinguishable, so can be classified differently by higher-order GNNs. Thanks to these properties, we can explicitly compare the performance of MPNNs with RNI to these higher-order models. + +Ensuring these properties in EXP is highly non-trivial, and the construction of this dataset is cumbersome. Fundamentally, every $( G _ { i } , H _ { i } )$ is carefully constructed on top of a basic building block, the core pair, such that the 2 cores underlie $G _ { i }$ and $H _ { i }$ , respectively. In this core pair, both cores are based on propositional clauses, such that one core is satisfiable and the other is not, and that these cores exclusively determine the satisfiability of $G _ { i }$ (resp., $H _ { i }$ ) and have graph encodings enabling all aforementioned properties. Core pairs, and their resulting graph instances in EXP are planar and are also carefully constrained to ensure they are 2-WL distinguishable. Hence, core pairs are key substructures within EXP, and distinguishing these cores is essential for good performance. + +Building on EXP, CEXP includes instances with varying expressiveness requirements. Specifically, CEXP is a standard EXP dataset where $50 \%$ of all satisfiable graph pairs are modified, such that they become 1-WL distinguishable from their unsatisfiable counterparts, only differing from these by a small number of added edges. Hence, CEXP consists of $50 \%$ “corrupted” data, which can be distinguished and learned by a standard MPNN model (1-WL), which we label CORRUPT, and $50 \%$ unmodified data, generated analogously to EXP, and requiring expressive power beyond 1- WL, which we refer to as $\overline { { \mathrm { E x p } } }$ . Thus, CEXP contains the same core structures as EXP, but these now lead to different SAT values in $\overline { { \mathrm { E x P } } }$ and CORRUPT, and this makes the overall learning task more challenging than learning $\overline { { \mathrm { E x p } } }$ or CORRUPT in isolation. Complete details of the overall data generation process for both datasets can be found in the appendix. + +# 6 EXPERIMENTAL EVALUATION + +In this section, we first evaluate the practical effect of RNI on MPNN expressiveness based on EXP, and compare MPNN-RNI against established higher-order GNNs. We then extend our empirical analysis to CEXP. Both experiments are conducted using the following models: + +- 1-WL GCN (1-GCN): A GCN with 8 distinct message passing iterations, ELU non-linearities (Clevert et al., 2016), 64-dimensional embeddings, and deterministic learnable initial node embeddings indicating node type. This model is guaranteed to achieve $50 \%$ accuracy on EXP. + +- GCN - Random node initialization (GCN-RNI): An analogous model to 1-GCN with an identical architecture, enhanced with RNI. We evaluate this model with four initialization distributions, namely the standard normal distribution $\mathcal { N } ( 0 , 1 )$ (N), the uniform distribution over $[ - 1 , 1 ]$ (U), Xavier normal (XN), and the Xavier uniform distribution (XU) (Glorot & Bengio, 2010). We denote the respective models $\mathbf { G C N - R N I } ( D )$ , where $D \in \{ \mathrm { N , U , X N , X U } \}$ . + +- GCN - Partial Random node initialization $( \mathbf { G C N - } x \% \mathbf { R N I } )$ : A GCN-RNI model, where only a percentage $x$ of initial node embedding dimensions are randomized. That is, for $d$ -dimensional node embeddings, GCN- $x \%$ RNI randomizes $\lfloor \frac { x d } { 1 0 0 } \rfloor$ dimensions, and sets the remaining dimensions deterministically from input features, namely, a one-hot representation of the two possible node types (literal and disjunction) in the input graph representation (see appendix for more details). We set $x$ to the extreme values 0 and $100 \%$ , $50 \%$ , as well as near-edge cases of $8 7 . 5 \%$ and $12 . 5 \%$ , respectively. + +- Provably Powerful Graph Network (PPGN) (Maron et al., 2019a): A higher-order GNN with 2-WL expressive power, and which requires quadratic memory relative to the number of graph nodes. We set up PPGN with eight 400-dimensional computational blocks. + +- 1-2-3-GCN-L: A higher-order GNN (Morris et al., 2019) emulating 2-WL on 3-tuples of nodes. 1-2-3-GCN-L operates at increasingly coarse node granularities, starting with single nodes and rising to 3-tuples. The model first computes all possible 3-tuples of nodes, then represents them as standard graph nodes. These nodes are connected to one another following the 2-WL neighborhood definition, i.e., tuples that exchange messages in 2-WL are connected by an edge in 3-GCN. 3-GCN-L implements a connected relaxation of 2-WL, in that only 3-tuples forming a connected graph are used, which comes at the cost of some theoretical guarantees. Nonetheless, the computation and representation of all tuples still imposes a severe overhead relative to MPNNs. We set up 1-2-3-GCN-L with 64-dimensional embeddings, 3 message passing iterations at level 1, 2 at level 2 and 8 at level 3. + +- 3-GCN: A modification to 1-2-3-GCN-L, such that (i) only the 3rd level is used, and (ii) the full 2-WL procedure is implemented, i.e., all 3-tuples are computed, as in standard 2-WL, rather than only the connected ones. + +6.1 EXPERIMENT 1: HOW DOES RNI IMPROVE MPNN EXPRESSIVENESS? + +In this experiment, we evaluate GCNs using different RNI settings on EXP, and compare with standard GNNs and higher-order models. Specifically, we generate an EXP dataset consisting of 600 graph pairs, and discuss this generation in more detail in the appendix. Then, we evaluate all models on EXP using 10-fold cross-validation. We train 3-GCN for 100 epochs per fold, and all other systems for 500 epochs. Mean test accuracy across all validation folds is measured and reported. + +Full test accuracy results for all models are reported in Table 1, and model convergence for 3-GCN and all GCN-RNI models are shown in Figure 2. In line with Theorem 4.1, GCN-RNI achieves a near-perfect performance on EXP, substantially surpassing $50 \%$ . Indeed, all fully randomized GCN-RNI models achieve a performance above $9 5 \%$ with all four RNI distributions. This finding supports observations made in related studies on RNI (Sato et al., 2020), which suggest that RNI enables (sub)structure detection beyond the theoretical limits of 1-WL. Empirically, we observed that GCN-RNI is highly sensitive to changes in learning rate, activation function, and/or randomization distribution, and required delicate tuning to achieve its best performance. + +Table 1: Accuracy results on EXP. + +
ModelTest Accuracy (%)
GCN-RNI(U)97.3 ± 2.55
GCN-RNI(N) GCN-RNI(XU)98.0 ± 1.85 97.0 ± 1.43
GCN-RNI(XN)96.6 ± 2.20
PPGN50.0
1-2-3-GCN-L50.0
3-GCN99.7 ± 0.004
+ +Surprisingly, PPGN does not achieve performance above $50 \%$ , despite being theoretically 2-WL expressive. Essentially, PPGN learns an approximation of 2-WL, based on power-sum multisymmetric polynomials (PMP), but fails to distinguish EXP graph pairs, despite extensive training. This suggests that PPGNs struggle to learn the required PMPs, and we could not improve these results, both for training and testing, with hyperparameter tuning. Furthermore, as mentioned in Section 3, PPGN requires exponentially many data samples in the size of the input graph (Puny et al., 2020) for learning. Hence, PPGN is likely struggling to discern between EXP graph pairs due to the smaller sample size and variability of the dataset. 1-2-3-GCN-L also only achieves $50 \%$ accuracy, which can be attributed to theoretical model limitations. Indeed, the local and connected algorithm drops necessary information to distinguish graph pairs, as it only considers 3-tuples of nodes that form a connected sub-graph. Thus, 1-2-3-GCN-L discards disconnected 3-tuples that crucially are where the difference between the EXP cores lies. This further highlights the difficulty of EXP instances, as even a relaxation of 2-WL costs the model the ability to achieve above-random performance. Note that 3-GCN achieves near-perfect performance, as it explicitly has the sufficient theoretical power needed for the task, irrespective of learning constraints, and must only learn appropriate injective aggregation functions for neighbor aggregation (Xu et al., 2019). + +![](images/bc876dd422d8f56b42cd4c08770bee5dbcdab007b0466e2888f1f7d72dd04bae.jpg) +Figure 2: Learning curves on EXP. + +In terms of model convergence, we observe that 3- GCN converges significantly faster than all GCNRNI models, for all randomization percentages. Indeed, 3-GCN only requires about 10 epochs to achieve its optimal performance, whereas GCN-RNI models all require in excess of 100 epochs. The rp Intuitively, the slower convergence of GCN-RNI can be attributed to a significantly harder learning task compared to 3- GCN: Whereas 3-GCN must learn from a deterministic set of node embeddings, and is naturally capable of discerning between dataset cores, GCN-RNI relies on RNI to discern between data points in EXP, via an artificial node ordering. This in turn implies that GCNRNI must first leverage RNI to detect structure, then subsequently learn robustness against the variability of RNI, which makes the learning task for GCN-RNI especially challenging. + +Our findings suggest that RNI can practically improve the expressiveness of MPNNs, and make them competitive with higher-order models, despite being significantly less demanding computationally. Indeed, for a typical EXP instance with 50 nodes, GCN-RNI only requires 3200 parameters (using + +![](images/9d002f0cbe123c2e07395bc5988e6354ce82697cff2e0ea6e3e87615d2e764cd.jpg) +Figure 3: Model convergence results for Experiment 2 on CEXP on all models. + +![](images/1b03f6be89d70f63c8d48b76ba763b79bb656d809a6afe63060e3a9ab29cd7ff.jpg) +(b) Learning curves for CEXP, split across EXP (/E) and CORRUPT (/C). + +(a) Learning curves for all GCN-RNI models and 3-GCN on CEXP. + +64-dimensional embeddings), whereas 3-GCN requires 1,254,400 parameters. Nonetheless, GCNRNI performs comparably to 3-GCN, and, unlike the latter model, can easily scale to larger instances exceeding the range used in our datasets. This increase in expressive power, however, comes at the cost of slower convergence. Even so, RNI proves to be a promising direction for building scalable yet powerful MPNNs. + +# 6.2 EXPERIMENT 2: HOW DOES RNI AFFECT MPNN ON MORE VARIABLE DATASETS? + +In Experiment 1, we observed that RNI practically improves the expressive power of GCNs over EXP. However, EXP is solely designed for expressiveness evaluation, and this leaves multiple questions open: How does RNI impact learning when data contains instances with varying expressiveness requirements, and how does RNI affect model generalization on more variable datasets? We experiment with CEXP to explicitly address these questions. + +Analogously to Experiment 1, we generate an EXP dataset with 600 pairs of graphs. Then, we create CEXP by selecting 300 graph pairs and modifying their satisfiable graph, yielding CORRUPT. CEXP is well-suited for evaluating the efficacy of RNI more holistically, as it allows (i) the evaluation of the contribution of RNI on EXP conjointly with a second learning task on CORRUPT involving very similar core structures, and (ii) a study of the effect of different degrees of randomization on overall and subset-specific model performance. + +In this experiment, we train GCN-RNI (with varying randomization degrees) and 3-GCN on CEXP, and compare their test accuracy across all cross-validation splits. For GCN-RNI models, we observe the effect of RNI on specifically learning $\overline { { \mathrm { E x p } } }$ and CORRUPT, and the interplay between these two tasks. In all experiments, we exclusively use normal distribution initialization, given its strong performance in Experiment 1. + +The learning curves of all GCN-RNI and 3-GCN on CEXP are shown in Figure 3a, and the same curves for the EXP and CORRUPT subsets are shown in Figure 3b. As on EXP, we observe that 3- GCN converges very quickly, exceeding $90 \%$ test accuracy within 25 epochs on CEXP. By contrast, GCN-RNI, for all randomization levels, converges much slower, around after 200 epochs, despite the small size of input graphs ( $\mathord { \sim } 7 0$ nodes at most). Furthermore, fully randomized GCN-RNI performs worse than partly randomized GCN-RNI models, particularly on CEXP, due to its weak performance on CORRUPT, as shown in Figure 3b. + +First, we observe that partial randomization can significantly improve model performance. This can clearly be seen on CEXP, in Figure 3a and Figure 3b, where GCN- $12 . 5 \%$ RNI and GCN-87.5%RNI achieve the best performance, by far outperforming GCN-RNI, which struggles on CORRUPT. This can be attributed to having a better inductive bias than a fully randomized model. Indeed, GCN$1 2 . 5 \% \mathrm { R N I }$ has mostly deterministic node embeddings, which simplifies learning over CORRUPT. + +This also applies to $\mathrm { G C N - } 8 7 . 5 \% \mathrm { R N }$ I, where the number of deterministic dimensions, though small, remains sufficient for learning over CORRUPT. Both models also benefit from randomization to perform strongly on $\overline { { \mathrm { E x p } } }$ , and have sufficient randomization to perform similarly to a fully randomized GCN. GCN- $1 2 . 5 \% \mathrm { R N I }$ and $\mathrm { G C N - } 8 7 . 5 \% \mathrm { R l }$ NI effectively achieve the best of both worlds on CEXP, leveraging inductive bias from deterministic node embeddings, while harnessing the power of random embeddings to perform strongly on $\overline { { \mathrm { E x p } } }$ . This is best shown in Figure 3b, where standard GCN fails to learn $\overline { { \mathrm { E x p } } }$ , fully randomized GCN-RNI struggles to learn CORRUPT, and the semi-randomized $G C N { - } 5 0 \% \mathrm { R l }$ NI achieves perfect performance on both subsets. Overall, this is a surprising finding, as it suggests that MPNNs can perform significantly better with partial, and even small, amounts of randomization. + +Second, we observe that the fully randomized GCN-RNI performs substantially worse than its partially randomized counterparts. Whereas fully randomized GCN-RNI only performs marginally worse on EXP (cf. Figure 2) than partially randomized models, this gap is very large on CEXP, primarily due to CORRUPT. This observation concurs with the earlier idea of inductive bias: Fully randomized GCN-RNI loses all node type information, which is valuable for making robust and consistent decisions, and therefore struggles to match 3-GCN and partially randomized models. Indeed, the model fails to achieve even $60 \%$ accuracy on CORRUPT, where other models are near perfect, and also relatively struggles on $\overline { { \mathrm { E x p } } }$ , only reaching $91 \%$ accuracy and converging slower. + +Third, all GCN-RNI models, at all randomization levels, converge significantly slower on both datasets than 3-GCN, similarly to Experiment 1. However, an interesting phenomenon can be seen on CEXP: All GCN-RNI models hover around $55 \%$ accuracy within the first 100 epochs over CEXP (cf. Figure 3a), suggesting a struggle jointly fitting both CORRUPT and $\overline { { \mathrm { E x p } } }$ , before these models ultimately improve. This, however, is not observed with 3-GCN. Unlike on EXP, randomness is not necessarily beneficial on CEXP, as it can hurt performance on CORRUPT. Hence, RNI-enhanced models must additionally learn to isolate deterministic dimensions for CORRUPT, and randomized dimensions for EXP. These findings consolidate the earlier observations made on EXP on the impact of RNI on MPNN learning behavior, and highlight that the variability and slower learning for RNI also hinges on the variability and complexity of the input dataset. + +Finally, we observe that both fully randomized GCN-RNI, and, surprisingly, 1-GCN, struggle to learn CORRUPT relative to partially randomized GCN-RNI. We can also observe that 1-GCN does not present a “struggle” phase, and begins improving consistently from the start of training. These observations can be attributed to key conceptual , but very distinct hindrances impeding both models. In the case of 1-GCN, the model is jointly trying to learn both EXP and CORRUPT, when it is proven that it cannot fit the former. This joint optimization severely hinders CORRUPT learning, as data pairs from both subsets are highly similar, and share identically generated UNSAT graphs (cf. Appendix). Hence, 1-GCN, in attempting to fit SAT graphs from both subsets, knowing it cannot distinguish EXP pairs, struggles to learn the simpler difference in CORRUPT pairs. For GCN-RNI, the model discards key type information, so must only rely on structural differences to learn CORRUPT, which impedes its convergence. All in all, this further consolidates the promise of partial RNI as a means to combine the strengths of both deterministic and random features. + +Further to the earlier two experiments, we also conducted analogous experiments using sparser analogs of the datasets EXP and CEXP. In these cases, we observed similar behavior, albeit with slower convergence overall. More details on these experiments can be found in the appendix. + +# 7 SUMMARY AND OUTLOOK + +We studied the expressive power of MPNNs with RNI, and showed that these are universal models. We empirically evaluated this model on carefully designed datasets, and observed that RNI practically improves the learning abilities of MPNNs for challenging data, though it does slow down model convergence owing to the need to learn robustness against random variability. Our work delivers a strong theoretical result, supported by empirical evaluation and practical insights, to rigorously quantify the effect of RNI on GNNs. Somewhat surprisingly, our experiments suggest that partial randomization may be the best strategy in most practical scenarios. An important direction for future work is to theoretically study the sensitivity of RNI to model architectures and initialization distributions, to yield a more complete understanding of the benefits and limitations of RNI. + +# REFERENCES + +Pablo Barcelo, Egor V. Kostylev, Mika ´ el Monet, Jorge P ¨ erez, Juan L. Reutter, and Juan Pablo Silva. ´ The logical expressiveness of graph neural networks. 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In Proceedings of the Twenty-Fourth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pp. 974–983, 2018. + +Jiaxuan You, Bowen Liu, Zhitao Ying, Vijay S. Pande, and Jure Leskovec. Graph convolutional policy network for goal-directed molecular graph generation. In Proceedings of the Thirty-First Annual Conference on Advances in Neural Information Processing Systems, (NeurIPS), pp. 6412– 6422, 2018. + +# A APPENDIX + +# A.1 PROPOSITIONAL LOGIC + +We briefly present propositional logic, which underpins the dataset generation. Let $S$ be a (finite) set $S$ of propositional variables. A literal is defined as $v$ , or $\bar { v }$ (resp., $\neg v$ ), where $v \in S$ . A disjunction of literals is a clause. The width of a clause is defined as the number of literals it contains. A formula $\varphi$ is in conjunctive normal form $( C N F )$ if it is a conjunction of clauses. A CNF has width $k$ if it contains clauses of width at most $k$ , and is referred to as a $k$ −CNF. To illustrate, the formula $\varphi = \left( x _ { 1 } \vee { } { \neg x } _ { 3 } \right) \wedge \left( x _ { 4 } \vee x _ { 1 } \right)$ is a CNF with clauses of width 2. + +An assignment $\nu : S \mapsto \{ 0 , 1 \}$ maps variables to False (0), or True (1), and satisfies $\varphi$ , which we denote by $\nu \models \varphi$ , in the usual sense, where $\vDash$ is propositional entailment. Given a propositional formula $\varphi$ , the satisfiability problem, commonly known as SAT, consists of determining whether $\varphi$ admits a satisfying assignment, and is NP-complete (Cook, 1971). + +# A.2 PROOF OF THEOREM 4.1 + +We first prove a Boolean version of the theorem. + +Lemma A.1. Let $n \geq 1$ , and let $f : { \mathcal { G } } _ { n } \to \{ 0 , 1 \}$ be an invariant Boolean function. Then, for all $\epsilon , \delta > 0$ there is a MPNN with RNI that $( \epsilon , \delta )$ -approximates $f$ . + +To prove this lemma, we use a logical characterization of the expressiveness of MPNNs, which we always assume to admit global readouts. Let $\complement$ be the extension of first-order predicate logic using counting quantifiers of the form $\exists ^ { \geq k } x$ for $k \geq 0$ , where $\exists ^ { \geq k } x \varphi ( x )$ means that there are at least $k$ elements x satisfying ϕ. + +For example, consider the formula + +$$ +\varphi ( x ) : = \lnot \exists ^ { \geq 3 } y \big ( E ( x , y ) \land \exists ^ { \geq 5 } z E ( y , z ) \big ) . +$$ + +This is a formula in the language of graphs; $E ( x , y )$ means that there is an edge between the nodes interpreting $x$ and $y$ . For a graph $G$ and a vertex $v \in V ( G )$ , we have $G \models \varphi ( v )$ ( $^ { 6 6 } G$ satisfies $\varphi$ if the variable $x$ is interpreted by the vertex $v '$ ”) if and only if $v$ has at most 2 neighbors in $G$ that have degree at least 5. + +We will not only consider formulas in the language of graphs, but also formulas in the language of colored graphs, where in addition to the binary edge relation we also have unary relations, that is, sets of nodes, which we may view as colors of the nodes. For example, the formula + +$$ +\psi ( x ) : = \exists ^ { \geq 4 } y { \bigl ( } E ( x , y ) \land R E D ( y ) { \bigr ) } +$$ + +says that node $x$ has at least 4 red neighbors (more precisely, neighbors in the unary relation $R E D$ ). Formally, we assume we have fixed infinite list $R _ { 1 } , R _ { 2 } , \ldots$ of color symbols that we may use in our formulas. Then a colored graph is a graph together with a mapping that assigns a finite set $\rho ( v )$ of colors $R _ { i }$ to each vertex (so we allow one vertex to have more than one, but only finitely many, colors). + +A sentence (of the logic $\complement$ or any other logic) is a formula without free variable. Thus a sentence expresses a property of a graph, which we can also view as a Boolean function. For a sentence $\varphi$ we denote this function by $[ [ \varphi ] ]$ . If $\varphi$ is a sentence in the language of (colored) graphs, then for every (colored) graph $G$ J we have $\mathbb { [ } \varphi ] ( G ) = 1$ if $G \models \varphi$ and $\mathbb { I } \varphi \mathbb { I } ( { \overline { { G } } } ) { \overline { { = 0 } } }$ otherwise. + +It is easy to see that $\complement$ is only a syntactic extension of first order logic FO—for every C-formula there is a logically equivalent $\mathsf { F O }$ -formula. To see this, note that we can simulate $\exists ^ { \geq k } x$ by $k$ ordinary existential quantifiers: $\exists ^ { \geq k } x$ is equivalent to $\exists x _ { 1 } \dots \exists x _ { k } { \Big ( } \bigwedge _ { 1 \leq i < j \leq k } x _ { i } \neq x _ { j } \wedge \bigwedge _ { 1 \leq i \leq k } \varphi ( x _ { i } ) { \Big ) }$ . However, counting quantifiers add expressiveness if we restrict the number of variables. The $\complement$ -formula $\exists ^ { \geq k } x$ $( x = x )$ ) (saying that there are at least $k$ vertices) with just one variable is not equivalent to any FO-formula using less than $k$ variables. By ${ \mathsf { C } } ^ { k }$ we denote the fragment of $\complement$ consisting of all formulas with at most $k$ variables. + +For example, the formula $\varphi ( x )$ in (A.1) is in ${ \mathsf C } ^ { 3 }$ , but not in $\mathsf { C } ^ { 2 }$ . But $\varphi ( x )$ is equivalent to the following formula $\varphi ^ { \prime } ( x )$ in $\mathsf { C } ^ { 2 }$ : + +$$ +\varphi ^ { \prime } ( x ) : = \neg \exists ^ { \geq 3 } y \big ( E ( x , y ) \land \exists ^ { \geq 5 } x E ( y , x ) \big ) . +$$ + +The fragments ${ \mathsf { C } } ^ { k }$ are interesting for us, because their expressiveness corresponds to that of $\left( k - 1 \right)$ - WL and hence to that of $k$ -GNNs. More precisely, for all $k \geq 2$ , two graphs $G$ and $H$ satisfy the same ${ \mathsf { C } } ^ { k }$ -sentences if and only if $\left( k - 1 \right)$ -WL does not distinguish them (Cai et al., 1992). By the results of (Morris et al., 2019; Xu et al., 2019) this implies, in particular, that two graphs are indistinguishable by all MPNNs if and only if they satisfy the same $\mathsf { C } ^ { 2 }$ -sentences. Barcelo et al. (2020) strengthened ´ this result and showed that every $\mathsf { C } ^ { 2 }$ -sentence can be simulated by an MPNN. + +Lemma A.2 (Barcelo et al. 2020) ´ . For every $\mathsf { C } ^ { 2 }$ -sentence $\varphi$ and every $\epsilon > 0$ there is an MPNN that $\epsilon$ -approximates $[ [ \varphi ] ]$ . + +Since here we are talking about deterministic MPNNs, there is no randomness involved, and we just say “ $\epsilon$ -approximates” instead of “ $\mathopen { } \mathclose \bgroup \left( \epsilon , 1 \aftergroup \egroup \right)$ -approximates”. + +Lemma A.2 not only holds for sentences in the language of graphs, but also for sentences in the language of colored graphs. Let us briefly discuss the way MPNNs access such colors. We encode the colors using one-hot vectors that are part of the initial states of the nodes. For example, if we have a formula that uses color symbols among $R _ { 1 } , \ldots , R _ { k }$ , then we reserve $k$ places in the initial state $\pmb { x } _ { v } = \left( x _ { v 1 } , \dots , x _ { v \ell } \right)$ of each vertex $v$ (say, for convenience, $x _ { v 1 } , \ldots , x _ { v k } )$ and we initialize $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { v } }$ by letting $x _ { v i } = 1$ if $v$ is in $R _ { i }$ and $x _ { v i } = 0$ otherwise. + +Let us call a colored graph $G$ individualized if for any two distinct vertices $v , w \in V ( G )$ the sets $\rho ( v ) , \rho ( w )$ of colors they have are distinct. Let us say that a sentence $\chi$ identifies a (colored) graph $G$ if for all (colored) graphs $H$ we have $H \models \chi$ if and only if $H$ is isomorphic to $G$ . + +Lemma A.3. For every individualized colored graph $G$ there is a $\mathsf { C } ^ { 2 }$ -sentence $\chi _ { G }$ that identifies $G$ + +Proof. Let $G$ be an individualized graph. For every vertex $v \in V ( G )$ , let + +$$ +\alpha _ { v } ( x ) : = \bigwedge _ { R \in \rho ( v ) } R ( x ) \wedge \bigwedge _ { R \in \{ R _ { 1 } , \ldots , R _ { k } \} \setminus \rho ( x ) } \neg R ( x ) . +$$ + +Then $v$ is the unique vertex of $G$ such that $G \models \alpha _ { v } ( v )$ . For every pair $v , w \in V ( G )$ of vertices, we let + +$$ +\begin{array} { r } { \beta _ { v w } ( x , y ) : = \left\{ \begin{array} { l l } { \alpha _ { v } ( x ) \wedge \alpha _ { w } ( y ) \wedge E ( x , y ) } & { \mathrm { i f ~ } ( v , w ) \in E ( G ) , } \\ { \alpha _ { v } ( x ) \wedge \alpha _ { w } ( y ) \wedge \neg E ( x , y ) } & { \mathrm { i f ~ } ( v , w ) \not \in E ( G ) . } \end{array} \right. } \end{array} +$$ + +We let + +$$ +\chi _ { G } : = \bigwedge _ { v \in V ( G ) } \left( \exists x \alpha _ { v } ( x ) \wedge \neg \exists ^ { \geq 2 } x \alpha _ { v } ( x ) \right) \wedge \bigwedge _ { v , w \in V ( G ) } \exists x \exists y \beta _ { v w } ( x , y ) . +$$ + +It is easy to see that $\chi _ { G }$ identifies $G$ . + +For $n , k \in \mathbb { N }$ , we let $\mathcal { G } _ { n , k }$ be the class of all individualized colored graphs that only use colors among $R _ { 1 } , \ldots , R _ { k }$ . + +Lemma A.4. Let $h : { \mathcal { G } } _ { n , k } \to \{ 0 , 1 \}$ be an invariant Boolean function. Then there exists a $\mathsf { C } ^ { 2 }$ - sentence $\psi _ { h }$ such that for all $G \in \mathcal { G } _ { n , k }$ it holds that $[ [ \psi _ { h } ] ] ( G ) = h ( G )$ . + +Proof. Let ${ \mathcal { H } } \subseteq { \mathcal { G } } _ { n , k }$ be the subset consisting of all graphs $H$ with $h ( H ) = 1$ . We let + +$$ +\psi _ { h } : = \bigvee _ { H \in \mathcal { H } } \chi _ { H } . +$$ + +We eliminate duplicates in the disjunction. Since up to isomorphism, the class $\mathcal { G } _ { n , k }$ is finite, this makes the disjunction finite and hence $\psi _ { h }$ well-defined. □ + +The restriction of a colored graph $G$ is the underlying plain graph, that is, the graph $G ^ { \vee }$ obtained from the colored graph $G$ by forgetting all the colors. Conversely, a colored graph $G ^ { \wedge }$ is an expansion of a plain graph $G$ if $G = ( G ^ { \wedge } ) ^ { \vee }$ . + +Corollary A.1. Let $f : { \mathcal { G } } _ { n } \to \{ 0 , 1 \}$ be an invariant Boolean function. Then there exists a $\mathsf { C } ^ { 2 }$ - sentence $\varphi _ { f } ^ { \wedge }$ (in the language of colored graphs) such that for all $G \in { \mathcal { G } } _ { n , k }$ it holds that $\mathbb { [ } \psi _ { f } ^ { \wedge } ] ( G ) =$ $f ( G ^ { \vee } )$ . + +Towards proving Lemma A.1, we fix an $n \geq 1$ and a $\epsilon , \delta > 0$ . We let + +$$ +c : = \left\lceil { \frac { 2 } { \delta } } \right\rceil \quad { \mathrm { a n d } } \quad k : = c ^ { 2 } \cdot n ^ { 3 } +$$ + +The technical details of the proof of Lemma A.1 and Theorem 4.1 depend on the exact choice of the random initialization and the activation functions used in the neural networks, but the idea is always the same. For simplicity, we assume that we initialize the states $\pmb { x } _ { v } = \left( x _ { v 1 } , \ldots , x _ { v \ell } \right)$ of all vertices to $( r _ { v } , 0 , \ldots , 0 )$ , where $r _ { v }$ for $v \in V ( G )$ are chosen independently uniformly at random from $[ 0 , 1 ]$ . As our activation function $\sigma$ , we choose the linearized sigmoid function defined by $\sigma ( x ) = \bar { 0 }$ for $x < 0$ , $\sigma ( x ) = x$ for $0 \leq x < 1$ , and $\sigma ( x ) = 1$ for $x \geq 1$ . + +Lemma A.5. Let $r _ { 1 } , \ldots , r _ { n }$ be chosen independently uniformly at random from the interval $[ 0 , 1 ]$ . For $1 \leq i \leq n$ and $1 \leq j \leq c \cdot n ^ { 2 }$ , let + +$$ +s _ { i j } : = k \cdot r _ { i } - \left( j - 1 \right) \cdot \frac { k } { c \cdot n ^ { 2 } } . +$$ + +Then with probability greater than $1 - \delta$ , the following conditions are satisfied. + +(ii) For all distinct $i , i ^ { \prime } \in \{ 1 , . . . , n \}$ there exists a $j \in \left\{ 1 , \dots , c \cdot n ^ { 2 } \right\}$ such that $\sigma ( s _ { i j } ) \neq \sigma ( s _ { i ^ { \prime } j } )$ + +Proof. For every $i$ , let $p _ { i } : = \lfloor r _ { i } \cdot k \rfloor$ . Since $k \cdot r _ { i }$ is uniformly random from the interval $[ 0 , k ]$ , the integer $p _ { i }$ is uniformly random from $\{ 0 , \ldots , k - 1 \}$ . Observe that $0 < \sigma ( s _ { i j } ) < 1$ only if $p _ { i } - ( j -$ $\textstyle 1 ) \cdot { \frac { k } { c \cdot n ^ { 2 } } } = 0$ (here we use the fact that $k$ is divisible by $c \cdot n ^ { 2 } .$ ). The probability that this happens is $\frac { 1 } { k }$ ⋅Thus, by the Union Bound, + +$$ +\operatorname* { P r } \left( \exists i , j : 0 < \sigma ( s _ { i j } ) < 1 \right) \leq \frac { c \cdot n ^ { 3 } } { k } . +$$ + +Now let $i , i ^ { \prime }$ be distinct and suppose that $\sigma ( s _ { i j } ) = \sigma ( s _ { i ^ { \prime } j } )$ for all $j$ . Then for all $j$ we have $s _ { i j } \leq$ $0 \iff s _ { i ^ { \prime } j } \leq 0$ and therefore $\lfloor s _ { i j } \rfloor \le 0 \iff \lfloor \stackrel { \sim } { s } _ { i ^ { \prime } j } \rfloor \le 0$ . This implies + +$$ +\forall j \in \{ 1 , \ldots , c \cdot n ^ { 2 } \} : \quad p _ { i } \leq ( j - 1 ) \cdot { \frac { k } { c \cdot n ^ { 2 } } } \Longleftrightarrow p _ { i ^ { \prime } } \leq ( j - 1 ) \cdot { \frac { k } { c \cdot n ^ { 2 } } } . +$$ + +Let $j ^ { * } \in \{ 1 , \ldots , c \cdot n ^ { 2 } \}$ such that $p _ { i } \in \left\{ \left( j ^ { * } - 1 \right) \cdot { \frac { k } { c \cdot n ^ { 2 } } } , \ldots , j ^ { * } \cdot { \frac { k } { c \cdot n ^ { 2 } } } - 1 \right\}$ . Then by (A.4) we have +$\begin{array} { r } { p _ { i } ^ { \prime } \in \left\{ \left( j ^ { * } - 1 \right) \cdot \frac { k } { c \cdot n ^ { 2 } } , \ldots , j ^ { * } \cdot \frac { k } { c \cdot n ^ { 2 } } - 1 \right\} } \end{array}$ . As $p _ { i ^ { \prime } }$ is independent of $p _ { i }$ and hence of $j ^ { * }$ , the probability +that this happens is at most $\begin{array} { r } { \frac { 1 } { k } \cdot \frac { k } { c \cdot n ^ { 2 } } = \frac { 1 } { c \cdot n ^ { 2 } } } \end{array}$ . This proves that for all distinct $i , i ^ { \prime }$ the probability that k1 +$\sigma ( s _ { i j } ) = \sigma ( s _ { i ^ { \prime } j } )$ is at most $\textstyle { \frac { 1 } { c \cdot n ^ { 2 } } }$ . Hence, again by the Union Bound, + +$$ +\operatorname* { P r } ( \exists i \neq i ^ { \prime } \forall j : \sigma ( s _ { i j } ) = \sigma ( s _ { i ^ { \prime } j } ) ) \leq \frac { 1 } { c } . +$$ + +(A.3) and (A.5) imply that the probability that either (i) or (ii) is violated is at most + +$$ +\frac { c \cdot n ^ { 3 } } { k } + \frac { 1 } { c } \leq \frac { 2 } { c } \leq \delta . +$$ + +Proof of Lemma A.1. For given function $f : { \mathcal { G } } _ { n } \to \{ 0 , 1 \}$ , we choose the sentence $\psi _ { f } ^ { \wedge }$ according to Corollary A.1. Applying Lemma A.2 to this sentence and $\epsilon$ , we obtain an MPNN $\ddot { \mathcal { N } } _ { f }$ that on a colored graph $G \in \mathcal { G } _ { n , k }$ computes an $\epsilon$ -approximation of $f ( G ^ { \vee } )$ . + +Without loss of generality, we assume that the vertex set of the input graph to our MPNN is $\{ 1 , \ldots , n \}$ . We choose $\ell$ (the dimension of the state vectors) in such a way that $\ell \geq c \cdot n ^ { 2 }$ and $\ell$ is at least as large as the dimension of the state vectors of $\mathcal { N } _ { f }$ . Recall that the state vectors are + +initialized as x(0i $\pmb { x } _ { i } ^ { ( 0 ) } = ( r _ { i } , 0 , \ldots , 0 )$ for values $r _ { i }$ chosen independently uniformly at random from the interval $[ 0 , 1 ]$ . + +passed) that maps x(0)i t o x( 1 )i = $\pmb { x } _ { i } ^ { ( 1 ) } = \bar { ( x _ { i 1 } ^ { ( 1 ) } , \dots , x _ { i \ell } ^ { ( 1 ) } ) }$ ly local transformation (no messages need to be with + +$$ +\begin{array} { r } { x _ { i j } ^ { ( 1 ) } = \left\{ \begin{array} { l l } { \sigma \Big ( \boldsymbol { k } \cdot \boldsymbol { r _ { i } } - \left( j - 1 \right) \cdot \frac { \boldsymbol { k } } { c \cdot n ^ { 2 } } \Big ) } & { \mathrm { f o r ~ } 1 \leq j \leq c \cdot n ^ { 2 } , } \\ { 0 } & { \mathrm { f o r ~ } c \cdot n ^ { 2 } + 1 \leq j \leq \ell . } \end{array} \right. } \end{array} +$$ + +Since we treat $k , c , n$ as constants, the mapping $\begin{array} { r } { r _ { i } \mapsto k \cdot r _ { i } - \left( j - 1 \right) \cdot \frac { k } { c \cdot n ^ { 2 } } } \end{array}$ is just a linear mapping applied to $r _ { i } = x _ { i 1 } ^ { ( 0 ) }$ . + +By Lemma A.5, with probability at least $1 - \delta$ , the vectors $\pmb { x } _ { i } ^ { ( 1 ) }$ are mutually distinct $\{ 0 , 1 \}$ -vectors, which we view as encoding a coloring of the input graph with colors from $R _ { 1 } , \ldots , R _ { k }$ . Let $G ^ { \wedge }$ be the resulting colored graph. Since the vectors $\mathbf { \bar { x } } _ { i } ^ { ( 0 ) }$ are mutually distinct, $G ^ { \wedge }$ is individualized and thus in the class $\mathcal { G } _ { n , k }$ . We now apply the MPNN $\mathcal { N } _ { f }$ , and it computes a value $\epsilon$ -close to $\mathbb { } [ \psi _ { f } ^ { \wedge } ] ( G ^ { \wedge } ) =$ $f ( ( G ^ { \wedge } ) ^ { \vee } ) = f ( G )$ . □ + +Proof of Theorem 4.1. Let $f : { \mathcal { G } } _ { n } \to \mathbb { R }$ be invariant. Since $\mathcal { G } _ { n }$ is finite, the range $Y : = f ( { \mathcal { G } } _ { n } )$ is finite. To be precise, we have $N : = | Y | \leq | \mathcal { G } _ { n } | = 2 ^ { { \binom { n } { 2 } } }$ . + +Say, $Y = \{ y _ { 1 } , \dots , y _ { N } \}$ . For $i = 1 , \ldots , N$ , let $g _ { i } : { \mathcal { G } } _ { n } \{ 0 , 1 \}$ be the Boolean function defined by + +$$ +g _ { i } ( G ) = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ } } f ( G ) = y _ { i } , } \\ { 0 } & { { \mathrm { o t h e r w i s e . } } } \end{array} \right. } +$$ + +Note that $g _ { i }$ is invariant. + +Let $\epsilon , \delta > 0$ and $\begin{array} { r } { \epsilon ^ { \prime } : = \frac { \epsilon } { \operatorname* { m a x } Y } } \end{array}$ and $\begin{array} { r } { \delta ^ { \prime } : = \frac { \delta } { N } } \end{array}$ . By Lemma A.1, for every $i \in \{ 1 , \ldots , N \}$ there is an MPNN with $\boldsymbol { \mathrm { R N I } } \mathcal { N } _ { i }$ that $( \epsilon ^ { \prime } , \delta )$ -approximates $g _ { i }$ . Putting all the ${ \mathcal { N } } _ { i }$ together, we obtain an invariant MPNN $\mathcal { N }$ that computes a function $g : { \mathcal { G } } _ { n } \to \{ 0 , 1 \} ^ { N }$ . We only need to apply the linear transformation + +$$ +\pmb { x } \mapsto \sum _ { i = 1 } ^ { N } x _ { i } \cdot y _ { i } +$$ + +to the output of $\mathcal { N }$ to obtain the desired approximation of $f$ . + +Remark 1. Obviously, our construction yields MPNNs with a prohibitively large state space. In particular, this is the case for the brute force step from Boolean to general functions. We doubt that there are much more efficient approximators, after all we make no assumption whatsoever on the function $f$ . + +The approximation of Boolean functions is more interesting. It may still happen that the GNNs get exponentially large in $n$ ; this seems unavoidable. However, the nice thing here is that our construction is very adaptive and tightly linked to the descriptive complexity of the function we want to approximate. This deserves a more thorough investigation, which we leave for future work. + +As opposed to other universality results for GNNs, our construction needs no higher-order tensors defined on tuples of nodes, with practically infeasible space requirements on all but very small graphs. Instead, the complexity of our construction goes entirely into the dimension of the state space. The advantage of this is that we can treat this dimension as a hyperparameter that we can easily adapt and that gives us more fine-grained control over the space requirements. Our experiments show that usually in practice a small dimension already yields very powerful networks. + +Remark 2. In our experiments, we found that a partial random initialization, which only assigns random values to a fraction of all node embedding vectors, often yields very good results, sometimes better than a full random initialization. There is plausibility to this from a theoretical perspective. For most graphs, we do not lose much by only initializing a small fraction of vertex embeddings, because in a few message-passing rounds GNNs can propagate the randomness and, referring our construction above, individualize the full input graph. On the other hand, we reduce the amount of noise our models have to handle when we only randomize partially. + +![](images/224f4caf68657c2ece8143ddc76017e80d9d914c8f0e7af17d004b811e9ebd17.jpg) +Figure 4: Illustration of planar embeddings for the formulas $\varphi _ { 1 }$ and $\varphi _ { 2 }$ for $n = 2$ . + +# A.3 DETAILS OF DATASET CONSTRUCTION + +There is an interesting universality result for functions defined on planar graphs. It is known that 3- WL can distinguish between any pair of planar graphs (Kiefer et al., 2019). Since 4-GCNs can simulate 3-WL, this implies that functions defined on planar graphs can be approximated by 4-GCNs. This result can be extended to much wider graph classes, including all graph classes excluding a fixed graph as a minor (Grohe, 2017). + +Inspired by this, we generate planar instances, and ensure that they can be distinguished by 2- WL, by carefully constraining these instances further. Hence, any GNN with 2-WL expressive power can approximate solutions to these planar instances. This, however, does not imply that these GNNs will solve EXP in practice, but only that an appropriate approximation function exists and can theoretically be learned. + +# A.3.1 CONSTRUCTION OF EXP + +We now explain the construction and composition of EXP. Fundamentally, EXP consists of two main components, (i) a pair of cores, which are non-isomorphic, planar, 1-WL indistinguishable, 2-WL distinguishable, and decide the satisfiability of every instance, and (ii) an additional randomly generated and satisfiable planar component, identically added to the core pair, to add variability to EXP and make learning more challenging. We first present both components, and then provide further details about graph encoding and planar embeddings. + +Core pair. In EXP, a core pair consists of two CNF formulas $\varphi _ { 1 } , \varphi _ { 2 }$ , both defined using $2 n$ variables, $n \in \mathbb { N } ^ { + }$ , such that $\varphi _ { 1 }$ is unsatisfiable and $\varphi _ { 2 }$ is satisfiable, and such that their graph encodings are 1-WL indistinguishable and planar. $\varphi _ { 1 }$ and $\varphi _ { 2 }$ are constructed using two structures which we refer to as variable chains and variable bridges respectively. + +A variable chain $\varphi _ { c h a i n }$ is defined over a set of $n \geq 2$ Boolean variables, and imposes that all variables be equally set. The variable chain can be defined in increasing or decreasing order over these variables. More specifically, given variables $x _ { i } , . . . , x _ { j }$ , + +$$ +\begin{array} { r l } & { \mathrm { C h a i n } _ { \mathrm { I n c } } ( i , j ) = \displaystyle \bigwedge _ { k = i } ^ { j - 1 } ( \bar { x _ { k } } \vee x _ { i + ( k + 1 ) } \% ( j - i + 1 ) ) , \mathrm { ~ a n d } } \\ & { \mathrm { C h a i n } _ { \mathrm { I n e c } } ( i , j ) = \displaystyle \bigwedge _ { k = i } ^ { j - 1 } ( x _ { k } \vee \bar { x } _ { i + ( k + 1 ) } \% ( j - i + 1 ) ) . } \end{array} +$$ + +Additionally, a variable bridge is defined over an even number of variables $x _ { 0 } , . . . , x _ { 2 n - 1 }$ , as + +$$ +\varphi _ { b r i d g e } = \bigwedge _ { i = 0 } ^ { n - 1 } { \big ( } ( x _ { i } \vee x _ { 2 n - 1 - i } ) \wedge ( { \bar { x } } _ { i } \vee { \bar { x } } _ { 2 n - 1 - i } ) { \big ) } . +$$ + +A variable bridge makes the variables it connects forcibly have opposite values, e.g., $x _ { 0 } = \bar { x _ { 1 } }$ for $n = 1$ . We denote a variable bridge over $x _ { 0 } , . . . , x _ { 2 n - 1 }$ as Bridge $( 2 n )$ . + +To get $\varphi _ { 1 }$ and $\varphi _ { 2 }$ , we define $\varphi _ { 1 }$ as a variable chain and bridge on all variables, yielding contrasting and unsatisfiable constraints. To define $\varphi _ { 2 }$ , we “cut” the chain in half, such that the first $n$ variables can differ from the latter $n$ , satisfying the bridge. The second half of the “cut” chain is then flipped to a decrementing order, which preserves the satisfiability of $\varphi _ { 2 }$ , but maintains the planarity of the resulting graph. More specifically, this yields: + +$$ +\begin{array} { r l } & { \varphi _ { 1 } = \mathbf { C h a i n } _ { \mathrm { I n c } } ( 0 , 2 n ) \wedge \mathbf { B r i d g e } ( 2 n ) , \mathrm { a n d } } \\ & { \varphi _ { 2 } = \mathbf { C h a i n } _ { \mathrm { I n c } } ( 0 , n ) \wedge \mathbf { C h a i n } _ { \mathrm { D e c } } ( n , 2 n ) \wedge \mathbf { B r i d g e } ( 2 n ) . } \end{array} +$$ + +Planar component. Following the generation of $\varphi _ { 1 }$ and $\varphi _ { 2 }$ , a disjoint satisfiable planar graph component $\varphi _ { \mathrm { p l a n a r } }$ is added. $\varphi _ { \mathrm { p l a n a r } }$ shares no variables or disjunctions with the cores, so is primarily introduced to create noise and make learning more challenging. $\varphi _ { \mathrm { p l a n a r } }$ is generated starting from random 2-connected (i.e., at least 2 edges must be removed to disconnect a component within the graph) bipartite planar graphs from the Plantri tool (Brinkmann et al., 2007), such that (i) the larger set of nodes in the graph is the variable set3, (ii) highly-connected disjunctions are split in a planaritypreserving fashion to maintain disjunction widths not exceeding 5, (iii) literal signs for variables are uniformly randomly assigned, and (iv) redundant disjunctions, if any, are removed. If this $\varphi _ { \mathrm { p l a n a r } }$ is satisfiable, then it is accepted and used. Otherwise, the formula is discarded and a new $\varphi _ { \mathrm { p l a n a r } }$ is analogously generated until a satisfiable formula is produced. + +Since the core pair and $\varphi _ { \mathrm { p l a n a r } }$ are disjoint, it is easy to deduce that the graph encoding of $\varphi _ { \mathrm { p l a n a r } } \wedge \varphi _ { 1 }$ and $\varphi _ { \mathrm { p l a n a r } } \wedge \varphi _ { 2 }$ are both planar and 1-WL indistinguishable. Furthermore, $\varphi _ { \mathrm { p l a n a r } } \wedge \varphi _ { 1 }$ is satisfiable, and $\varphi _ { \mathrm { p l a n a r } } \wedge \varphi _ { 2 }$ is not. Hence, the introduction of $\varphi _ { \mathrm { p l a n a r } }$ maintains all the desirable core properties, all while making any generated EXP dataset more challenging. + +The structural properties of the cores, combined with the combinatorial difficulty of SAT, make EXP a challenging dataset. For example, even minor formula changes, such as flipping a literal, can lead to a change in the SAT outcome, which enables the creation of near-identical, yet semantically different instances. Moreover, SAT is NP-complete (Cook, 1971), and remains so on planar instances (Hunt III et al., 1998). Hence, EXP is cast to be challenging, both from an expressiveness and computational perspective. + +Remark 3. Intuitively, $\varphi _ { 1 }$ and $\varphi _ { 2 }$ , generated as described, can be distinguished by 2-WL, as 2-WL can detect the break in cycles resulting from the aforementioned “cut”. In other words, 2-WL can identify that the chain has been broken in between these two formulas, and thus will return distinct colourings. Hence, $\varphi _ { 1 }$ and $\varphi _ { 2 }$ can be distinguished by 3-GCNs. + +Graph encoding. We use the following graph encoding, denoted by Enc: (i) Every variable is encoded by two nodes, representing its positive and negative literals, and connected by an edge, (ii) Every disjunction is represented by a node, and an edge connects a literal node to a disjunction node if the literal appears in the disjunction, and (iii) Variable and disjunction nodes are encoded with different types. We opt for this encoding, as it is commonly used in the literature (Selsam et al., 2019), and is sufficient, for the sake of our empirical evaluation, to yield planar encodings for EXP graph pairs. + +Planar embeddings for core pair. We show planar embeddings for $E n c ( \varphi _ { 1 } )$ and $E n c ( \varphi _ { 2 } )$ for $n = 2$ in Figure 4, and these embeddings can naturally be extended to any $n$ . $E n c ( \varphi _ { 1 } )$ and $E n c ( \varphi _ { 2 } )$ can also be shown to be 1-WL indistinguishable. This can be observed intuitively, as node neighborhoods in both graphs are identical and very regular: all variable nodes are connected to exactly one other variable node and two disjunction nodes, and all disjunction nodes are connected to exactly two variables. + +# A.3.2 CONSTRUCTION OF CEXP + +Given a EXP dataset with $N$ pairs of graphs, we create CEXP by selecting $N / 2$ graph pairs and modifying them to yield CORRUPT. The unmodified graph pairs are therefore exactly identical in type to EXP instances, and we refer to these instances within CEXP as $\overline { { \mathrm { E x p } } }$ . + +For every graph pair, we discard the satisfiable graph and construct a new graph from a copy of the unsatisfiable graph as follows. + +1. Randomly introduce new literals to the existing disjunctions of the copy of the unsatisfiable graph, such that no redundancies are created (i.e., adding $x$ to a disjunction when $x$ or $\bar { x }$ is already present), until 3 literals are added and the formula becomes satisfiable. Literal addition is done by creating new edges in the graph between disjunction and literal nodes. To do this, disjunctions with less than 5 literals are uniformly randomly selected, and the literal to add is uniformly randomly sampled from the set of all non-redundant literals given the selected disjunction. 2. Once a satisfiable formula is reached, iterate sequentially over all added edges, and eliminate any edge whose removal does not restore unsatisfiability. This ensures that a minimal number of new edges, relative to the original unsatisfiable graph, are added. + +Observe that these modifications have several interesting effects on the dataset. First, they preserve the existing UNSAT core nodes and edges, while flipping the satisfiability of their overall formulas, which makes the learning task go beyond structure identification. Second, they introduce significant new variability to the dataset, in that the planar component and cores can share edges. Finally, they make the graph pairs 1-WL distinguishable, which gives standard GNNs a chance to perform well on CORRUPT. + +# A.3.3 DATASET GENERATION FOR EXPERIMENTS + +To create the EXP dataset, we randomly generate 600 core pairs, where $n$ (cf. Appendix A.3) is uniformly randomly set between 2 and 4 inclusive. Then, we generate the additional planar component using Plantri, such that $5 0 0 \varphi _ { \mathrm { p l a n a r } }$ formulas are generated from 12-node planar bipartite planar graphs, and the remaining 100 from planar bipartite graphs with 15 nodes. + +This generation process implies that every formula has a number of variables ranging between 10 (4 core variables when $n = 2$ plus a minimum 6 variables from the larger bipartite set during $\varphi _ { 1 }$ planar generation from 12-node graphs) and 22 variables (8 core variables for $n = 4$ plus a maximally-sized variable subset of 14 nodes for $\varphi _ { \mathrm { p l a n a r } }$ generation from 15-node graphs). + +Furthermore, the number of disjunctions also ranges from 10 (8 core disjunctions for $n = 2$ plus the minimum 2 disjunctions for the case where $\varphi _ { \mathrm { p l a n a r } }$ , generated from 12-node graphs, has 10 variables and 2 disjunctions) to 30 disjunctions (16 core disjunctions for $n = 4$ plus at most 14 disjunctions for the case where $\varphi _ { \mathrm { p l a n a r } }$ , generated from 15-node graphs, initially has 8 variables and 7 disjunctions, which can at most lead to 14 final disjunctions following step (ii)). + +In this subsection, we investigate the variability of GCN-RNI learning across validation folds, and do so with a representative model and dataset, namely the semi-randomized GCN$5 0 \% \mathrm { R N I }$ model and the standard EXP dataset. The standard deviation of the test accuracy of $\mathrm { G C N } { - } 5 0 \% \mathrm { F }$ RNI over EXP, across all 10 cross-validation folds relative to the number of epochs, is shown in Figure 5. From this figure, we see that standard deviation spikes sharply at the start of training, and only begins dropping after 100 epochs. This suggests that the learning behavior of $G C N { - } 5 0 \%$ RNI is quite variable, sometimes requiring few epochs to converge, and in other cases requiring a very high number of epochs. Furthermore, standard deviation converges to almost zero following 200 epochs, corresponding to the phase where all + +![](images/6721e91c6ef6ad166351a12e8a09d04875d6a5590bdce0dbf328d56a14ad16f7.jpg) +Figure 5: Standard deviation of test accuracy over all 10 validation splits of GCN- $50 \%$ RNI on EXP. + +validation folds have achieved near-perfect test performance. From these findings, we further confirm that RNI introduces volatility to GCN training, this time manifesting in variable convergence times across validation folds, but that this volatility does not ultimately hinder convergence and performance, as all folds eventually reach satisfactory performance within a reasonable amount of epochs, and subsequently stabilize at this level. + +# A.5 ADDITIONAL EXPERIMENTS + +In addition to the experiments in the main body of the paper, we additionally evaluate RNI on sparser analog datasets to EXP and CEXP, namely SPARSEEXP and SPARSECEXP. These datasets only contain $2 5 \%$ of the number of instances of their original counterparts, and are used to study the behavior and impact of RNI when data is sparse. + +# A.5.1 EXPERIMENT 1 ON SPARSEEXP + +In this experiment, we generate SPARSEEXP analogously to EXP, except that this dataset only consists of 150 graph pairs, i.e., 300 graphs in total. We then train 3-GCN for 200 epochs, and all other systems for 1000 epochs on SPARSEEXP, as opposed to 100 and 500 respectively for EXP, to give all evaluated models a better opportunity to compensate for the smaller dataset size. We show the learning curves for all models on SPARSEEXP, and reproduce the original figure for EXP, in Figure 6 for easier comparison. + +First, we observe that all models converge slower on SPARSEEXP compared to EXP. This is not surprising, as a lower data availability makes learning a well-performing function slower and more challenging. More specifically, sparsity implies that (i) fewer weight updates are made per epoch, and (ii) these updates are of lower quality, as they are computed from a less representative and complete dataset. Nonetheless, the same relative convergence patterns between GCN-RNI models and 3-GCN are also visible in this setting, further highighting the increased convergence time required by GCN-RNI models. + +We also observe that all GCN-RNI models, though also eventually converging, do so in a more volatile fashion. Indeed, GCN-RNI models suffer from the sparseness of the dataset, as this makes them more sensitive to RNI. As a result, these models require more training to effectively learn robustness against RNI values, and learn this from a smaller sample set, increasing their variability further. Moreover, the nature of SPARSEEXP makes learning more difficult, as it fully relies on RNI for MPNNs to have a chance of achieving above-random performance, and thus encourages MPNNs to fit specific RNI values. Hence, RNI introduces significant volatility and variability to training, particularly with sparser data, and requires substantial training and epochs for GCN-RNI models to effectively develop a robustness to RNI instantiations. + +![](images/3e34f5c14f46d45fe3c84ed3ad0d37c02529c9ba1994fbc1b2a2e78da9469c34.jpg) + +![](images/d46c799afbb58cde215b1e1e6653e44f15833296971fcff521f1e04053e34a83.jpg) + +![](images/304db04ca525ded70671e1d8bb10d83ae0d8a2cb6de77ce8d9d70971cc2489a1.jpg) +Figure 6: Model convergence results for Experiment 1 on the datasets EXP and SPARSEEXP. + +![](images/5049be94ec77ea9ee403ae2c147fdec86bc0cad7769fd2e158608f1be76376c0.jpg) + +(a) Learning curves for all GCN-RNI models and 3-GCN on CEXP. + +(b) Learning curves for CEXP, split across EXP $\left( / \mathrm { E } \right)$ and CORRUPT (/C). + +![](images/0e3f99163a4ce7abb7da4ef05db4d2d7798c4b79c8ae66352eaa87f363b990de.jpg) +Figure 7: Model convergence results for Experiment 2 on CEXP and SPARSECEXP. + +![](images/580ee443cb99c23aa3dd0992f0ec654652fdf2b78035ea953af789aafe04f02a.jpg) + +(c) Learning curves for all GCN-RNI models and 3-GCN on SPARSECEXP. + +(d) Learning curves for SPARSECEXP, split across EXP (/E) and CORRUPT (/C). + +# A.5.2 EXPERIMENT 2 ON SPARSECEXP + +Analogously to Experiment 1, we generate a SPARSECEXP dataset similarly to CEXP, but only generate 150 graph pairs. Then, we select 75 graph pairs and modify them, as described in Appendix A.3.2. We report the learning curves for all models on SPARSECEXP, as well as the original curves for CEXP from the main body, in Figure 7. + +Table 2: Hyper-parameter configurations for all GCN-RNIand 3-GCN experiments. + +
DatasetEXPCEXP
入p入p
GCN1×10-4N/A1×10-4N/A
GCN-12.5%RNI2×10-4N2×10-4N
GCN-50%RNI2×10-4N2×10-4N
GCN-87.5%RNI2×10-4N5×10-4N
GCN-RNI5×10-4N5×10-4N
3-GCN5×10-4N/A2×10-4N/A
+ +Table 3: Performance of GCN-RNI models on the EXP dataset with the hyperbolic tangent activation function. + +
ModelTesting Accuracy (%)
GCN-RNI(U)92.7 ± 5.61
GCN-RNI(N)96.0 ± 2.11
GCN-RNI(XU)64.6 ± 19.9
GCN-RNI(XN)63.0 ± 20.9
+ +As in the previous subsection, similar behavior is observed on SPARSECEXP compared with CEXP, only differing by slower convergence in the former case. However, we note that the “struggle” phase described in the main paper, which only occurs during the first 100 epochs over CEXP, lasts for around 500 epochs on SPARSEEXP. Intuitively, this “struggle” phenomenon is due to conflicting learning requirements, stemming from CORRUPT and EXP, which effectively require models to “isolate” deterministic dimensions for CORRUPT, and other randomized dimensions for $\overline { { \mathrm { E x p } } }$ . This in itself is already challenging on CEXP, but is made even more difficult on SPARSECEXP due to its sparsity. Indeed, sparsity makes that further samples are needed in expectation to find a reasonable solution, leading to a lengthy “struggle” phase, in which both CORRUPT and $\overline { { \mathrm { E x p } } }$ data points conflict with one another during optimization. + +# A.6 HYPER-PARAMETER DETAILS + +All GCN models with (partially or completely) deterministic initial node embeddings map a 2- dimensional one-hot encoding of node type (literal or disjunction) to a $k$ -dimensional embedding space, where $k$ corresponds to the dimensionality of the deterministic embeddings. Furthermore, the final prediction for every graph is computed by aggregating all node embeddings following message passing using the max function, and then passing the result through a multi-layer perceptron of 3 layers with dimensionality $x$ , 32 and 2 respectively, where $x$ is the embedding dimensionality used in the given model. The activation function for the first two MLP layers is the ELU function (Clevert et al., 2016), and the softmax function is used to make a final prediction at the final MLP layer. + +All neural networks in this work are optimized using the Adam optimizer (Kingma & Ba, 2015). All training is conducted with a fixed learning rate $\lambda$ , for fairer comparison between all models. Initially, decaying learning rates were used, but these were discarded, as they yielded sub-optimal convergence for all GCN-RNI models. Finally, all experiments were run on a V100 GPU. Detailed hyper-parameters, namely learning rate $\lambda$ and RNI distribution $p$ , per model on every evaluation dataset are shown in Table 2. + +# A.6.1 RESULTS FOR GCN-RNI WITH HYPERBOLIC TANGENT ACTIVATION + +In addition to experimenting with the RNI probability distribution, we also experimented with different activation functions for the GCN message passing iterations. Results are shown in Table 3. Performance with tanh is significantly more variable across distributions than ELU, which shows that RNI can be highly sensitive to practical choices of hyper-parameters. \ No newline at end of file diff --git a/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_content_list.json b/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..fc3e7980191995c654fb32e4e5ac118aac7891cb --- /dev/null +++ b/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_content_list.json @@ -0,0 +1,2760 @@ +[ + { + "type": "text", + "text": "THE SURPRISING POWER OF GRAPH NEURAL NETWORKS WITH RANDOM NODE INITIALIZATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 184, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph neural networks (GNNs) are effective models for representation learning on graph-structured data. However, standard GNNs are limited in their expressive power, as they cannot distinguish graphs beyond the capability of the WeisfeilerLeman (1-WL) graph isomorphism heuristic. This limitation motivated a large body of work, including higher-order GNNs, which are provably more powerful models. To date, higher-order invariant and equivariant networks are the only models with known universality results, but these results are practically hindered by prohibitive computational complexity. Thus, despite their limitations, standard GNNs are commonly used, due to their strong practical performance. In practice, GNNs have shown a promising performance when enhanced with random node initialization (RNI), where the idea is to train and run the models with randomized initial node features. In this paper, we analyze the expressive power of GNNs with RNI, and pose the following question: are GNNs with RNI more expressive than GNNs? We prove that this is indeed the case, by showing that GNNs with RNI are universal, a first such result for GNNs not relying on computationally demanding higher-order properties. We then empirically analyze the effect of RNI on GNNs, based on carefully constructed datasets. Our empirical findings support the superior performance of GNNs with RNI over standard GNNs. In fact, we demonstrate that the performance of GNNs with RNI is often comparable with or better than that of higher-order GNNs, while keeping the much lower memory requirements of standard GNNs. However, this improvement typically comes at the cost of slower model convergence. Somewhat surprisingly, we found that the convergence rate and the accuracy of the models can be improved by using only a partial random initialization regime. ", + "bbox": [ + 232, + 265, + 764, + 597 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 622, + 336, + 637 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph neural networks (GNNs) (Scarselli et al., 2009; Gori et al., 2005) are neural architectures designed for learning functions over graph-structured data, and naturally encode desirable properties such as permutation invariance (resp., equivariance) relative to graph nodes, and node-level computation based on message passing between these nodes. These properties provide GNNs with a strong inductive bias, enabling them to effectively learn and combine both local and global graph features (Battaglia et al., 2018). As a result, GNNs have been applied to a multitude of tasks, ranging from protein classification (Gilmer et al., 2017) and synthesis (You et al., 2018), protein-protein interaction (Fout et al., 2017), and social network analysis (Hamilton et al., 2017), to recommender systems (Ying et al., 2018) and combinatorial optimization (Bengio et al., 2018; Selsam et al., 2019). ", + "bbox": [ + 174, + 652, + 823, + 777 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, popular GNN architectures, primarily based on message passing (MPNNs), are limited in their expressive power. In particular, MPNNs are at most as powerful as the Weisfeiler-Leman (1-WL) graph isomorphism heuristic (Morris et al., 2019; Xu et al., 2019), and thus cannot discern between several families of non-isomorphic graphs, e.g., sets of regular graphs (Cai et al., 1992). To address this limitation, alternative GNN architectures with provably higher expressive power than MPNNs have been proposed. These models, which we refer to as higher-order GNNs, are inspired by the more powerful generalization of 1-WL to $k$ −tuples of nodes, known as $k$ -WL (Grohe, 2017). These models are the only GNNs with an established universality result, but these models are computationally very demanding. As a result, MPNNs, despite their limited expressiveness, remain the standard GNN model for graph learning applications. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In a parallel development, MPNNs have recently achieved significant empirical improvements using random node initialization (RNI), through which initial graph node embeddings are randomly set. Indeed, RNI has enabled MPNNs to distinguish instances that 1-WL cannot distinguish, and is proven to enable better approximation of a class of combinatorial problems (Sato et al., 2020). However, the effect of RNI on the expressive power of GNNs has not yet been comprehensively studied, and its impact on the inductive capacity and learning ability of GNNs remains unclear. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we thoroughly study the impact of RNI on MPNNs. First, we prove that MPNNs enhanced with RNI are universal, in the sense that they can approximate every function defined on graphs of any fixed order. This follows from a logical characterisation of the expressiveness of MPNNs (Barcelo et al., 2020) combined with an argument on order-invariant definability. Our ´ result strongly contrasts with existing 1-WL limitations for deterministic MPNNs, and provides a foundation for developing very expressive and memory-efficient MPNN models. ", + "bbox": [ + 174, + 194, + 825, + 277 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To empirically verify our theoretical findings, we carry out a careful empirical study to quantify the practical impact of RNI. To this end, we design EXP, a synthetic dataset requiring 2-WL expressive power for models to achieve above-random performance, and run MPNNs with RNI on it, to observe how well and how easily this model can learn and generalize based on this dataset. Then, we propose CEXP, a modification of EXP with partially 1-WL distinguishable data, and evaluate the same questions in this more variable setting. Overall, the contributions of this paper are as follows: ", + "bbox": [ + 174, + 285, + 825, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "- We prove that MPNNs with RNI are universal, a significant improvement over the 1-WL limit of standard MPNNs and, to our knowledge, a first universality result for memory-efficient GNNs. \n- We introduce two carefully designed datasets, EXP and CEXP, based on graph pairs only distinguishable by 2-WL or higher, to rigorously evaluate the impact of RNI. \n- Using these datasets, we thoroughly analyze the effects of RNI on MPNN, and observe that (i) MPNNs with RNI can closely match the performance of higher-order GNNs, (ii) the improved performance of MPNNs with RNI comes at the cost of slower convergence (compared to higherorder GNNs), and (iii) using a partial random initialization regime over node features typically improves convergence rate and the accuracy of the models. \n- We additionally perform the same experiments with analog, sparser datasets, with longer training, and observe similar behavior, but more volatility. ", + "bbox": [ + 171, + 381, + 826, + 553 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 GRAPH NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 580, + 444, + 595 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Graph neural networks (GNNs) (Gori et al., 2005; Scarselli et al., 2009) are neural architectures dedicated to learning functions over graph-structured data. In a GNN, nodes in the input graph are assigned vector representations, which are updated iteratively through series of invariant or equivariant computational layers. We recall message passing neural networks (MPNNs) (Gilmer et al., 2017), a popular family of GNN models, and its expressive power in relation to the WeisfeilerLeman graph isomorphism heuristic. We discuss alternative GNN models in Section 3; for a broader coverage, we refer the reader to the literature (Hamilton, 2020). ", + "bbox": [ + 173, + 612, + 825, + 710 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In MPNNs, node representations aggregate messages from their neighboring nodes, and use this information to iteratively update their representations. Formally, given a node $x$ , its vector representation $v _ { x , t }$ at time $t$ , and its neighborhood $N ( x )$ , a message passing update can be written as: ", + "bbox": [ + 174, + 717, + 823, + 760 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/1891c78d9e9eba3f5c35eee8496bd9ba075bd2b9f17c08536b623b5ef9d89f17.jpg", + "text": "$$\nv _ { x , t + 1 } = c o m b i n e { \\Big ( } v _ { x , t } , a g g r e g a t e { \\big ( } \\{ v _ { y , t } | y \\in N ( x ) \\} { \\big ) } { \\Big ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 315, + 768, + 681, + 794 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where combine and aggregate are functions, and aggregate is typically permutation-invariant. Once message passing is complete, the final node representations are then used to compute target outputs. Prominent message passing GNN architectures include graph convolutional networks (GCNs) (Kipf & Welling, 2017) and gated graph neural networks (GGNNs) (Li et al., 2016). ", + "bbox": [ + 174, + 801, + 825, + 858 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "It is well-known that standard MPNNs have the same power as the 1-dimensional Weisfeiler-Leman algorithm (1-WL) (Xu et al., 2019; Morris et al., 2019). This entails that two nodes in a graph cannot be distinguished if 1-WL does not distinguish them, and neither can two graphs be distinguished if 1-WL cannot distinguish them. ", + "bbox": [ + 174, + 864, + 823, + 921 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Consider the graphs $G$ and $H$ shown in Figure 1. 1-WL cannot distinguish any two nodes of the graph $G$ . Thus, for example, the invariant function $f : V ( G ) \\to \\mathbb { R }$ that maps all nodes in the 4-cycle of $G$ to 1 and all nodes in the triangle to 0 is not expressible (or approximable) by a GNN. Moreover, 1-WL cannot distinguish the graphs $G , H$ , even though they are obviously non-isomorphic, so no classifier based on the node embeddings computed by an MPNN can distinguish the two graphs. ", + "bbox": [ + 174, + 103, + 535, + 229 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/aa77d9397242894aa279436cb0ba0d6163929d466561f5dcda8966db0ef5a676.jpg", + "image_caption": [ + "Figure 1: $G$ and $H$ are indistinguishable by 1-WL and hence by (1-WL) GNNs. " + ], + "image_footnote": [], + "bbox": [ + 566, + 106, + 812, + 189 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A somewhat trivial limitation in the expressiveness of MPNNs is that information is only propagated along edges, and hence can never be shared between distinct connected components of a graph (Barcelo et al., 2020; Xu et al., 2019). An easy way to overcome this limitation is by adding ´ global readouts, that is, permutation-invariant functions that aggregate the current states of all nodes1. Throughout the paper, we therefore focus on MPNNs with global readouts (also called aggregatecombine GNNs with global readout, i.e., ACR-GNNs (Barcelo et al., 2020)). ´ ", + "bbox": [ + 174, + 236, + 825, + 319 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 RELATED WORK & MOTIVATION ", + "text_level": 1, + "bbox": [ + 176, + 338, + 478, + 354 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Developing more expressive GNNs is an active research area due to the prominence of GNNs for relational learning (Hamilton et al., 2017) and combinatorial optimization (Bengio et al., 2018). As mentioned earlier, standard GNN models are at most as expressive as 1-WL (Morris et al., 2019; Xu et al., 2019), and thus cannot distinguish between non-isomorphic input instances. In this section, we describe theoretical results quantifying the expressive power of existing GNNs. ", + "bbox": [ + 174, + 369, + 825, + 440 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Higher-order GNNs. We recall the following families of higher-order GNN models: ", + "bbox": [ + 176, + 446, + 728, + 462 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "- Invariant (resp., equivariant) graph networks: Invariant (resp., equivariant) graph networks (Maron et al., 2019b) represent graphs as a tensor where node adjacency is directly encoded, and implicitly pass information between nodes through invariant (resp., equivariant) computational blocks. Hence, these models are themselves invariant (resp., equivariant), a desirable property for computations on graphs. Following intermediate blocks, higher-order tensors are typically returned, and the order of these tensors correlates directly with the expressive power of the overall model. Indeed, invariant networks (Maron et al., 2019c), and later equivariant networks (Keriven & Peyre, 2019), are shown to be universal, but with tensor orders of ´ $\\dot { O } ( | V | ^ { 2 } )$ , where $| V |$ denotes the number of graph nodes. Furthermore, invariant (resp., equivariant) networks with intermediate tensor order $k$ are shown to be equivalent in power to $\\left( k - 1 \\right)$ -WL (Maron et al., 2019a), which is strictly more expressive as $k$ increases (Cai et al., 1992). Therefore, such universal higher-order models require intractably-sized intermediate tensors in practice. ", + "bbox": [ + 173, + 472, + 825, + 638 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "- Higher-order MPNNs: The $k { \\mathrm { - } } \\mathbf { W } \\mathbf { L }$ hierarchy has been directly emulated in GNNs, such that these models learn embeddings for tuples of nodes, and perform messaging passing between them, as opposed to individual nodes. This approach has yielded models such as $k$ -GNNs (Morris et al., 2019). $k$ -GNNs have $\\left( k - 1 \\right)$ -WL expressive power,2 but need $O ( | V | ^ { k } )$ memory to run, leading to excessive memory requirements. ", + "bbox": [ + 174, + 642, + 825, + 712 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "- Provably powerful graph networks (PPGNs): PPGN is an invariant GNN (Maron et al., 2019a), based on “blocks” of multilayer perceptrons (MLPs) and matrix multiplication, which theoretically has 2-WL expressive power, and only requires memory $O ( | V | ^ { 2 } )$ (compared to $O ( | V | ^ { 3 } )$ for 3-GNNs). However, PPGN theoretically requires exponentially many samples in the number of graph nodes to learn necessary functions for 2-WL expressiveness (Puny et al., 2020). ", + "bbox": [ + 174, + 715, + 825, + 785 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "GNNs with random node initialization. MPNNs have been enhanced with random node initialization (Sato et al., 2020), such that the model trains and runs with partially randomized initial node features. These models, denoted rGNNs, are shown to near-optimally approximate solutions to specific combinatorial optimization problems, and are able to distinguish between 1-WL indistinguishable graph pairs. rGNNs can also detect characteristic sub-graphs in an input graph with high probability. Nonetheless, it remains open as to how much expressive power is exactly gained through RNI, and, in general, whether a GNN model that is universal, scalable, and structure-preserving, can be developed. Our work strongly improves the theoretical result of Sato et al. (2020), as it shows universality of MPNNs with RNI, and thus that arbitrary real-valued functions over graphs can be learned by MPNNs with the help of RNI. On the empirical side, we highlight the power of RNI in a significantly more challenging setting than rGNN, using a target function (SAT) beyond their theoretical scope. Indeed, for SAT, approximation is known to be hard, and fixed local structures are not useful for prediction. ", + "bbox": [ + 174, + 796, + 821, + 824 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 256 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Similar work to RNI has also been conducted in terms of randomly adding features from a predetermined set of colors (Dasoulas et al., 2020) to disambiguate between nodes. This model, known as CLIP, is similar in spirit to RNI, in that it introduces randomness to node representations, but explicitly makes graphs distinguishable by construction. By contrast, we study random features produced by RNI, which (i) are not designed a priori to distinguish nodes, (ii) do not explicitly introduce a fixed underlying structure, and (iii) yield potentially infinitely many representations for a single graph. In this more general setting, we nonetheless show that RNI adds expressive power to distinguish between nodes with high probability, leads to a universality result, and performs strongly in challenging problem settings. ", + "bbox": [ + 174, + 263, + 825, + 388 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 RANDOM NODE INITIALIZATION MAKES GNNS UNIVERSAL ", + "text_level": 1, + "bbox": [ + 174, + 410, + 705, + 425 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We present the main result of the paper, showing that random node initialization significantly increases the expressiveness and makes MPNNs universal, in a natural sense. Our work is a first positive result for the universality of MPNNs. This result is not based on a new model, but rather on random initialization of node features, which is widely used in practice, and in this respect, it also serves as a theoretical justification for models that are successfully employed in practice. ", + "bbox": [ + 174, + 441, + 825, + 511 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It may appear somewhat surprising, and even counter-intuitive, that randomly initializing node features, on its own, would deliver such a gain in expressiveness. In fact, on the surface, random initialization no longer preserves the invariance of MPNNs, since the result of the computation of an MPNN with RNI not only depends on the structure (i.e., the isomorphism type) of the input graph, but also on the random initialization. The broader picture is, however, rather subtle, as we can view such a model as computing a random variable (or as generating an output distribution), and this random variable would still be invariant. This means that the outcome of the computation of an MPNN with RNI does still not depend on the specific representation of the input graph, which fundamentally maintains invariance. Indeed, random features vary around a mean which, in expectation, will inform GNN predictions, and is identical across all nodes as randomization is i.i.d. However, the variability between different samples, and the variability of a random sample relative to this mean, enable graph discrimination and improve expressiveness. Hence, in expectation, all samples over training and evaluation fluctuate around a unique value, preserving invariance, whereas single-sample variance achieves the improved expressiveness. ", + "bbox": [ + 174, + 518, + 825, + 713 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Formally, let ${ \\mathcal { G } } _ { n }$ be the class of all $n$ -vertex graphs, i.e., graphs that consist of at most $n$ vertices, and let $f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }$ . We say that $f$ is invariant if for isomorphic graphs $G , H \\ \\in \\ G _ { n }$ it holds that $f ( G ) = f ( H )$ . We say that a randomized function $\\mathcal { X }$ that associates with every graph $G \\in \\mathcal G _ { n }$ a random variable $\\mathcal { X } ( G )$ is an $( \\epsilon , \\delta )$ -approximation of $f$ if for all $G \\in \\mathcal G _ { n }$ it holds that $\\operatorname* { P r } \\left( | f ( G ) - \\mathcal { X } ( G ) | \\leq \\epsilon \\right) \\geq 1 - \\delta$ . Note that MPNNs with RNI compute functions $\\mathcal { X }$ of this type. If $\\mathcal { X }$ is computed by an MPNN $\\mathcal { N }$ with RNI, we say that $\\mathcal { N } \\left( \\epsilon , \\delta \\right)$ -approximates $f$ . ", + "bbox": [ + 174, + 719, + 825, + 808 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 4.1 (Universal approximation). Let $n \\geq 1$ , and let $f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }$ be invariant. Then, for all $\\epsilon , \\delta > 0$ , there is a MPNN with RNI that $( \\epsilon , \\delta )$ -approximates $f$ . ", + "bbox": [ + 173, + 811, + 823, + 840 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For ease of presentation, we state the theorem only for real-valued functions, but it can be extended to equivariant functions defined on the nodes of a graph (that is, functions $f$ mapping graphs $G$ to vectors $\\pmb { x } \\in \\mathbb { R } ^ { V ( G ) }$ with the property that for every permutation $\\pi$ of $V ( G )$ it holds that $f ( G ^ { \\pi } ) = f ( G ) ^ { \\pi } )$ and even higher-order equivariant functions. The result can also be extended to weighted graphs, but then the function $f$ that we approximate needs to be continuous. ", + "bbox": [ + 174, + 852, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To prove Theorem 4.1, we first show that MPNNs with RNI can capture arbitrary Boolean functions, by building on the result of Barcelo et al. (2020), which states that any logical sentence in ´ $\\mathsf { C } ^ { 2 }$ can be captured by an MPNN with global readout. The logic $\\complement$ is the extension of first-order predicate logic using counting quantifiers of the form $\\exists ^ { \\geq k } x$ for $k \\geq 0$ , where $\\exists ^ { \\geq k } x \\varphi ( x )$ means that there are at least $k$ elements $x$ satisfying $\\varphi$ , and $\\mathsf { C } ^ { 2 }$ is the two-variable fragment of $\\mathrm { c }$ (see appendix for further details). We establish that any graph with identifying node features, which we call individualized graphs, can be represented by a sentence in $\\mathsf { C } ^ { 2 }$ . Then, we extend this result to sets of individualized graphs, and thus to Boolean functions mapping these sets to True, by showing that these functions are represented by a $\\mathsf { C } ^ { 2 }$ sentence, namely the disjunction of all constituent graph sentences. Following this, we provide a construction with node embeddings based on random node initialization, and show that, with high probability, RNI makes the input graph individualized. Thus, with high probability, RNI makes that MPNNs learn a Boolean function over individualized graphs. Since all such functions can be captured by a sentence in $\\mathsf { C } ^ { 2 }$ , and an MPNN with a global readout can capture any Boolean function by Barcelo et al. (2020), we conclude that MPNNs with RNI can capture arbitrary ´ Boolean functions. Finally, the result is extended to real-valued functions via a natural mapping, yielding universality. ", + "bbox": [ + 174, + 103, + 825, + 337 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The concrete implications of Theorem 4.1 can be summarized as follows: First, MPNNs enhanced with RNI are able to distinguish individual graphs, already with an embedding dimensionality polynomial in the inverse of desired confidence $\\delta$ , namely $O ( n ^ { 2 } \\delta ^ { - 1 } )$ , where $n$ is the number of graph nodes. Second, our universality results holds also with partial RNI. More specifically, it already holds with only one randomized dimension. Third, although Theorem 4.1 can potentially result in very large constructions, we note that it is very adaptive and tightly linked to the descriptive complexity of the function that we want to approximate. That is, for a more restricted class of functions, there may be more efficient constructions, and our proof does not rely on a particular construction. This deserves a more thorough investigation, which we leave for future work. Finally, our construction provides a logical characterization of the power of MPNNs with RNI, and substantiates the means through which randomization yields expressiveness improvements. This construction therefore also serves as a basis for a more logically grounded theoretical study of randomized MPNN models, based on particular architectural or parametric choices. ", + "bbox": [ + 174, + 343, + 825, + 523 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Similarly to other universality results, Theorem 4.1 can potentially result in very large constructions. This is a simple consequence of the general nature of such universality results: Theorem 4.1 applies to families of functions, describing problems of arbitrary computational complexity, including problems that are computationally hard (even to approximate). Thus, a practically more relevant aspect is to empirically verify the formal statement, and test the capacity of MPNNs with RNI, in comparison to higher-order GNNs. Higher-order GNNs typically suffer from prohibitive space requirements, which is not the case for MPNNs with RNI, and this already makes them more viable in practice. As we discuss later in detail, our experiments demonstrate that MPNNs with RNI indeed combine expressiveness with efficiency in practice. ", + "bbox": [ + 174, + 531, + 825, + 656 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 DATASETS FOR EVALUATING THE EXPRESSIVE POWER OF GNNS ", + "text_level": 1, + "bbox": [ + 176, + 684, + 745, + 702 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "GNN models are evaluated on prominent real-world datasets, such as IMDB, TU, and Proteins (Kersting et al., 2016). These datasets are not tailored for evaluating the expressive power of GNNs, as they do not contain instances or edge cases requiring expressiveness beyond 1-WL. In fact, higherorder models only marginally outperform MPNNs on these datasets (Maron et al., 2019a; Dwivedi et al., 2020), which further highlights their unsuitability for expressiveness evaluation. ", + "bbox": [ + 174, + 722, + 825, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We develop the datasets EXP and CEXP. EXP is designed to explicitly evaluate the expressiveness of GNN models, and consists of a set of graph instances $\\left\\{ { G _ { 1 } \\dots { \\bar { G } } _ { n } , \\dot { H } _ { 1 } \\dots { \\cal H } _ { n } } \\right\\}$ , such that each instance is a graph encoding of a propositional formula. The classification task is to determine whether the formula is satisfiable (SAT). Each pair $( G _ { i } , H _ { i } )$ respects the following properties: (i) $G _ { i }$ and $H _ { i }$ are non-isomorphic, (ii) $G _ { i }$ and $H _ { i }$ have different SAT outcomes, that is, $G _ { i }$ encodes a satisfiable formula, while $H _ { i }$ encodes an unsatisfiable formula, (iii) $G _ { i }$ and $H _ { i }$ are 1-WL indistinguishable, so are guaranteed to be classified in the same way by standard MPNNs, and (iv) $G _ { i }$ and $H _ { i }$ are 2-WL distinguishable, so can be classified differently by higher-order GNNs. Thanks to these properties, we can explicitly compare the performance of MPNNs with RNI to these higher-order models. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Ensuring these properties in EXP is highly non-trivial, and the construction of this dataset is cumbersome. Fundamentally, every $( G _ { i } , H _ { i } )$ is carefully constructed on top of a basic building block, the core pair, such that the 2 cores underlie $G _ { i }$ and $H _ { i }$ , respectively. In this core pair, both cores are based on propositional clauses, such that one core is satisfiable and the other is not, and that these cores exclusively determine the satisfiability of $G _ { i }$ (resp., $H _ { i }$ ) and have graph encodings enabling all aforementioned properties. Core pairs, and their resulting graph instances in EXP are planar and are also carefully constrained to ensure they are 2-WL distinguishable. Hence, core pairs are key substructures within EXP, and distinguishing these cores is essential for good performance. ", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Building on EXP, CEXP includes instances with varying expressiveness requirements. Specifically, CEXP is a standard EXP dataset where $50 \\%$ of all satisfiable graph pairs are modified, such that they become 1-WL distinguishable from their unsatisfiable counterparts, only differing from these by a small number of added edges. Hence, CEXP consists of $50 \\%$ “corrupted” data, which can be distinguished and learned by a standard MPNN model (1-WL), which we label CORRUPT, and $50 \\%$ unmodified data, generated analogously to EXP, and requiring expressive power beyond 1- WL, which we refer to as $\\overline { { \\mathrm { E x p } } }$ . Thus, CEXP contains the same core structures as EXP, but these now lead to different SAT values in $\\overline { { \\mathrm { E x P } } }$ and CORRUPT, and this makes the overall learning task more challenging than learning $\\overline { { \\mathrm { E x p } } }$ or CORRUPT in isolation. Complete details of the overall data generation process for both datasets can be found in the appendix. ", + "bbox": [ + 174, + 222, + 825, + 363 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 EXPERIMENTAL EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 383, + 452, + 400 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we first evaluate the practical effect of RNI on MPNN expressiveness based on EXP, and compare MPNN-RNI against established higher-order GNNs. We then extend our empirical analysis to CEXP. Both experiments are conducted using the following models: ", + "bbox": [ + 176, + 412, + 823, + 455 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "- 1-WL GCN (1-GCN): A GCN with 8 distinct message passing iterations, ELU non-linearities (Clevert et al., 2016), 64-dimensional embeddings, and deterministic learnable initial node embeddings indicating node type. This model is guaranteed to achieve $50 \\%$ accuracy on EXP. ", + "bbox": [ + 174, + 467, + 823, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "- GCN - Random node initialization (GCN-RNI): An analogous model to 1-GCN with an identical architecture, enhanced with RNI. We evaluate this model with four initialization distributions, namely the standard normal distribution $\\mathcal { N } ( 0 , 1 )$ (N), the uniform distribution over $[ - 1 , 1 ]$ (U), Xavier normal (XN), and the Xavier uniform distribution (XU) (Glorot & Bengio, 2010). We denote the respective models $\\mathbf { G C N - R N I } ( D )$ , where $D \\in \\{ \\mathrm { N , U , X N , X U } \\}$ . ", + "bbox": [ + 174, + 511, + 825, + 582 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "- GCN - Partial Random node initialization $( \\mathbf { G C N - } x \\% \\mathbf { R N I } )$ : A GCN-RNI model, where only a percentage $x$ of initial node embedding dimensions are randomized. That is, for $d$ -dimensional node embeddings, GCN- $x \\%$ RNI randomizes $\\lfloor \\frac { x d } { 1 0 0 } \\rfloor$ dimensions, and sets the remaining dimensions deterministically from input features, namely, a one-hot representation of the two possible node types (literal and disjunction) in the input graph representation (see appendix for more details). We set $x$ to the extreme values 0 and $100 \\%$ , $50 \\%$ , as well as near-edge cases of $8 7 . 5 \\%$ and $12 . 5 \\%$ , respectively. ", + "bbox": [ + 174, + 585, + 825, + 684 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "- Provably Powerful Graph Network (PPGN) (Maron et al., 2019a): A higher-order GNN with 2-WL expressive power, and which requires quadratic memory relative to the number of graph nodes. We set up PPGN with eight 400-dimensional computational blocks. ", + "bbox": [ + 174, + 686, + 825, + 729 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "- 1-2-3-GCN-L: A higher-order GNN (Morris et al., 2019) emulating 2-WL on 3-tuples of nodes. 1-2-3-GCN-L operates at increasingly coarse node granularities, starting with single nodes and rising to 3-tuples. The model first computes all possible 3-tuples of nodes, then represents them as standard graph nodes. These nodes are connected to one another following the 2-WL neighborhood definition, i.e., tuples that exchange messages in 2-WL are connected by an edge in 3-GCN. 3-GCN-L implements a connected relaxation of 2-WL, in that only 3-tuples forming a connected graph are used, which comes at the cost of some theoretical guarantees. Nonetheless, the computation and representation of all tuples still imposes a severe overhead relative to MPNNs. We set up 1-2-3-GCN-L with 64-dimensional embeddings, 3 message passing iterations at level 1, 2 at level 2 and 8 at level 3. ", + "bbox": [ + 176, + 733, + 825, + 872 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "- 3-GCN: A modification to 1-2-3-GCN-L, such that (i) only the 3rd level is used, and (ii) the full 2-WL procedure is implemented, i.e., all 3-tuples are computed, as in standard 2-WL, rather than only the connected ones. ", + "bbox": [ + 176, + 876, + 825, + 917 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.1 EXPERIMENT 1: HOW DOES RNI IMPROVE MPNN EXPRESSIVENESS? ", + "bbox": [ + 174, + 103, + 696, + 117 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this experiment, we evaluate GCNs using different RNI settings on EXP, and compare with standard GNNs and higher-order models. Specifically, we generate an EXP dataset consisting of 600 graph pairs, and discuss this generation in more detail in the appendix. Then, we evaluate all models on EXP using 10-fold cross-validation. We train 3-GCN for 100 epochs per fold, and all other systems for 500 epochs. Mean test accuracy across all validation folds is measured and reported. ", + "bbox": [ + 173, + 131, + 825, + 202 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Full test accuracy results for all models are reported in Table 1, and model convergence for 3-GCN and all GCN-RNI models are shown in Figure 2. In line with Theorem 4.1, GCN-RNI achieves a near-perfect performance on EXP, substantially surpassing $50 \\%$ . Indeed, all fully randomized GCN-RNI models achieve a performance above $9 5 \\%$ with all four RNI distributions. This finding supports observations made in related studies on RNI (Sato et al., 2020), which suggest that RNI enables (sub)structure detection beyond the theoretical limits of 1-WL. Empirically, we observed that GCN-RNI is highly sensitive to changes in learning rate, activation function, and/or randomization distribution, and required delicate tuning to achieve its best performance. ", + "bbox": [ + 176, + 209, + 549, + 402 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/8e39b157b9b1c4e26382f2283262d684738bb36f6ae20784dfe694cbe3ee2387.jpg", + "table_caption": [ + "Table 1: Accuracy results on EXP. " + ], + "table_footnote": [], + "table_body": "
ModelTest Accuracy (%)
GCN-RNI(U)97.3 ± 2.55
GCN-RNI(N) GCN-RNI(XU)98.0 ± 1.85 97.0 ± 1.43
GCN-RNI(XN)96.6 ± 2.20
PPGN50.0
1-2-3-GCN-L50.0
3-GCN99.7 ± 0.004
", + "bbox": [ + 571, + 251, + 808, + 398 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Surprisingly, PPGN does not achieve performance above $50 \\%$ , despite being theoretically 2-WL expressive. Essentially, PPGN learns an approximation of 2-WL, based on power-sum multisymmetric polynomials (PMP), but fails to distinguish EXP graph pairs, despite extensive training. This suggests that PPGNs struggle to learn the required PMPs, and we could not improve these results, both for training and testing, with hyperparameter tuning. Furthermore, as mentioned in Section 3, PPGN requires exponentially many data samples in the size of the input graph (Puny et al., 2020) for learning. Hence, PPGN is likely struggling to discern between EXP graph pairs due to the smaller sample size and variability of the dataset. 1-2-3-GCN-L also only achieves $50 \\%$ accuracy, which can be attributed to theoretical model limitations. Indeed, the local and connected algorithm drops necessary information to distinguish graph pairs, as it only considers 3-tuples of nodes that form a connected sub-graph. Thus, 1-2-3-GCN-L discards disconnected 3-tuples that crucially are where the difference between the EXP cores lies. This further highlights the difficulty of EXP instances, as even a relaxation of 2-WL costs the model the ability to achieve above-random performance. Note that 3-GCN achieves near-perfect performance, as it explicitly has the sufficient theoretical power needed for the task, irrespective of learning constraints, and must only learn appropriate injective aggregation functions for neighbor aggregation (Xu et al., 2019). ", + "bbox": [ + 174, + 410, + 825, + 632 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/bc876dd422d8f56b42cd4c08770bee5dbcdab007b0466e2888f1f7d72dd04bae.jpg", + "image_caption": [ + "Figure 2: Learning curves on EXP. " + ], + "image_footnote": [], + "bbox": [ + 186, + 657, + 436, + 827 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In terms of model convergence, we observe that 3- GCN converges significantly faster than all GCNRNI models, for all randomization percentages. Indeed, 3-GCN only requires about 10 epochs to achieve its optimal performance, whereas GCN-RNI models all require in excess of 100 epochs. The rp Intuitively, the slower convergence of GCN-RNI can be attributed to a significantly harder learning task compared to 3- GCN: Whereas 3-GCN must learn from a deterministic set of node embeddings, and is naturally capable of discerning between dataset cores, GCN-RNI relies on RNI to discern between data points in EXP, via an artificial node ordering. This in turn implies that GCNRNI must first leverage RNI to detect structure, then subsequently learn robustness against the variability of RNI, which makes the learning task for GCN-RNI especially challenging. ", + "bbox": [ + 468, + 638, + 823, + 875 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our findings suggest that RNI can practically improve the expressiveness of MPNNs, and make them competitive with higher-order models, despite being significantly less demanding computationally. Indeed, for a typical EXP instance with 50 nodes, GCN-RNI only requires 3200 parameters (using ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/9d002f0cbe123c2e07395bc5988e6354ce82697cff2e0ea6e3e87615d2e764cd.jpg", + "image_caption": [ + "Figure 3: Model convergence results for Experiment 2 on CEXP on all models. " + ], + "image_footnote": [], + "bbox": [ + 205, + 102, + 462, + 301 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1b03f6be89d70f63c8d48b76ba763b79bb656d809a6afe63060e3a9ab29cd7ff.jpg", + "image_caption": [ + "(b) Learning curves for CEXP, split across EXP (/E) and CORRUPT (/C). " + ], + "image_footnote": [], + "bbox": [ + 534, + 102, + 789, + 301 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(a) Learning curves for all GCN-RNI models and 3-GCN on CEXP. ", + "bbox": [ + 184, + 308, + 485, + 333 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "64-dimensional embeddings), whereas 3-GCN requires 1,254,400 parameters. Nonetheless, GCNRNI performs comparably to 3-GCN, and, unlike the latter model, can easily scale to larger instances exceeding the range used in our datasets. This increase in expressive power, however, comes at the cost of slower convergence. Even so, RNI proves to be a promising direction for building scalable yet powerful MPNNs. ", + "bbox": [ + 174, + 388, + 825, + 458 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 EXPERIMENT 2: HOW DOES RNI AFFECT MPNN ON MORE VARIABLE DATASETS? ", + "text_level": 1, + "bbox": [ + 174, + 474, + 779, + 489 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Experiment 1, we observed that RNI practically improves the expressive power of GCNs over EXP. However, EXP is solely designed for expressiveness evaluation, and this leaves multiple questions open: How does RNI impact learning when data contains instances with varying expressiveness requirements, and how does RNI affect model generalization on more variable datasets? We experiment with CEXP to explicitly address these questions. ", + "bbox": [ + 174, + 501, + 823, + 570 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Analogously to Experiment 1, we generate an EXP dataset with 600 pairs of graphs. Then, we create CEXP by selecting 300 graph pairs and modifying their satisfiable graph, yielding CORRUPT. CEXP is well-suited for evaluating the efficacy of RNI more holistically, as it allows (i) the evaluation of the contribution of RNI on EXP conjointly with a second learning task on CORRUPT involving very similar core structures, and (ii) a study of the effect of different degrees of randomization on overall and subset-specific model performance. ", + "bbox": [ + 174, + 578, + 825, + 662 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this experiment, we train GCN-RNI (with varying randomization degrees) and 3-GCN on CEXP, and compare their test accuracy across all cross-validation splits. For GCN-RNI models, we observe the effect of RNI on specifically learning $\\overline { { \\mathrm { E x p } } }$ and CORRUPT, and the interplay between these two tasks. In all experiments, we exclusively use normal distribution initialization, given its strong performance in Experiment 1. ", + "bbox": [ + 174, + 670, + 825, + 741 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The learning curves of all GCN-RNI and 3-GCN on CEXP are shown in Figure 3a, and the same curves for the EXP and CORRUPT subsets are shown in Figure 3b. As on EXP, we observe that 3- GCN converges very quickly, exceeding $90 \\%$ test accuracy within 25 epochs on CEXP. By contrast, GCN-RNI, for all randomization levels, converges much slower, around after 200 epochs, despite the small size of input graphs ( $\\mathord { \\sim } 7 0$ nodes at most). Furthermore, fully randomized GCN-RNI performs worse than partly randomized GCN-RNI models, particularly on CEXP, due to its weak performance on CORRUPT, as shown in Figure 3b. ", + "bbox": [ + 173, + 747, + 825, + 847 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "First, we observe that partial randomization can significantly improve model performance. This can clearly be seen on CEXP, in Figure 3a and Figure 3b, where GCN- $12 . 5 \\%$ RNI and GCN-87.5%RNI achieve the best performance, by far outperforming GCN-RNI, which struggles on CORRUPT. This can be attributed to having a better inductive bias than a fully randomized model. Indeed, GCN$1 2 . 5 \\% \\mathrm { R N I }$ has mostly deterministic node embeddings, which simplifies learning over CORRUPT. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "This also applies to $\\mathrm { G C N - } 8 7 . 5 \\% \\mathrm { R N }$ I, where the number of deterministic dimensions, though small, remains sufficient for learning over CORRUPT. Both models also benefit from randomization to perform strongly on $\\overline { { \\mathrm { E x p } } }$ , and have sufficient randomization to perform similarly to a fully randomized GCN. GCN- $1 2 . 5 \\% \\mathrm { R N I }$ and $\\mathrm { G C N - } 8 7 . 5 \\% \\mathrm { R l }$ NI effectively achieve the best of both worlds on CEXP, leveraging inductive bias from deterministic node embeddings, while harnessing the power of random embeddings to perform strongly on $\\overline { { \\mathrm { E x p } } }$ . This is best shown in Figure 3b, where standard GCN fails to learn $\\overline { { \\mathrm { E x p } } }$ , fully randomized GCN-RNI struggles to learn CORRUPT, and the semi-randomized $G C N { - } 5 0 \\% \\mathrm { R l }$ NI achieves perfect performance on both subsets. Overall, this is a surprising finding, as it suggests that MPNNs can perform significantly better with partial, and even small, amounts of randomization. ", + "bbox": [ + 174, + 103, + 825, + 246 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Second, we observe that the fully randomized GCN-RNI performs substantially worse than its partially randomized counterparts. Whereas fully randomized GCN-RNI only performs marginally worse on EXP (cf. Figure 2) than partially randomized models, this gap is very large on CEXP, primarily due to CORRUPT. This observation concurs with the earlier idea of inductive bias: Fully randomized GCN-RNI loses all node type information, which is valuable for making robust and consistent decisions, and therefore struggles to match 3-GCN and partially randomized models. Indeed, the model fails to achieve even $60 \\%$ accuracy on CORRUPT, where other models are near perfect, and also relatively struggles on $\\overline { { \\mathrm { E x p } } }$ , only reaching $91 \\%$ accuracy and converging slower. ", + "bbox": [ + 174, + 253, + 825, + 366 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Third, all GCN-RNI models, at all randomization levels, converge significantly slower on both datasets than 3-GCN, similarly to Experiment 1. However, an interesting phenomenon can be seen on CEXP: All GCN-RNI models hover around $55 \\%$ accuracy within the first 100 epochs over CEXP (cf. Figure 3a), suggesting a struggle jointly fitting both CORRUPT and $\\overline { { \\mathrm { E x p } } }$ , before these models ultimately improve. This, however, is not observed with 3-GCN. Unlike on EXP, randomness is not necessarily beneficial on CEXP, as it can hurt performance on CORRUPT. Hence, RNI-enhanced models must additionally learn to isolate deterministic dimensions for CORRUPT, and randomized dimensions for EXP. These findings consolidate the earlier observations made on EXP on the impact of RNI on MPNN learning behavior, and highlight that the variability and slower learning for RNI also hinges on the variability and complexity of the input dataset. ", + "bbox": [ + 174, + 372, + 825, + 515 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, we observe that both fully randomized GCN-RNI, and, surprisingly, 1-GCN, struggle to learn CORRUPT relative to partially randomized GCN-RNI. We can also observe that 1-GCN does not present a “struggle” phase, and begins improving consistently from the start of training. These observations can be attributed to key conceptual , but very distinct hindrances impeding both models. In the case of 1-GCN, the model is jointly trying to learn both EXP and CORRUPT, when it is proven that it cannot fit the former. This joint optimization severely hinders CORRUPT learning, as data pairs from both subsets are highly similar, and share identically generated UNSAT graphs (cf. Appendix). Hence, 1-GCN, in attempting to fit SAT graphs from both subsets, knowing it cannot distinguish EXP pairs, struggles to learn the simpler difference in CORRUPT pairs. For GCN-RNI, the model discards key type information, so must only rely on structural differences to learn CORRUPT, which impedes its convergence. All in all, this further consolidates the promise of partial RNI as a means to combine the strengths of both deterministic and random features. ", + "bbox": [ + 174, + 522, + 825, + 690 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Further to the earlier two experiments, we also conducted analogous experiments using sparser analogs of the datasets EXP and CEXP. In these cases, we observed similar behavior, albeit with slower convergence overall. More details on these experiments can be found in the appendix. ", + "bbox": [ + 174, + 699, + 825, + 739 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 SUMMARY AND OUTLOOK ", + "text_level": 1, + "bbox": [ + 178, + 765, + 424, + 781 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We studied the expressive power of MPNNs with RNI, and showed that these are universal models. We empirically evaluated this model on carefully designed datasets, and observed that RNI practically improves the learning abilities of MPNNs for challenging data, though it does slow down model convergence owing to the need to learn robustness against random variability. Our work delivers a strong theoretical result, supported by empirical evaluation and practical insights, to rigorously quantify the effect of RNI on GNNs. Somewhat surprisingly, our experiments suggest that partial randomization may be the best strategy in most practical scenarios. An important direction for future work is to theoretically study the sensitivity of RNI to model architectures and initialization distributions, to yield a more complete understanding of the benefits and limitations of RNI. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 285, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Pablo Barcelo, Egor V. Kostylev, Mika ´ el Monet, Jorge P ¨ erez, Juan L. Reutter, and Juan Pablo Silva. ´ The logical expressiveness of graph neural networks. In Proceedings of the Eighth International Conference on Learning Representations , (ICLR), 2020. ", + "bbox": [ + 176, + 126, + 823, + 169 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Peter W. Battaglia, Jessica B. 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", + "bbox": [ + 173, + 859, + 825, + 916 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 PROPOSITIONAL LOGIC ", + "text_level": 1, + "bbox": [ + 176, + 135, + 382, + 150 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We briefly present propositional logic, which underpins the dataset generation. Let $S$ be a (finite) set $S$ of propositional variables. A literal is defined as $v$ , or $\\bar { v }$ (resp., $\\neg v$ ), where $v \\in S$ . A disjunction of literals is a clause. The width of a clause is defined as the number of literals it contains. A formula $\\varphi$ is in conjunctive normal form $( C N F )$ if it is a conjunction of clauses. A CNF has width $k$ if it contains clauses of width at most $k$ , and is referred to as a $k$ −CNF. To illustrate, the formula $\\varphi = \\left( x _ { 1 } \\vee { } { \\neg x } _ { 3 } \\right) \\wedge \\left( x _ { 4 } \\vee x _ { 1 } \\right)$ is a CNF with clauses of width 2. ", + "bbox": [ + 173, + 161, + 825, + 246 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "An assignment $\\nu : S \\mapsto \\{ 0 , 1 \\}$ maps variables to False (0), or True (1), and satisfies $\\varphi$ , which we denote by $\\nu \\models \\varphi$ , in the usual sense, where $\\vDash$ is propositional entailment. Given a propositional formula $\\varphi$ , the satisfiability problem, commonly known as SAT, consists of determining whether $\\varphi$ admits a satisfying assignment, and is NP-complete (Cook, 1971). ", + "bbox": [ + 174, + 252, + 825, + 309 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 PROOF OF THEOREM 4.1 ", + "text_level": 1, + "bbox": [ + 176, + 328, + 388, + 342 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We first prove a Boolean version of the theorem. ", + "bbox": [ + 174, + 354, + 491, + 369 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Lemma A.1. Let $n \\geq 1$ , and let $f : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\}$ be an invariant Boolean function. Then, for all $\\epsilon , \\delta > 0$ there is a MPNN with RNI that $( \\epsilon , \\delta )$ -approximates $f$ . ", + "bbox": [ + 174, + 375, + 823, + 405 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "To prove this lemma, we use a logical characterization of the expressiveness of MPNNs, which we always assume to admit global readouts. Let $\\complement$ be the extension of first-order predicate logic using counting quantifiers of the form $\\exists ^ { \\geq k } x$ for $k \\geq 0$ , where $\\exists ^ { \\geq k } x \\varphi ( x )$ means that there are at least $k$ elements x satisfying ϕ. ", + "bbox": [ + 174, + 415, + 825, + 473 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For example, consider the formula ", + "bbox": [ + 174, + 479, + 401, + 494 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c9c3375acbd81f2ad8311375c2d6464f2be05cbe5ff5a6aabf71a6ace53e12c1.jpg", + "text": "$$\n\\varphi ( x ) : = \\lnot \\exists ^ { \\geq 3 } y \\big ( E ( x , y ) \\land \\exists ^ { \\geq 5 } z E ( y , z ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 502, + 635, + 522 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "This is a formula in the language of graphs; $E ( x , y )$ means that there is an edge between the nodes interpreting $x$ and $y$ . For a graph $G$ and a vertex $v \\in V ( G )$ , we have $G \\models \\varphi ( v )$ ( $^ { 6 6 } G$ satisfies $\\varphi$ if the variable $x$ is interpreted by the vertex $v '$ ”) if and only if $v$ has at most 2 neighbors in $G$ that have degree at least 5. ", + "bbox": [ + 174, + 531, + 825, + 588 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We will not only consider formulas in the language of graphs, but also formulas in the language of colored graphs, where in addition to the binary edge relation we also have unary relations, that is, sets of nodes, which we may view as colors of the nodes. For example, the formula ", + "bbox": [ + 174, + 594, + 825, + 637 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/8a4c2726f461aabb49cb9b23306f0a1fe42393897a437e35d9a9aa46ce297de8.jpg", + "text": "$$\n\\psi ( x ) : = \\exists ^ { \\geq 4 } y { \\bigl ( } E ( x , y ) \\land R E D ( y ) { \\bigr ) }\n$$", + "text_format": "latex", + "bbox": [ + 383, + 646, + 614, + 666 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "says that node $x$ has at least 4 red neighbors (more precisely, neighbors in the unary relation $R E D$ ). Formally, we assume we have fixed infinite list $R _ { 1 } , R _ { 2 } , \\ldots$ of color symbols that we may use in our formulas. Then a colored graph is a graph together with a mapping that assigns a finite set $\\rho ( v )$ of colors $R _ { i }$ to each vertex (so we allow one vertex to have more than one, but only finitely many, colors). ", + "bbox": [ + 173, + 674, + 825, + 744 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A sentence (of the logic $\\complement$ or any other logic) is a formula without free variable. Thus a sentence expresses a property of a graph, which we can also view as a Boolean function. For a sentence $\\varphi$ we denote this function by $[ [ \\varphi ] ]$ . If $\\varphi$ is a sentence in the language of (colored) graphs, then for every (colored) graph $G$ J we have $\\mathbb { [ } \\varphi ] ( G ) = 1$ if $G \\models \\varphi$ and $\\mathbb { I } \\varphi \\mathbb { I } ( { \\overline { { G } } } ) { \\overline { { = 0 } } }$ otherwise. ", + "bbox": [ + 173, + 751, + 825, + 808 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "It is easy to see that $\\complement$ is only a syntactic extension of first order logic FO—for every C-formula there is a logically equivalent $\\mathsf { F O }$ -formula. To see this, note that we can simulate $\\exists ^ { \\geq k } x$ by $k$ ordinary existential quantifiers: $\\exists ^ { \\geq k } x$ is equivalent to $\\exists x _ { 1 } \\dots \\exists x _ { k } { \\Big ( } \\bigwedge _ { 1 \\leq i < j \\leq k } x _ { i } \\neq x _ { j } \\wedge \\bigwedge _ { 1 \\leq i \\leq k } \\varphi ( x _ { i } ) { \\Big ) }$ . However, counting quantifiers add expressiveness if we restrict the number of variables. The $\\complement$ -formula $\\exists ^ { \\geq k } x$ $( x = x )$ ) (saying that there are at least $k$ vertices) with just one variable is not equivalent to any FO-formula using less than $k$ variables. By ${ \\mathsf { C } } ^ { k }$ we denote the fragment of $\\complement$ consisting of all formulas with at most $k$ variables. ", + "bbox": [ + 173, + 813, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For example, the formula $\\varphi ( x )$ in (A.1) is in ${ \\mathsf C } ^ { 3 }$ , but not in $\\mathsf { C } ^ { 2 }$ . But $\\varphi ( x )$ is equivalent to the following formula $\\varphi ^ { \\prime } ( x )$ in $\\mathsf { C } ^ { 2 }$ : ", + "bbox": [ + 174, + 101, + 823, + 135 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/e5977154c37ccfd32ccf6ed566d062c810ee6a61f2e2c2ff8e24484d12801827.jpg", + "text": "$$\n\\varphi ^ { \\prime } ( x ) : = \\neg \\exists ^ { \\geq 3 } y \\big ( E ( x , y ) \\land \\exists ^ { \\geq 5 } x E ( y , x ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 141, + 637, + 161 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The fragments ${ \\mathsf { C } } ^ { k }$ are interesting for us, because their expressiveness corresponds to that of $\\left( k - 1 \\right)$ - WL and hence to that of $k$ -GNNs. More precisely, for all $k \\geq 2$ , two graphs $G$ and $H$ satisfy the same ${ \\mathsf { C } } ^ { k }$ -sentences if and only if $\\left( k - 1 \\right)$ -WL does not distinguish them (Cai et al., 1992). By the results of (Morris et al., 2019; Xu et al., 2019) this implies, in particular, that two graphs are indistinguishable by all MPNNs if and only if they satisfy the same $\\mathsf { C } ^ { 2 }$ -sentences. Barcelo et al. (2020) strengthened ´ this result and showed that every $\\mathsf { C } ^ { 2 }$ -sentence can be simulated by an MPNN. ", + "bbox": [ + 173, + 169, + 825, + 262 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma A.2 (Barcelo et al. 2020) ´ . For every $\\mathsf { C } ^ { 2 }$ -sentence $\\varphi$ and every $\\epsilon > 0$ there is an MPNN that $\\epsilon$ -approximates $[ [ \\varphi ] ]$ . ", + "bbox": [ + 173, + 265, + 825, + 297 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Since here we are talking about deterministic MPNNs, there is no randomness involved, and we just say “ $\\epsilon$ -approximates” instead of “ $\\mathopen { } \\mathclose \\bgroup \\left( \\epsilon , 1 \\aftergroup \\egroup \\right)$ -approximates”. ", + "bbox": [ + 174, + 306, + 821, + 337 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma A.2 not only holds for sentences in the language of graphs, but also for sentences in the language of colored graphs. Let us briefly discuss the way MPNNs access such colors. We encode the colors using one-hot vectors that are part of the initial states of the nodes. For example, if we have a formula that uses color symbols among $R _ { 1 } , \\ldots , R _ { k }$ , then we reserve $k$ places in the initial state $\\pmb { x } _ { v } = \\left( x _ { v 1 } , \\dots , x _ { v \\ell } \\right)$ of each vertex $v$ (say, for convenience, $x _ { v 1 } , \\ldots , x _ { v k } )$ and we initialize $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { v } }$ by letting $x _ { v i } = 1$ if $v$ is in $R _ { i }$ and $x _ { v i } = 0$ otherwise. ", + "bbox": [ + 173, + 342, + 825, + 428 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let us call a colored graph $G$ individualized if for any two distinct vertices $v , w \\in V ( G )$ the sets $\\rho ( v ) , \\rho ( w )$ of colors they have are distinct. Let us say that a sentence $\\chi$ identifies a (colored) graph $G$ if for all (colored) graphs $H$ we have $H \\models \\chi$ if and only if $H$ is isomorphic to $G$ . ", + "bbox": [ + 174, + 433, + 825, + 477 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma A.3. For every individualized colored graph $G$ there is a $\\mathsf { C } ^ { 2 }$ -sentence $\\chi _ { G }$ that identifies $G$ ", + "bbox": [ + 171, + 479, + 821, + 497 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. Let $G$ be an individualized graph. For every vertex $v \\in V ( G )$ , let ", + "bbox": [ + 176, + 511, + 650, + 526 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/14150dbb1d09a00b3cd650c16ed3ee5f40b409a0a643ae5a65981aab74e27d20.jpg", + "text": "$$\n\\alpha _ { v } ( x ) : = \\bigwedge _ { R \\in \\rho ( v ) } R ( x ) \\wedge \\bigwedge _ { R \\in \\{ R _ { 1 } , \\ldots , R _ { k } \\} \\setminus \\rho ( x ) } \\neg R ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 531, + 663, + 561 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Then $v$ is the unique vertex of $G$ such that $G \\models \\alpha _ { v } ( v )$ . For every pair $v , w \\in V ( G )$ of vertices, we let ", + "bbox": [ + 173, + 568, + 821, + 594 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/43e2cf135e329f955b0399861f2b2f4df74a84e6ce5284180abd191b73cad377.jpg", + "text": "$$\n\\begin{array} { r } { \\beta _ { v w } ( x , y ) : = \\left\\{ \\begin{array} { l l } { \\alpha _ { v } ( x ) \\wedge \\alpha _ { w } ( y ) \\wedge E ( x , y ) } & { \\mathrm { i f ~ } ( v , w ) \\in E ( G ) , } \\\\ { \\alpha _ { v } ( x ) \\wedge \\alpha _ { w } ( y ) \\wedge \\neg E ( x , y ) } & { \\mathrm { i f ~ } ( v , w ) \\not \\in E ( G ) . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 593, + 704, + 630 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We let ", + "bbox": [ + 174, + 632, + 218, + 646 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/75c1e753957f39e0c74303f9c314cf0dbd543bde4e128cecc4c52ee14b75e28c.jpg", + "text": "$$\n\\chi _ { G } : = \\bigwedge _ { v \\in V ( G ) } \\left( \\exists x \\alpha _ { v } ( x ) \\wedge \\neg \\exists ^ { \\geq 2 } x \\alpha _ { v } ( x ) \\right) \\wedge \\bigwedge _ { v , w \\in V ( G ) } \\exists x \\exists y \\beta _ { v w } ( x , y ) .\n$$", + "text_format": "latex", + "bbox": [ + 272, + 642, + 725, + 674 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "It is easy to see that $\\chi _ { G }$ identifies $G$ . ", + "bbox": [ + 173, + 676, + 413, + 691 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For $n , k \\in \\mathbb { N }$ , we let $\\mathcal { G } _ { n , k }$ be the class of all individualized colored graphs that only use colors among $R _ { 1 } , \\ldots , R _ { k }$ . ", + "bbox": [ + 171, + 705, + 825, + 734 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma A.4. Let $h : { \\mathcal { G } } _ { n , k } \\to \\{ 0 , 1 \\}$ be an invariant Boolean function. Then there exists a $\\mathsf { C } ^ { 2 }$ - sentence $\\psi _ { h }$ such that for all $G \\in \\mathcal { G } _ { n , k }$ it holds that $[ [ \\psi _ { h } ] ] ( G ) = h ( G )$ . ", + "bbox": [ + 171, + 739, + 825, + 771 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. Let ${ \\mathcal { H } } \\subseteq { \\mathcal { G } } _ { n , k }$ be the subset consisting of all graphs $H$ with $h ( H ) = 1$ . We let ", + "bbox": [ + 171, + 784, + 727, + 799 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/49874527490f55ab52810e93895774282e09833140c26bcfe589949691871a8f.jpg", + "text": "$$\n\\psi _ { h } : = \\bigvee _ { H \\in \\mathcal { H } } \\chi _ { H } .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 804, + 549, + 832 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We eliminate duplicates in the disjunction. Since up to isomorphism, the class $\\mathcal { G } _ { n , k }$ is finite, this makes the disjunction finite and hence $\\psi _ { h }$ well-defined. □ ", + "bbox": [ + 173, + 837, + 825, + 866 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The restriction of a colored graph $G$ is the underlying plain graph, that is, the graph $G ^ { \\vee }$ obtained from the colored graph $G$ by forgetting all the colors. Conversely, a colored graph $G ^ { \\wedge }$ is an expansion of a plain graph $G$ if $G = ( G ^ { \\wedge } ) ^ { \\vee }$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Corollary A.1. Let $f : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\}$ be an invariant Boolean function. Then there exists a $\\mathsf { C } ^ { 2 }$ - sentence $\\varphi _ { f } ^ { \\wedge }$ (in the language of colored graphs) such that for all $G \\in { \\mathcal { G } } _ { n , k }$ it holds that $\\mathbb { [ } \\psi _ { f } ^ { \\wedge } ] ( G ) =$ $f ( G ^ { \\vee } )$ . ", + "bbox": [ + 173, + 102, + 825, + 148 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Towards proving Lemma A.1, we fix an $n \\geq 1$ and a $\\epsilon , \\delta > 0$ . We let ", + "bbox": [ + 173, + 159, + 616, + 175 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ba7e9ba578d79f745c90f449340b738ae19ffb6bc9a251db3c8484f2bf543f85.jpg", + "text": "$$\nc : = \\left\\lceil { \\frac { 2 } { \\delta } } \\right\\rceil \\quad { \\mathrm { a n d } } \\quad k : = c ^ { 2 } \\cdot n ^ { 3 }\n$$", + "text_format": "latex", + "bbox": [ + 405, + 180, + 593, + 213 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The technical details of the proof of Lemma A.1 and Theorem 4.1 depend on the exact choice of the random initialization and the activation functions used in the neural networks, but the idea is always the same. For simplicity, we assume that we initialize the states $\\pmb { x } _ { v } = \\left( x _ { v 1 } , \\ldots , x _ { v \\ell } \\right)$ of all vertices to $( r _ { v } , 0 , \\ldots , 0 )$ , where $r _ { v }$ for $v \\in V ( G )$ are chosen independently uniformly at random from $[ 0 , 1 ]$ . As our activation function $\\sigma$ , we choose the linearized sigmoid function defined by $\\sigma ( x ) = \\bar { 0 }$ for $x < 0$ , $\\sigma ( x ) = x$ for $0 \\leq x < 1$ , and $\\sigma ( x ) = 1$ for $x \\geq 1$ . ", + "bbox": [ + 173, + 217, + 826, + 303 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma A.5. Let $r _ { 1 } , \\ldots , r _ { n }$ be chosen independently uniformly at random from the interval $[ 0 , 1 ]$ . For $1 \\leq i \\leq n$ and $1 \\leq j \\leq c \\cdot n ^ { 2 }$ , let ", + "bbox": [ + 173, + 306, + 821, + 335 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0258962bac425b6802425ddc9855ec6eef3641e804a5d99f7b6570262f6d3558.jpg", + "text": "$$\ns _ { i j } : = k \\cdot r _ { i } - \\left( j - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 340, + 593, + 372 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Then with probability greater than $1 - \\delta$ , the following conditions are satisfied. ", + "bbox": [ + 176, + 377, + 689, + 393 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(ii) For all distinct $i , i ^ { \\prime } \\in \\{ 1 , . . . , n \\}$ there exists a $j \\in \\left\\{ 1 , \\dots , c \\cdot n ^ { 2 } \\right\\}$ such that $\\sigma ( s _ { i j } ) \\neq \\sigma ( s _ { i ^ { \\prime } j } )$ ", + "bbox": [ + 202, + 428, + 821, + 445 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. For every $i$ , let $p _ { i } : = \\lfloor r _ { i } \\cdot k \\rfloor$ . Since $k \\cdot r _ { i }$ is uniformly random from the interval $[ 0 , k ]$ , the integer $p _ { i }$ is uniformly random from $\\{ 0 , \\ldots , k - 1 \\}$ . Observe that $0 < \\sigma ( s _ { i j } ) < 1$ only if $p _ { i } - ( j -$ $\\textstyle 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } = 0$ (here we use the fact that $k$ is divisible by $c \\cdot n ^ { 2 } .$ ). The probability that this happens is $\\frac { 1 } { k }$ ⋅Thus, by the Union Bound, ", + "bbox": [ + 174, + 459, + 825, + 518 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/bd4afacfa46bc02b6c3ee163416d9fae134482461bb962ca484f5fed1aa068ed.jpg", + "text": "$$\n\\operatorname* { P r } \\left( \\exists i , j : 0 < \\sigma ( s _ { i j } ) < 1 \\right) \\leq \\frac { c \\cdot n ^ { 3 } } { k } .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 525, + 619, + 558 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Now let $i , i ^ { \\prime }$ be distinct and suppose that $\\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } )$ for all $j$ . Then for all $j$ we have $s _ { i j } \\leq$ $0 \\iff s _ { i ^ { \\prime } j } \\leq 0$ and therefore $\\lfloor s _ { i j } \\rfloor \\le 0 \\iff \\lfloor \\stackrel { \\sim } { s } _ { i ^ { \\prime } j } \\rfloor \\le 0$ . This implies ", + "bbox": [ + 168, + 564, + 825, + 594 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/cb2e1e4fdb6329bb8691bc30d59a92c0ea0650ad0c4d973f46995b676839b4f6.jpg", + "text": "$$\n\\forall j \\in \\{ 1 , \\ldots , c \\cdot n ^ { 2 } \\} : \\quad p _ { i } \\leq ( j - 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } \\Longleftrightarrow p _ { i ^ { \\prime } } \\leq ( j - 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 599, + 730, + 631 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Let $j ^ { * } \\in \\{ 1 , \\ldots , c \\cdot n ^ { 2 } \\}$ such that $p _ { i } \\in \\left\\{ \\left( j ^ { * } - 1 \\right) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } , \\ldots , j ^ { * } \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } - 1 \\right\\}$ . Then by (A.4) we have \n$\\begin{array} { r } { p _ { i } ^ { \\prime } \\in \\left\\{ \\left( j ^ { * } - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } , \\ldots , j ^ { * } \\cdot \\frac { k } { c \\cdot n ^ { 2 } } - 1 \\right\\} } \\end{array}$ . As $p _ { i ^ { \\prime } }$ is independent of $p _ { i }$ and hence of $j ^ { * }$ , the probability \nthat this happens is at most $\\begin{array} { r } { \\frac { 1 } { k } \\cdot \\frac { k } { c \\cdot n ^ { 2 } } = \\frac { 1 } { c \\cdot n ^ { 2 } } } \\end{array}$ . This proves that for all distinct $i , i ^ { \\prime }$ the probability that k1 \n$\\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } )$ is at most $\\textstyle { \\frac { 1 } { c \\cdot n ^ { 2 } } }$ . Hence, again by the Union Bound, ", + "bbox": [ + 173, + 638, + 826, + 720 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/dac60406468b443641503a34a679b80846b803faea6fccbc9b5b1bc8a71abf25.jpg", + "text": "$$\n\\operatorname* { P r } ( \\exists i \\neq i ^ { \\prime } \\forall j : \\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } ) ) \\leq \\frac { 1 } { c } .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 727, + 627, + 757 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(A.3) and (A.5) imply that the probability that either (i) or (ii) is violated is at most ", + "bbox": [ + 173, + 762, + 717, + 779 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/69eac52e7ccaccfbf44b2a2ca33d0511400584990b8b4dbbfa3a5384a4218f2a.jpg", + "text": "$$\n\\frac { c \\cdot n ^ { 3 } } { k } + \\frac { 1 } { c } \\leq \\frac { 2 } { c } \\leq \\delta .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 784, + 562, + 818 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof of Lemma A.1. For given function $f : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\}$ , we choose the sentence $\\psi _ { f } ^ { \\wedge }$ according to Corollary A.1. Applying Lemma A.2 to this sentence and $\\epsilon$ , we obtain an MPNN $\\ddot { \\mathcal { N } } _ { f }$ that on a colored graph $G \\in \\mathcal { G } _ { n , k }$ computes an $\\epsilon$ -approximation of $f ( G ^ { \\vee } )$ . ", + "bbox": [ + 173, + 830, + 825, + 876 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Without loss of generality, we assume that the vertex set of the input graph to our MPNN is $\\{ 1 , \\ldots , n \\}$ . We choose $\\ell$ (the dimension of the state vectors) in such a way that $\\ell \\geq c \\cdot n ^ { 2 }$ and $\\ell$ is at least as large as the dimension of the state vectors of $\\mathcal { N } _ { f }$ . Recall that the state vectors are ", + "bbox": [ + 173, + 881, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "initialized as x(0i $\\pmb { x } _ { i } ^ { ( 0 ) } = ( r _ { i } , 0 , \\ldots , 0 )$ for values $r _ { i }$ chosen independently uniformly at random from the interval $[ 0 , 1 ]$ . ", + "bbox": [ + 174, + 101, + 825, + 133 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "passed) that maps x(0)i t o x( 1 )i = $\\pmb { x } _ { i } ^ { ( 1 ) } = \\bar { ( x _ { i 1 } ^ { ( 1 ) } , \\dots , x _ { i \\ell } ^ { ( 1 ) } ) }$ ly local transformation (no messages need to be with ", + "bbox": [ + 169, + 138, + 823, + 172 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f491397e8a04115d9493e751e43fe8d3424c020ba3e5659d872ec83ef1a90ff4.jpg", + "text": "$$\n\\begin{array} { r } { x _ { i j } ^ { ( 1 ) } = \\left\\{ \\begin{array} { l l } { \\sigma \\Big ( \\boldsymbol { k } \\cdot \\boldsymbol { r _ { i } } - \\left( j - 1 \\right) \\cdot \\frac { \\boldsymbol { k } } { c \\cdot n ^ { 2 } } \\Big ) } & { \\mathrm { f o r ~ } 1 \\leq j \\leq c \\cdot n ^ { 2 } , } \\\\ { 0 } & { \\mathrm { f o r ~ } c \\cdot n ^ { 2 } + 1 \\leq j \\leq \\ell . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 179, + 681, + 222 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since we treat $k , c , n$ as constants, the mapping $\\begin{array} { r } { r _ { i } \\mapsto k \\cdot r _ { i } - \\left( j - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } } \\end{array}$ is just a linear mapping applied to $r _ { i } = x _ { i 1 } ^ { ( 0 ) }$ . ", + "bbox": [ + 174, + 229, + 820, + 266 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "By Lemma A.5, with probability at least $1 - \\delta$ , the vectors $\\pmb { x } _ { i } ^ { ( 1 ) }$ are mutually distinct $\\{ 0 , 1 \\}$ -vectors, which we view as encoding a coloring of the input graph with colors from $R _ { 1 } , \\ldots , R _ { k }$ . Let $G ^ { \\wedge }$ be the resulting colored graph. Since the vectors $\\mathbf { \\bar { x } } _ { i } ^ { ( 0 ) }$ are mutually distinct, $G ^ { \\wedge }$ is individualized and thus in the class $\\mathcal { G } _ { n , k }$ . We now apply the MPNN $\\mathcal { N } _ { f }$ , and it computes a value $\\epsilon$ -close to $\\mathbb { } [ \\psi _ { f } ^ { \\wedge } ] ( G ^ { \\wedge } ) =$ $f ( ( G ^ { \\wedge } ) ^ { \\vee } ) = f ( G )$ . □ ", + "bbox": [ + 173, + 272, + 825, + 353 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof of Theorem 4.1. Let $f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }$ be invariant. Since $\\mathcal { G } _ { n }$ is finite, the range $Y : = f ( { \\mathcal { G } } _ { n } )$ is finite. To be precise, we have $N : = | Y | \\leq | \\mathcal { G } _ { n } | = 2 ^ { { \\binom { n } { 2 } } }$ . ", + "bbox": [ + 173, + 366, + 823, + 400 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Say, $Y = \\{ y _ { 1 } , \\dots , y _ { N } \\}$ . For $i = 1 , \\ldots , N$ , let $g _ { i } : { \\mathcal { G } } _ { n } \\{ 0 , 1 \\}$ be the Boolean function defined by ", + "bbox": [ + 173, + 405, + 807, + 421 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f8d35281f86440cd78ab828e31e4926334bf5f34e161f63493791264032c0240.jpg", + "text": "$$\ng _ { i } ( G ) = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } f ( G ) = y _ { i } , } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 406, + 428, + 588, + 464 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Note that $g _ { i }$ is invariant. ", + "bbox": [ + 173, + 472, + 333, + 486 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Let $\\epsilon , \\delta > 0$ and $\\begin{array} { r } { \\epsilon ^ { \\prime } : = \\frac { \\epsilon } { \\operatorname* { m a x } Y } } \\end{array}$ and $\\begin{array} { r } { \\delta ^ { \\prime } : = \\frac { \\delta } { N } } \\end{array}$ . By Lemma A.1, for every $i \\in \\{ 1 , \\ldots , N \\}$ there is an MPNN with $\\boldsymbol { \\mathrm { R N I } } \\mathcal { N } _ { i }$ that $( \\epsilon ^ { \\prime } , \\delta )$ -approximates $g _ { i }$ . Putting all the ${ \\mathcal { N } } _ { i }$ together, we obtain an invariant MPNN $\\mathcal { N }$ that computes a function $g : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\} ^ { N }$ . We only need to apply the linear transformation ", + "bbox": [ + 173, + 491, + 825, + 539 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d3d99d70fdddcbd9c1bdb870b842520c2da7540f0984497e7388c45560c7696b.jpg", + "text": "$$\n\\pmb { x } \\mapsto \\sum _ { i = 1 } ^ { N } x _ { i } \\cdot y _ { i }\n$$", + "text_format": "latex", + "bbox": [ + 450, + 545, + 545, + 583 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "to the output of $\\mathcal { N }$ to obtain the desired approximation of $f$ . ", + "bbox": [ + 173, + 588, + 566, + 604 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Remark 1. Obviously, our construction yields MPNNs with a prohibitively large state space. In particular, this is the case for the brute force step from Boolean to general functions. We doubt that there are much more efficient approximators, after all we make no assumption whatsoever on the function $f$ . ", + "bbox": [ + 173, + 611, + 825, + 669 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The approximation of Boolean functions is more interesting. It may still happen that the GNNs get exponentially large in $n$ ; this seems unavoidable. However, the nice thing here is that our construction is very adaptive and tightly linked to the descriptive complexity of the function we want to approximate. This deserves a more thorough investigation, which we leave for future work. ", + "bbox": [ + 173, + 674, + 825, + 732 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "As opposed to other universality results for GNNs, our construction needs no higher-order tensors defined on tuples of nodes, with practically infeasible space requirements on all but very small graphs. Instead, the complexity of our construction goes entirely into the dimension of the state space. The advantage of this is that we can treat this dimension as a hyperparameter that we can easily adapt and that gives us more fine-grained control over the space requirements. Our experiments show that usually in practice a small dimension already yields very powerful networks. ", + "bbox": [ + 173, + 737, + 825, + 821 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Remark 2. In our experiments, we found that a partial random initialization, which only assigns random values to a fraction of all node embedding vectors, often yields very good results, sometimes better than a full random initialization. There is plausibility to this from a theoretical perspective. For most graphs, we do not lose much by only initializing a small fraction of vertex embeddings, because in a few message-passing rounds GNNs can propagate the randomness and, referring our construction above, individualize the full input graph. On the other hand, we reduce the amount of noise our models have to handle when we only randomize partially. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/224f4caf68657c2ece8143ddc76017e80d9d914c8f0e7af17d004b811e9ebd17.jpg", + "image_caption": [ + "Figure 4: Illustration of planar embeddings for the formulas $\\varphi _ { 1 }$ and $\\varphi _ { 2 }$ for $n = 2$ . " + ], + "image_footnote": [], + "bbox": [ + 284, + 98, + 714, + 541 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3 DETAILS OF DATASET CONSTRUCTION ", + "text_level": 1, + "bbox": [ + 174, + 604, + 485, + 618 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "There is an interesting universality result for functions defined on planar graphs. It is known that 3- WL can distinguish between any pair of planar graphs (Kiefer et al., 2019). Since 4-GCNs can simulate 3-WL, this implies that functions defined on planar graphs can be approximated by 4-GCNs. This result can be extended to much wider graph classes, including all graph classes excluding a fixed graph as a minor (Grohe, 2017). ", + "bbox": [ + 174, + 635, + 825, + 705 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Inspired by this, we generate planar instances, and ensure that they can be distinguished by 2- WL, by carefully constraining these instances further. Hence, any GNN with 2-WL expressive power can approximate solutions to these planar instances. This, however, does not imply that these GNNs will solve EXP in practice, but only that an appropriate approximation function exists and can theoretically be learned. ", + "bbox": [ + 174, + 712, + 825, + 781 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3.1 CONSTRUCTION OF EXP ", + "text_level": 1, + "bbox": [ + 176, + 809, + 400, + 823 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We now explain the construction and composition of EXP. Fundamentally, EXP consists of two main components, (i) a pair of cores, which are non-isomorphic, planar, 1-WL indistinguishable, 2-WL distinguishable, and decide the satisfiability of every instance, and (ii) an additional randomly generated and satisfiable planar component, identically added to the core pair, to add variability to EXP and make learning more challenging. We first present both components, and then provide further details about graph encoding and planar embeddings. ", + "bbox": [ + 174, + 837, + 825, + 921 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Core pair. In EXP, a core pair consists of two CNF formulas $\\varphi _ { 1 } , \\varphi _ { 2 }$ , both defined using $2 n$ variables, $n \\in \\mathbb { N } ^ { + }$ , such that $\\varphi _ { 1 }$ is unsatisfiable and $\\varphi _ { 2 }$ is satisfiable, and such that their graph encodings are 1-WL indistinguishable and planar. $\\varphi _ { 1 }$ and $\\varphi _ { 2 }$ are constructed using two structures which we refer to as variable chains and variable bridges respectively. ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A variable chain $\\varphi _ { c h a i n }$ is defined over a set of $n \\geq 2$ Boolean variables, and imposes that all variables be equally set. The variable chain can be defined in increasing or decreasing order over these variables. More specifically, given variables $x _ { i } , . . . , x _ { j }$ , ", + "bbox": [ + 174, + 166, + 825, + 209 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5a443b8e1d851aa053f60a9652603c292767f2223ecc0cf224db16d61bf1de93.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { C h a i n } _ { \\mathrm { I n c } } ( i , j ) = \\displaystyle \\bigwedge _ { k = i } ^ { j - 1 } ( \\bar { x _ { k } } \\vee x _ { i + ( k + 1 ) } \\% ( j - i + 1 ) ) , \\mathrm { ~ a n d } } \\\\ & { \\mathrm { C h a i n } _ { \\mathrm { I n e c } } ( i , j ) = \\displaystyle \\bigwedge _ { k = i } ^ { j - 1 } ( x _ { k } \\vee \\bar { x } _ { i + ( k + 1 ) } \\% ( j - i + 1 ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 336, + 218, + 663, + 294 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Additionally, a variable bridge is defined over an even number of variables $x _ { 0 } , . . . , x _ { 2 n - 1 }$ , as ", + "bbox": [ + 173, + 300, + 772, + 315 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/3d2890cb3545abacee0aed89cd8d925528982926abce8d5f6f2af3d238b2937d.jpg", + "text": "$$\n\\varphi _ { b r i d g e } = \\bigwedge _ { i = 0 } ^ { n - 1 } { \\big ( } ( x _ { i } \\vee x _ { 2 n - 1 - i } ) \\wedge ( { \\bar { x } } _ { i } \\vee { \\bar { x } } _ { 2 n - 1 - i } ) { \\big ) } .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 323, + 658, + 359 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A variable bridge makes the variables it connects forcibly have opposite values, e.g., $x _ { 0 } = \\bar { x _ { 1 } }$ for $n = 1$ . We denote a variable bridge over $x _ { 0 } , . . . , x _ { 2 n - 1 }$ as Bridge $( 2 n )$ . ", + "bbox": [ + 171, + 367, + 823, + 397 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "To get $\\varphi _ { 1 }$ and $\\varphi _ { 2 }$ , we define $\\varphi _ { 1 }$ as a variable chain and bridge on all variables, yielding contrasting and unsatisfiable constraints. To define $\\varphi _ { 2 }$ , we “cut” the chain in half, such that the first $n$ variables can differ from the latter $n$ , satisfying the bridge. The second half of the “cut” chain is then flipped to a decrementing order, which preserves the satisfiability of $\\varphi _ { 2 }$ , but maintains the planarity of the resulting graph. More specifically, this yields: ", + "bbox": [ + 173, + 402, + 825, + 473 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/a2e58547bb169db7ae266da9d0b79cb00beba7b954b1d7d862a337bd8fff6e3c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\varphi _ { 1 } = \\mathbf { C h a i n } _ { \\mathrm { I n c } } ( 0 , 2 n ) \\wedge \\mathbf { B r i d g e } ( 2 n ) , \\mathrm { a n d } } \\\\ & { \\varphi _ { 2 } = \\mathbf { C h a i n } _ { \\mathrm { I n c } } ( 0 , n ) \\wedge \\mathbf { C h a i n } _ { \\mathrm { D e c } } ( n , 2 n ) \\wedge \\mathbf { B r i d g e } ( 2 n ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 479, + 676, + 517 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Planar component. Following the generation of $\\varphi _ { 1 }$ and $\\varphi _ { 2 }$ , a disjoint satisfiable planar graph component $\\varphi _ { \\mathrm { p l a n a r } }$ is added. $\\varphi _ { \\mathrm { p l a n a r } }$ shares no variables or disjunctions with the cores, so is primarily introduced to create noise and make learning more challenging. $\\varphi _ { \\mathrm { p l a n a r } }$ is generated starting from random 2-connected (i.e., at least 2 edges must be removed to disconnect a component within the graph) bipartite planar graphs from the Plantri tool (Brinkmann et al., 2007), such that (i) the larger set of nodes in the graph is the variable set3, (ii) highly-connected disjunctions are split in a planaritypreserving fashion to maintain disjunction widths not exceeding 5, (iii) literal signs for variables are uniformly randomly assigned, and (iv) redundant disjunctions, if any, are removed. If this $\\varphi _ { \\mathrm { p l a n a r } }$ is satisfiable, then it is accepted and used. Otherwise, the formula is discarded and a new $\\varphi _ { \\mathrm { p l a n a r } }$ is analogously generated until a satisfiable formula is produced. ", + "bbox": [ + 173, + 531, + 825, + 671 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since the core pair and $\\varphi _ { \\mathrm { p l a n a r } }$ are disjoint, it is easy to deduce that the graph encoding of $\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 1 }$ and $\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 2 }$ are both planar and 1-WL indistinguishable. Furthermore, $\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 1 }$ is satisfiable, and $\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 2 }$ is not. Hence, the introduction of $\\varphi _ { \\mathrm { p l a n a r } }$ maintains all the desirable core properties, all while making any generated EXP dataset more challenging. ", + "bbox": [ + 174, + 676, + 823, + 733 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The structural properties of the cores, combined with the combinatorial difficulty of SAT, make EXP a challenging dataset. For example, even minor formula changes, such as flipping a literal, can lead to a change in the SAT outcome, which enables the creation of near-identical, yet semantically different instances. Moreover, SAT is NP-complete (Cook, 1971), and remains so on planar instances (Hunt III et al., 1998). Hence, EXP is cast to be challenging, both from an expressiveness and computational perspective. ", + "bbox": [ + 173, + 739, + 825, + 824 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Remark 3. Intuitively, $\\varphi _ { 1 }$ and $\\varphi _ { 2 }$ , generated as described, can be distinguished by 2-WL, as 2-WL can detect the break in cycles resulting from the aforementioned “cut”. In other words, 2-WL can identify that the chain has been broken in between these two formulas, and thus will return distinct colourings. Hence, $\\varphi _ { 1 }$ and $\\varphi _ { 2 }$ can be distinguished by 3-GCNs. ", + "bbox": [ + 174, + 840, + 825, + 897 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Graph encoding. We use the following graph encoding, denoted by Enc: (i) Every variable is encoded by two nodes, representing its positive and negative literals, and connected by an edge, (ii) Every disjunction is represented by a node, and an edge connects a literal node to a disjunction node if the literal appears in the disjunction, and (iii) Variable and disjunction nodes are encoded with different types. We opt for this encoding, as it is commonly used in the literature (Selsam et al., 2019), and is sufficient, for the sake of our empirical evaluation, to yield planar encodings for EXP graph pairs. ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Planar embeddings for core pair. We show planar embeddings for $E n c ( \\varphi _ { 1 } )$ and $E n c ( \\varphi _ { 2 } )$ for $n = 2$ in Figure 4, and these embeddings can naturally be extended to any $n$ . $E n c ( \\varphi _ { 1 } )$ and $E n c ( \\varphi _ { 2 } )$ can also be shown to be 1-WL indistinguishable. This can be observed intuitively, as node neighborhoods in both graphs are identical and very regular: all variable nodes are connected to exactly one other variable node and two disjunction nodes, and all disjunction nodes are connected to exactly two variables. ", + "bbox": [ + 173, + 219, + 825, + 303 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.3.2 CONSTRUCTION OF CEXP ", + "text_level": 1, + "bbox": [ + 176, + 321, + 411, + 337 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Given a EXP dataset with $N$ pairs of graphs, we create CEXP by selecting $N / 2$ graph pairs and modifying them to yield CORRUPT. The unmodified graph pairs are therefore exactly identical in type to EXP instances, and we refer to these instances within CEXP as $\\overline { { \\mathrm { E x p } } }$ . ", + "bbox": [ + 174, + 348, + 825, + 390 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For every graph pair, we discard the satisfiable graph and construct a new graph from a copy of the unsatisfiable graph as follows. ", + "bbox": [ + 174, + 398, + 823, + 426 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "1. Randomly introduce new literals to the existing disjunctions of the copy of the unsatisfiable graph, such that no redundancies are created (i.e., adding $x$ to a disjunction when $x$ or $\\bar { x }$ is already present), until 3 literals are added and the formula becomes satisfiable. Literal addition is done by creating new edges in the graph between disjunction and literal nodes. To do this, disjunctions with less than 5 literals are uniformly randomly selected, and the literal to add is uniformly randomly sampled from the set of all non-redundant literals given the selected disjunction. 2. Once a satisfiable formula is reached, iterate sequentially over all added edges, and eliminate any edge whose removal does not restore unsatisfiability. This ensures that a minimal number of new edges, relative to the original unsatisfiable graph, are added. ", + "bbox": [ + 212, + 440, + 825, + 587 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Observe that these modifications have several interesting effects on the dataset. First, they preserve the existing UNSAT core nodes and edges, while flipping the satisfiability of their overall formulas, which makes the learning task go beyond structure identification. Second, they introduce significant new variability to the dataset, in that the planar component and cores can share edges. Finally, they make the graph pairs 1-WL distinguishable, which gives standard GNNs a chance to perform well on CORRUPT. ", + "bbox": [ + 174, + 601, + 825, + 684 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.3.3 DATASET GENERATION FOR EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 178, + 703, + 524, + 717 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To create the EXP dataset, we randomly generate 600 core pairs, where $n$ (cf. Appendix A.3) is uniformly randomly set between 2 and 4 inclusive. Then, we generate the additional planar component using Plantri, such that $5 0 0 \\varphi _ { \\mathrm { p l a n a r } }$ formulas are generated from 12-node planar bipartite planar graphs, and the remaining 100 from planar bipartite graphs with 15 nodes. ", + "bbox": [ + 174, + 728, + 825, + 784 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "This generation process implies that every formula has a number of variables ranging between 10 (4 core variables when $n = 2$ plus a minimum 6 variables from the larger bipartite set during $\\varphi _ { 1 }$ planar generation from 12-node graphs) and 22 variables (8 core variables for $n = 4$ plus a maximally-sized variable subset of 14 nodes for $\\varphi _ { \\mathrm { p l a n a r } }$ generation from 15-node graphs). ", + "bbox": [ + 174, + 791, + 825, + 848 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Furthermore, the number of disjunctions also ranges from 10 (8 core disjunctions for $n = 2$ plus the minimum 2 disjunctions for the case where $\\varphi _ { \\mathrm { p l a n a r } }$ , generated from 12-node graphs, has 10 variables and 2 disjunctions) to 30 disjunctions (16 core disjunctions for $n = 4$ plus at most 14 disjunctions for the case where $\\varphi _ { \\mathrm { p l a n a r } }$ , generated from 15-node graphs, initially has 8 variables and 7 disjunctions, which can at most lead to 14 final disjunctions following step (ii)). ", + "bbox": [ + 176, + 854, + 823, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In this subsection, we investigate the variability of GCN-RNI learning across validation folds, and do so with a representative model and dataset, namely the semi-randomized GCN$5 0 \\% \\mathrm { R N I }$ model and the standard EXP dataset. The standard deviation of the test accuracy of $\\mathrm { G C N } { - } 5 0 \\% \\mathrm { F }$ RNI over EXP, across all 10 cross-validation folds relative to the number of epochs, is shown in Figure 5. From this figure, we see that standard deviation spikes sharply at the start of training, and only begins dropping after 100 epochs. This suggests that the learning behavior of $G C N { - } 5 0 \\%$ RNI is quite variable, sometimes requiring few epochs to converge, and in other cases requiring a very high number of epochs. Furthermore, standard deviation converges to almost zero following 200 epochs, corresponding to the phase where all ", + "bbox": [ + 174, + 132, + 483, + 381 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/6721e91c6ef6ad166351a12e8a09d04875d6a5590bdce0dbf328d56a14ad16f7.jpg", + "image_caption": [ + "Figure 5: Standard deviation of test accuracy over all 10 validation splits of GCN- $50 \\%$ RNI on EXP. " + ], + "image_footnote": [], + "bbox": [ + 519, + 148, + 797, + 325 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "validation folds have achieved near-perfect test performance. From these findings, we further confirm that RNI introduces volatility to GCN training, this time manifesting in variable convergence times across validation folds, but that this volatility does not ultimately hinder convergence and performance, as all folds eventually reach satisfactory performance within a reasonable amount of epochs, and subsequently stabilize at this level. ", + "bbox": [ + 174, + 382, + 825, + 450 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.5 ADDITIONAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 473, + 413, + 487 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In addition to the experiments in the main body of the paper, we additionally evaluate RNI on sparser analog datasets to EXP and CEXP, namely SPARSEEXP and SPARSECEXP. These datasets only contain $2 5 \\%$ of the number of instances of their original counterparts, and are used to study the behavior and impact of RNI when data is sparse. ", + "bbox": [ + 174, + 501, + 825, + 556 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.5.1 EXPERIMENT 1 ON SPARSEEXP ", + "text_level": 1, + "bbox": [ + 176, + 577, + 447, + 592 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this experiment, we generate SPARSEEXP analogously to EXP, except that this dataset only consists of 150 graph pairs, i.e., 300 graphs in total. We then train 3-GCN for 200 epochs, and all other systems for 1000 epochs on SPARSEEXP, as opposed to 100 and 500 respectively for EXP, to give all evaluated models a better opportunity to compensate for the smaller dataset size. We show the learning curves for all models on SPARSEEXP, and reproduce the original figure for EXP, in Figure 6 for easier comparison. ", + "bbox": [ + 174, + 603, + 825, + 686 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "First, we observe that all models converge slower on SPARSEEXP compared to EXP. This is not surprising, as a lower data availability makes learning a well-performing function slower and more challenging. More specifically, sparsity implies that (i) fewer weight updates are made per epoch, and (ii) these updates are of lower quality, as they are computed from a less representative and complete dataset. Nonetheless, the same relative convergence patterns between GCN-RNI models and 3-GCN are also visible in this setting, further highighting the increased convergence time required by GCN-RNI models. ", + "bbox": [ + 174, + 694, + 825, + 791 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We also observe that all GCN-RNI models, though also eventually converging, do so in a more volatile fashion. Indeed, GCN-RNI models suffer from the sparseness of the dataset, as this makes them more sensitive to RNI. As a result, these models require more training to effectively learn robustness against RNI values, and learn this from a smaller sample set, increasing their variability further. Moreover, the nature of SPARSEEXP makes learning more difficult, as it fully relies on RNI for MPNNs to have a chance of achieving above-random performance, and thus encourages MPNNs to fit specific RNI values. Hence, RNI introduces significant volatility and variability to training, particularly with sparser data, and requires substantial training and epochs for GCN-RNI models to effectively develop a robustness to RNI instantiations. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/3e34f5c14f46d45fe3c84ed3ad0d37c02529c9ba1994fbc1b2a2e78da9469c34.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 207, + 102, + 460, + 271 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/d46c799afbb58cde215b1e1e6653e44f15833296971fcff521f1e04053e34a83.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 532, + 101, + 789, + 272 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/304db04ca525ded70671e1d8bb10d83ae0d8a2cb6de77ce8d9d70971cc2489a1.jpg", + "image_caption": [ + "Figure 6: Model convergence results for Experiment 1 on the datasets EXP and SPARSEEXP. " + ], + "image_footnote": [], + "bbox": [ + 207, + 325, + 460, + 523 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/5049be94ec77ea9ee403ae2c147fdec86bc0cad7769fd2e158608f1be76376c0.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 534, + 325, + 789, + 522 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(a) Learning curves for all GCN-RNI models and 3-GCN on CEXP. ", + "bbox": [ + 184, + 530, + 485, + 554 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(b) Learning curves for CEXP, split across EXP $\\left( / \\mathrm { E } \\right)$ and CORRUPT (/C). ", + "bbox": [ + 511, + 530, + 813, + 555 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/0e3f99163a4ce7abb7da4ef05db4d2d7798c4b79c8ae66352eaa87f363b990de.jpg", + "image_caption": [ + "Figure 7: Model convergence results for Experiment 2 on CEXP and SPARSECEXP. " + ], + "image_footnote": [], + "bbox": [ + 204, + 560, + 464, + 757 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/580ee443cb99c23aa3dd0992f0ec654652fdf2b78035ea953af789aafe04f02a.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 529, + 560, + 789, + 756 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(c) Learning curves for all GCN-RNI models and 3-GCN on SPARSECEXP. ", + "bbox": [ + 184, + 763, + 485, + 789 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(d) Learning curves for SPARSECEXP, split across EXP (/E) and CORRUPT (/C). ", + "bbox": [ + 511, + 762, + 813, + 790 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "A.5.2 EXPERIMENT 2 ON SPARSECEXP ", + "text_level": 1, + "bbox": [ + 178, + 843, + 460, + 858 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Analogously to Experiment 1, we generate a SPARSECEXP dataset similarly to CEXP, but only generate 150 graph pairs. Then, we select 75 graph pairs and modify them, as described in Appendix A.3.2. We report the learning curves for all models on SPARSECEXP, as well as the original curves for CEXP from the main body, in Figure 7. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/f95922159a6172a46a0928eb99e81682db9f3ff11e214b16b98558982a655af3.jpg", + "table_caption": [ + "Table 2: Hyper-parameter configurations for all GCN-RNIand 3-GCN experiments. " + ], + "table_footnote": [], + "table_body": "
DatasetEXPCEXP
入p入p
GCN1×10-4N/A1×10-4N/A
GCN-12.5%RNI2×10-4N2×10-4N
GCN-50%RNI2×10-4N2×10-4N
GCN-87.5%RNI2×10-4N5×10-4N
GCN-RNI5×10-4N5×10-4N
3-GCN5×10-4N/A2×10-4N/A
", + "bbox": [ + 307, + 126, + 684, + 258 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/45afe8cb6efa15f221b488c7c5af1b80f06a2dc603e90554fd21b679092adecd.jpg", + "table_caption": [ + "Table 3: Performance of GCN-RNI models on the EXP dataset with the hyperbolic tangent activation function. " + ], + "table_footnote": [], + "table_body": "
ModelTesting Accuracy (%)
GCN-RNI(U)92.7 ± 5.61
GCN-RNI(N)96.0 ± 2.11
GCN-RNI(XU)64.6 ± 19.9
GCN-RNI(XN)63.0 ± 20.9
", + "bbox": [ + 352, + 311, + 640, + 397 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "As in the previous subsection, similar behavior is observed on SPARSECEXP compared with CEXP, only differing by slower convergence in the former case. However, we note that the “struggle” phase described in the main paper, which only occurs during the first 100 epochs over CEXP, lasts for around 500 epochs on SPARSEEXP. Intuitively, this “struggle” phenomenon is due to conflicting learning requirements, stemming from CORRUPT and EXP, which effectively require models to “isolate” deterministic dimensions for CORRUPT, and other randomized dimensions for $\\overline { { \\mathrm { E x p } } }$ . This in itself is already challenging on CEXP, but is made even more difficult on SPARSECEXP due to its sparsity. Indeed, sparsity makes that further samples are needed in expectation to find a reasonable solution, leading to a lengthy “struggle” phase, in which both CORRUPT and $\\overline { { \\mathrm { E x p } } }$ data points conflict with one another during optimization. ", + "bbox": [ + 173, + 422, + 825, + 566 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "A.6 HYPER-PARAMETER DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 583, + 421, + 597 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "All GCN models with (partially or completely) deterministic initial node embeddings map a 2- dimensional one-hot encoding of node type (literal or disjunction) to a $k$ -dimensional embedding space, where $k$ corresponds to the dimensionality of the deterministic embeddings. Furthermore, the final prediction for every graph is computed by aggregating all node embeddings following message passing using the max function, and then passing the result through a multi-layer perceptron of 3 layers with dimensionality $x$ , 32 and 2 respectively, where $x$ is the embedding dimensionality used in the given model. The activation function for the first two MLP layers is the ELU function (Clevert et al., 2016), and the softmax function is used to make a final prediction at the final MLP layer. ", + "bbox": [ + 174, + 609, + 825, + 720 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "All neural networks in this work are optimized using the Adam optimizer (Kingma & Ba, 2015). All training is conducted with a fixed learning rate $\\lambda$ , for fairer comparison between all models. Initially, decaying learning rates were used, but these were discarded, as they yielded sub-optimal convergence for all GCN-RNI models. Finally, all experiments were run on a V100 GPU. Detailed hyper-parameters, namely learning rate $\\lambda$ and RNI distribution $p$ , per model on every evaluation dataset are shown in Table 2. ", + "bbox": [ + 174, + 727, + 825, + 810 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "A.6.1 RESULTS FOR GCN-RNI WITH HYPERBOLIC TANGENT ACTIVATION ", + "text_level": 1, + "bbox": [ + 174, + 827, + 697, + 840 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In addition to experimenting with the RNI probability distribution, we also experimented with different activation functions for the GCN message passing iterations. Results are shown in Table 3. Performance with tanh is significantly more variable across distributions than ELU, which shows that RNI can be highly sensitive to practical choices of hyper-parameters. ", + "bbox": [ + 176, + 851, + 825, + 906 + ], + "page_idx": 20 + } +] \ No newline at end of file diff --git a/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_middle.json b/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..8e9560be1e4f781374a390ffd00d44684ce114ec --- /dev/null +++ b/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_middle.json @@ -0,0 +1,74021 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 107, + 79, + 504, + 96 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 504, + 96 + ], + "score": 1.0, + "content": "THE SURPRISING POWER OF GRAPH NEURAL NET-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 450, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 450, + 118 + ], + "score": 1.0, + "content": "WORKS WITH RANDOM NODE INITIALIZATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 142, + 210, + 468, + 473 + ], + "lines": [ + { + "bbox": [ + 142, + 210, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 142, + 210, + 469, + 223 + ], + "score": 1.0, + "content": "Graph neural networks (GNNs) are effective models for representation learning", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 470, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 470, + 234 + ], + "score": 1.0, + "content": "on graph-structured data. However, standard GNNs are limited in their expressive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 469, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 469, + 244 + ], + "score": 1.0, + "content": "power, as they cannot distinguish graphs beyond the capability of the Weisfeiler-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "score": 1.0, + "content": "Leman (1-WL) graph isomorphism heuristic. 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In this paper, we analyze the expressive power of GNNs", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "score": 1.0, + "content": "with RNI, and pose the following question: are GNNs with RNI more expressive", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "than GNNs? 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Our empirical findings support", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 469, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 469, + 408 + ], + "score": 1.0, + "content": "the superior performance of GNNs with RNI over standard GNNs. In fact, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "score": 1.0, + "content": "demonstrate that the performance of GNNs with RNI is often comparable with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 417, + 470, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 470, + 432 + ], + "score": 1.0, + "content": "or better than that of higher-order GNNs, while keeping the much lower memory", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 430, + 470, + 442 + ], + "spans": [ + { + "bbox": [ + 141, + 430, + 470, + 442 + ], + "score": 1.0, + "content": "requirements of standard GNNs. 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These properties provide GNNs with a strong", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "inductive bias, enabling them to effectively learn and combine both local and global graph features", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "(Battaglia et al., 2018). 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In particular, MPNNs are at most as powerful as the Weisfeiler-Leman", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "(1-WL) graph isomorphism heuristic (Morris et al., 2019; Xu et al., 2019), and thus cannot discern", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "between several families of non-isomorphic graphs, e.g., sets of regular graphs (Cai et al., 1992).", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "To address this limitation, alternative GNN architectures with provably higher expressive power", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "than MPNNs have been proposed. These models, which we refer to as higher-order GNNs, are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 331, + 700 + ], + "score": 1.0, + "content": "inspired by the more powerful generalization of 1-WL to", + "type": "text" + }, + { + "bbox": [ + 331, + 688, + 338, + 698 + ], + "score": 0.63, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 687, + 445, + 700 + ], + "score": 1.0, + "content": "−tuples of nodes, known as", + "type": "text" + }, + { + "bbox": [ + 446, + 688, + 452, + 698 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "-WL (Grohe,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "2017). These models are the only GNNs with an established universality result, but these models are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "computationally very demanding. 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However, standard GNNs are limited in their expressive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 469, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 469, + 244 + ], + "score": 1.0, + "content": "power, as they cannot distinguish graphs beyond the capability of the Weisfeiler-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "score": 1.0, + "content": "Leman (1-WL) graph isomorphism heuristic. This limitation motivated a large", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "score": 1.0, + "content": "body of work, including higher-order GNNs, which are provably more powerful", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 469, + 277 + ], + "score": 1.0, + "content": "models. To date, higher-order invariant and equivariant networks are the only", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 289 + ], + "score": 1.0, + "content": "models with known universality results, but these results are practically hindered", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "score": 1.0, + "content": "by prohibitive computational complexity. Thus, despite their limitations, standard", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 297, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 470, + 311 + ], + "score": 1.0, + "content": "GNNs are commonly used, due to their strong practical performance. In practice,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 309, + 469, + 320 + ], + "spans": [ + { + "bbox": [ + 142, + 309, + 469, + 320 + ], + "score": 1.0, + "content": "GNNs have shown a promising performance when enhanced with random node", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 320, + 469, + 331 + ], + "spans": [ + { + "bbox": [ + 142, + 320, + 469, + 331 + ], + "score": 1.0, + "content": "initialization (RNI), where the idea is to train and run the models with randomized", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 470, + 344 + ], + "score": 1.0, + "content": "initial node features. In this paper, we analyze the expressive power of GNNs", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "score": 1.0, + "content": "with RNI, and pose the following question: are GNNs with RNI more expressive", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "than GNNs? We prove that this is indeed the case, by showing that GNNs with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "score": 1.0, + "content": "RNI are universal, a first such result for GNNs not relying on computationally", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 469, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 469, + 386 + ], + "score": 1.0, + "content": "demanding higher-order properties. We then empirically analyze the effect of RNI", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 385, + 470, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 385, + 470, + 398 + ], + "score": 1.0, + "content": "on GNNs, based on carefully constructed datasets. Our empirical findings support", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 469, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 469, + 408 + ], + "score": 1.0, + "content": "the superior performance of GNNs with RNI over standard GNNs. In fact, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "score": 1.0, + "content": "demonstrate that the performance of GNNs with RNI is often comparable with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 417, + 470, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 470, + 432 + ], + "score": 1.0, + "content": "or better than that of higher-order GNNs, while keeping the much lower memory", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 430, + 470, + 442 + ], + "spans": [ + { + "bbox": [ + 141, + 430, + 470, + 442 + ], + "score": 1.0, + "content": "requirements of standard GNNs. However, this improvement typically comes at", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 441, + 469, + 453 + ], + "spans": [ + { + "bbox": [ + 141, + 441, + 469, + 453 + ], + "score": 1.0, + "content": "the cost of slower model convergence. Somewhat surprisingly, we found that the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 451, + 470, + 464 + ], + "spans": [ + { + "bbox": [ + 141, + 451, + 470, + 464 + ], + "score": 1.0, + "content": "convergence rate and the accuracy of the models can be improved by using only a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 463, + 290, + 475 + ], + "spans": [ + { + "bbox": [ + 141, + 463, + 290, + 475 + ], + "score": 1.0, + "content": "partial random initialization regime.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 16.5, + "bbox_fs": [ + 141, + 210, + 470, + 475 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 206, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 208, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 208, + 509 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 504, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 504, + 529 + ], + "score": 1.0, + "content": "Graph neural networks (GNNs) (Scarselli et al., 2009; Gori et al., 2005) are neural architectures de-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "signed for learning functions over graph-structured data, and naturally encode desirable properties", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "such as permutation invariance (resp., equivariance) relative to graph nodes, and node-level compu-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "tation based on message passing between these nodes. These properties provide GNNs with a strong", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "inductive bias, enabling them to effectively learn and combine both local and global graph features", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "(Battaglia et al., 2018). As a result, GNNs have been applied to a multitude of tasks, ranging from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "protein classification (Gilmer et al., 2017) and synthesis (You et al., 2018), protein-protein interac-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 607 + ], + "score": 1.0, + "content": "tion (Fout et al., 2017), and social network analysis (Hamilton et al., 2017), to recommender systems", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 605, + 477, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 477, + 617 + ], + "score": 1.0, + "content": "(Ying et al., 2018) and combinatorial optimization (Bengio et al., 2018; Selsam et al., 2019).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 517, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "However, popular GNN architectures, primarily based on message passing (MPNNs), are limited", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "in their expressive power. In particular, MPNNs are at most as powerful as the Weisfeiler-Leman", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "(1-WL) graph isomorphism heuristic (Morris et al., 2019; Xu et al., 2019), and thus cannot discern", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "between several families of non-isomorphic graphs, e.g., sets of regular graphs (Cai et al., 1992).", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "To address this limitation, alternative GNN architectures with provably higher expressive power", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "than MPNNs have been proposed. These models, which we refer to as higher-order GNNs, are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 331, + 700 + ], + "score": 1.0, + "content": "inspired by the more powerful generalization of 1-WL to", + "type": "text" + }, + { + "bbox": [ + 331, + 688, + 338, + 698 + ], + "score": 0.63, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 687, + 445, + 700 + ], + "score": 1.0, + "content": "−tuples of nodes, known as", + "type": "text" + }, + { + "bbox": [ + 446, + 688, + 452, + 698 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "-WL (Grohe,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "2017). These models are the only GNNs with an established universality result, but these models are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "computationally very demanding. As a result, MPNNs, despite their limited expressiveness, remain", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 336, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 336, + 733 + ], + "score": 1.0, + "content": "the standard GNN model for graph learning applications.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 622, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "In a parallel development, MPNNs have recently achieved significant empirical improvements using", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "random node initialization (RNI), through which initial graph node embeddings are randomly set.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "Indeed, RNI has enabled MPNNs to distinguish instances that 1-WL cannot distinguish, and is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "proven to enable better approximation of a class of combinatorial problems (Sato et al., 2020).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "However, the effect of RNI on the expressive power of GNNs has not yet been comprehensively", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 487, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 487, + 150 + ], + "score": 1.0, + "content": "studied, and its impact on the inductive capacity and learning ability of GNNs remains unclear.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "In this paper, we thoroughly study the impact of RNI on MPNNs. First, we prove that MPNNs", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "enhanced with RNI are universal, in the sense that they can approximate every function defined", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "on graphs of any fixed order. This follows from a logical characterisation of the expressiveness", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "of MPNNs (Barcelo et al., 2020) combined with an argument on order-invariant definability. Our ´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "result strongly contrasts with existing 1-WL limitations for deterministic MPNNs, and provides a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 430, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 430, + 222 + ], + "score": 1.0, + "content": "foundation for developing very expressive and memory-efficient MPNN models.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "To empirically verify our theoretical findings, we carry out a careful empirical study to quantify the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "practical impact of RNI. To this end, we design EXP, a synthetic dataset requiring 2-WL expressive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 104, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "power for models to achieve above-random performance, and run MPNNs with RNI on it, to ob-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "serve how well and how easily this model can learn and generalize based on this dataset. Then, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "propose CEXP, a modification of EXP with partially 1-WL distinguishable data, and evaluate the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 502, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 502, + 293 + ], + "score": 1.0, + "content": "same questions in this more variable setting. Overall, the contributions of this paper are as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 105, + 302, + 506, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "- We prove that MPNNs with RNI are universal, a significant improvement over the 1-WL limit of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 113, + 313, + 494, + 326 + ], + "spans": [ + { + "bbox": [ + 113, + 313, + 494, + 326 + ], + "score": 1.0, + "content": "standard MPNNs and, to our knowledge, a first universality result for memory-efficient GNNs.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "- We introduce two carefully designed datasets, EXP and CEXP, based on graph pairs only distin-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 114, + 340, + 401, + 353 + ], + "spans": [ + { + "bbox": [ + 114, + 340, + 401, + 353 + ], + "score": 1.0, + "content": "guishable by 2-WL or higher, to rigorously evaluate the impact of RNI.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "- Using these datasets, we thoroughly analyze the effects of RNI on MPNN, and observe that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 114, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 114, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "(i) MPNNs with RNI can closely match the performance of higher-order GNNs, (ii) the improved", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 113, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 113, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "performance of MPNNs with RNI comes at the cost of slower convergence (compared to higher-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 114, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 114, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "order GNNs), and (iii) using a partial random initialization regime over node features typically", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 113, + 401, + 352, + 412 + ], + "spans": [ + { + "bbox": [ + 113, + 401, + 352, + 412 + ], + "score": 1.0, + "content": "improves convergence rate and the accuracy of the models.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 109, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 109, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "- We additionally perform the same experiments with analog, sparser datasets, with longer training,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 427, + 312, + 439 + ], + "spans": [ + { + "bbox": [ + 114, + 427, + 312, + 439 + ], + "score": 1.0, + "content": "and observe similar behavior, but more volatility.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 272, + 472 + ], + "lines": [ + { + "bbox": [ + 104, + 457, + 275, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 275, + 475 + ], + "score": 1.0, + "content": "2 GRAPH NEURAL NETWORKS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "Graph neural networks (GNNs) (Gori et al., 2005; Scarselli et al., 2009) are neural architectures", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "dedicated to learning functions over graph-structured data. In a GNN, nodes in the input graph", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "are assigned vector representations, which are updated iteratively through series of invariant or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 531 + ], + "score": 1.0, + "content": "equivariant computational layers. We recall message passing neural networks (MPNNs) (Gilmer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "et al., 2017), a popular family of GNN models, and its expressive power in relation to the Weisfeiler-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 552 + ], + "score": 1.0, + "content": "Leman graph isomorphism heuristic. We discuss alternative GNN models in Section 3; for a broader", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 552, + 362, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 362, + 563 + ], + "score": 1.0, + "content": "coverage, we refer the reader to the literature (Hamilton, 2020).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 504, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "In MPNNs, node representations aggregate messages from their neighboring nodes, and use this", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 416, + 592 + ], + "score": 1.0, + "content": "information to iteratively update their representations. Formally, given a node", + "type": "text" + }, + { + "bbox": [ + 416, + 581, + 423, + 589 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 579, + 505, + 592 + ], + "score": 1.0, + "content": ", its vector represen-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 480, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 131, + 603 + ], + "score": 1.0, + "content": "tation", + "type": "text" + }, + { + "bbox": [ + 132, + 592, + 148, + 602 + ], + "score": 0.88, + "content": "v _ { x , t }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 591, + 179, + 603 + ], + "score": 1.0, + "content": "at time", + "type": "text" + }, + { + "bbox": [ + 180, + 591, + 184, + 600 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 591, + 274, + 603 + ], + "score": 1.0, + "content": ", and its neighborhood", + "type": "text" + }, + { + "bbox": [ + 275, + 590, + 299, + 602 + ], + "score": 0.92, + "content": "N ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 591, + 480, + 603 + ], + "score": 1.0, + "content": ", a message passing update can be written as:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 609, + 417, + 629 + ], + "lines": [ + { + "bbox": [ + 193, + 609, + 417, + 629 + ], + "spans": [ + { + "bbox": [ + 193, + 609, + 417, + 629 + ], + "score": 0.92, + "content": "v _ { x , t + 1 } = c o m b i n e { \\Big ( } v _ { x , t } , a g g r e g a t e { \\big ( } \\{ v _ { y , t } | y \\in N ( x ) \\} { \\big ) } { \\Big ) } ,", + "type": "interline_equation", + "image_path": "1891c78d9e9eba3f5c35eee8496bd9ba075bd2b9f17c08536b623b5ef9d89f17.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 193, + 609, + 417, + 629 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 649 + ], + "score": 1.0, + "content": "where combine and aggregate are functions, and aggregate is typically permutation-invariant. Once", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 647, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 659 + ], + "score": 1.0, + "content": "message passing is complete, the final node representations are then used to compute target outputs.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "Prominent message passing GNN architectures include graph convolutional networks (GCNs) (Kipf", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 420, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 420, + 681 + ], + "score": 1.0, + "content": "& Welling, 2017) and gated graph neural networks (GGNNs) (Li et al., 2016).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 504, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "It is well-known that standard MPNNs have the same power as the 1-dimensional Weisfeiler-Leman", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "score": 1.0, + "content": "algorithm (1-WL) (Xu et al., 2019; Morris et al., 2019). This entails that two nodes in a graph cannot", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 707, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 720 + ], + "score": 1.0, + "content": "be distinguished if 1-WL does not distinguish them, and neither can two graphs be distinguished if", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 718, + 233, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 233, + 731 + ], + "score": 1.0, + "content": "1-WL cannot distinguish them.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "In a parallel development, MPNNs have recently achieved significant empirical improvements using", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "random node initialization (RNI), through which initial graph node embeddings are randomly set.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "Indeed, RNI has enabled MPNNs to distinguish instances that 1-WL cannot distinguish, and is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "proven to enable better approximation of a class of combinatorial problems (Sato et al., 2020).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "However, the effect of RNI on the expressive power of GNNs has not yet been comprehensively", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 487, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 487, + 150 + ], + "score": 1.0, + "content": "studied, and its impact on the inductive capacity and learning ability of GNNs remains unclear.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 81, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "In this paper, we thoroughly study the impact of RNI on MPNNs. First, we prove that MPNNs", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "enhanced with RNI are universal, in the sense that they can approximate every function defined", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "on graphs of any fixed order. This follows from a logical characterisation of the expressiveness", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "of MPNNs (Barcelo et al., 2020) combined with an argument on order-invariant definability. Our ´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "result strongly contrasts with existing 1-WL limitations for deterministic MPNNs, and provides a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 430, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 430, + 222 + ], + "score": 1.0, + "content": "foundation for developing very expressive and memory-efficient MPNN models.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 154, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "To empirically verify our theoretical findings, we carry out a careful empirical study to quantify the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "practical impact of RNI. To this end, we design EXP, a synthetic dataset requiring 2-WL expressive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 104, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "power for models to achieve above-random performance, and run MPNNs with RNI on it, to ob-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "serve how well and how easily this model can learn and generalize based on this dataset. Then, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "propose CEXP, a modification of EXP with partially 1-WL distinguishable data, and evaluate the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 502, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 502, + 293 + ], + "score": 1.0, + "content": "same questions in this more variable setting. Overall, the contributions of this paper are as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 225, + 506, + 293 + ] + }, + { + "type": "list", + "bbox": [ + 105, + 302, + 506, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "- We prove that MPNNs with RNI are universal, a significant improvement over the 1-WL limit of", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 113, + 313, + 494, + 326 + ], + "spans": [ + { + "bbox": [ + 113, + 313, + 494, + 326 + ], + "score": 1.0, + "content": "standard MPNNs and, to our knowledge, a first universality result for memory-efficient GNNs.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "- We introduce two carefully designed datasets, EXP and CEXP, based on graph pairs only distin-", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 340, + 401, + 353 + ], + "spans": [ + { + "bbox": [ + 114, + 340, + 401, + 353 + ], + "score": 1.0, + "content": "guishable by 2-WL or higher, to rigorously evaluate the impact of RNI.", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "- Using these datasets, we thoroughly analyze the effects of RNI on MPNN, and observe that", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 114, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "(i) MPNNs with RNI can closely match the performance of higher-order GNNs, (ii) the improved", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 113, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 113, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "performance of MPNNs with RNI comes at the cost of slower convergence (compared to higher-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 114, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 114, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "order GNNs), and (iii) using a partial random initialization regime over node features typically", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 113, + 401, + 352, + 412 + ], + "spans": [ + { + "bbox": [ + 113, + 401, + 352, + 412 + ], + "score": 1.0, + "content": "improves convergence rate and the accuracy of the models.", + "type": "text" + } + ], + "index": 26, + "is_list_end_line": true + }, + { + "bbox": [ + 109, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 109, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "- We additionally perform the same experiments with analog, sparser datasets, with longer training,", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 427, + 312, + 439 + ], + "spans": [ + { + "bbox": [ + 114, + 427, + 312, + 439 + ], + "score": 1.0, + "content": "and observe similar behavior, but more volatility.", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + } + ], + "index": 23, + "bbox_fs": [ + 105, + 301, + 506, + 439 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 272, + 472 + ], + "lines": [ + { + "bbox": [ + 104, + 457, + 275, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 275, + 475 + ], + "score": 1.0, + "content": "2 GRAPH NEURAL NETWORKS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "Graph neural networks (GNNs) (Gori et al., 2005; Scarselli et al., 2009) are neural architectures", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "dedicated to learning functions over graph-structured data. In a GNN, nodes in the input graph", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "are assigned vector representations, which are updated iteratively through series of invariant or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 531 + ], + "score": 1.0, + "content": "equivariant computational layers. We recall message passing neural networks (MPNNs) (Gilmer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "et al., 2017), a popular family of GNN models, and its expressive power in relation to the Weisfeiler-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 552 + ], + "score": 1.0, + "content": "Leman graph isomorphism heuristic. We discuss alternative GNN models in Section 3; for a broader", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 552, + 362, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 362, + 563 + ], + "score": 1.0, + "content": "coverage, we refer the reader to the literature (Hamilton, 2020).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 486, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 504, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "In MPNNs, node representations aggregate messages from their neighboring nodes, and use this", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 416, + 592 + ], + "score": 1.0, + "content": "information to iteratively update their representations. Formally, given a node", + "type": "text" + }, + { + "bbox": [ + 416, + 581, + 423, + 589 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 579, + 505, + 592 + ], + "score": 1.0, + "content": ", its vector represen-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 480, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 131, + 603 + ], + "score": 1.0, + "content": "tation", + "type": "text" + }, + { + "bbox": [ + 132, + 592, + 148, + 602 + ], + "score": 0.88, + "content": "v _ { x , t }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 591, + 179, + 603 + ], + "score": 1.0, + "content": "at time", + "type": "text" + }, + { + "bbox": [ + 180, + 591, + 184, + 600 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 591, + 274, + 603 + ], + "score": 1.0, + "content": ", and its neighborhood", + "type": "text" + }, + { + "bbox": [ + 275, + 590, + 299, + 602 + ], + "score": 0.92, + "content": "N ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 591, + 480, + 603 + ], + "score": 1.0, + "content": ", a message passing update can be written as:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 568, + 505, + 603 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 609, + 417, + 629 + ], + "lines": [ + { + "bbox": [ + 193, + 609, + 417, + 629 + ], + "spans": [ + { + "bbox": [ + 193, + 609, + 417, + 629 + ], + "score": 0.92, + "content": "v _ { x , t + 1 } = c o m b i n e { \\Big ( } v _ { x , t } , a g g r e g a t e { \\big ( } \\{ v _ { y , t } | y \\in N ( x ) \\} { \\big ) } { \\Big ) } ,", + "type": "interline_equation", + "image_path": "1891c78d9e9eba3f5c35eee8496bd9ba075bd2b9f17c08536b623b5ef9d89f17.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 193, + 609, + 417, + 629 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 649 + ], + "score": 1.0, + "content": "where combine and aggregate are functions, and aggregate is typically permutation-invariant. Once", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 647, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 659 + ], + "score": 1.0, + "content": "message passing is complete, the final node representations are then used to compute target outputs.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "Prominent message passing GNN architectures include graph convolutional networks (GCNs) (Kipf", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 420, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 420, + 681 + ], + "score": 1.0, + "content": "& Welling, 2017) and gated graph neural networks (GGNNs) (Li et al., 2016).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 635, + 506, + 681 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 504, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "It is well-known that standard MPNNs have the same power as the 1-dimensional Weisfeiler-Leman", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "score": 1.0, + "content": "algorithm (1-WL) (Xu et al., 2019; Morris et al., 2019). This entails that two nodes in a graph cannot", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 707, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 720 + ], + "score": 1.0, + "content": "be distinguished if 1-WL does not distinguish them, and neither can two graphs be distinguished if", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 718, + 233, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 233, + 731 + ], + "score": 1.0, + "content": "1-WL cannot distinguish them.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 685, + 506, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 328, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 329, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 187, + 94 + ], + "score": 1.0, + "content": "Consider the graphs", + "type": "text" + }, + { + "bbox": [ + 188, + 83, + 197, + 93 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 82, + 214, + 94 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 83, + 225, + 92 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 82, + 329, + 94 + ], + "score": 1.0, + "content": "shown in Figure 1. 1-WL", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 329, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 290, + 106 + ], + "score": 1.0, + "content": "cannot distinguish any two nodes of the graph", + "type": "text" + }, + { + "bbox": [ + 291, + 94, + 300, + 104 + ], + "score": 0.73, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 93, + 329, + 106 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 329, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 250, + 116 + ], + "score": 1.0, + "content": "for example, the invariant function", + "type": "text" + }, + { + "bbox": [ + 250, + 105, + 309, + 116 + ], + "score": 0.93, + "content": "f : V ( G ) \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 104, + 329, + 116 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 329, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 235, + 126 + ], + "score": 1.0, + "content": "maps all nodes in the 4-cycle of", + "type": "text" + }, + { + "bbox": [ + 236, + 116, + 245, + 125 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 115, + 329, + 126 + ], + "score": 1.0, + "content": "to 1 and all nodes in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 328, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 328, + 139 + ], + "score": 1.0, + "content": "the triangle to 0 is not expressible (or approximable) by", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 329, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 329, + 149 + ], + "score": 1.0, + "content": "a GNN. Moreover, 1-WL cannot distinguish the graphs", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 148, + 328, + 160 + ], + "spans": [ + { + "bbox": [ + 107, + 149, + 129, + 160 + ], + "score": 0.89, + "content": "G , H", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 148, + 328, + 160 + ], + "score": 1.0, + "content": ", even though they are obviously non-isomorphic,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 329, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 329, + 171 + ], + "score": 1.0, + "content": "so no classifier based on the node embeddings com-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 313, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 313, + 183 + ], + "score": 1.0, + "content": "puted by an MPNN can distinguish the two graphs.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "image", + "bbox": [ + 347, + 84, + 497, + 150 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 347, + 84, + 497, + 150 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 347, + 84, + 497, + 150 + ], + "spans": [ + { + "bbox": [ + 347, + 84, + 497, + 150 + ], + "score": 0.963, + "type": "image", + "image_path": "aa77d9397242894aa279436cb0ba0d6163929d466561f5dcda8966db0ef5a676.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 347, + 84, + 497, + 117.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 347, + 117.0, + 497, + 150.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 336, + 157, + 504, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 335, + 157, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 335, + 157, + 378, + 170 + ], + "score": 1.0, + "content": "Figure 1:", + "type": "text" + }, + { + "bbox": [ + 378, + 158, + 387, + 168 + ], + "score": 0.72, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 157, + 406, + 170 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 158, + 417, + 167 + ], + "score": 0.73, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 157, + 505, + 170 + ], + "score": 1.0, + "content": "are indistinguishable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 335, + 169, + 493, + 181 + ], + "spans": [ + { + "bbox": [ + 335, + 169, + 493, + 181 + ], + "score": 1.0, + "content": "by 1-WL and hence by (1-WL) GNNs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "A somewhat trivial limitation in the expressiveness of MPNNs is that information is only propagated", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "along edges, and hence can never be shared between distinct connected components of a graph", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "(Barcelo et al., 2020; Xu et al., 2019). An easy way to overcome this limitation is by adding ´ global", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 220, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 504, + 232 + ], + "score": 1.0, + "content": "readouts, that is, permutation-invariant functions that aggregate the current states of all nodes1.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "Throughout the paper, we therefore focus on MPNNs with global readouts (also called aggregate-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 242, + 412, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 412, + 254 + ], + "score": 1.0, + "content": "combine GNNs with global readout, i.e., ACR-GNNs (Barcelo et al., 2020)). ´", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 268, + 293, + 281 + ], + "lines": [ + { + "bbox": [ + 104, + 267, + 295, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 295, + 284 + ], + "score": 1.0, + "content": "3 RELATED WORK & MOTIVATION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "Developing more expressive GNNs is an active research area due to the prominence of GNNs for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "relational learning (Hamilton et al., 2017) and combinatorial optimization (Bengio et al., 2018). As", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "mentioned earlier, standard GNN models are at most as expressive as 1-WL (Morris et al., 2019; Xu", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "score": 1.0, + "content": "et al., 2019), and thus cannot distinguish between non-isomorphic input instances. In this section,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 337, + 438, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 438, + 350 + ], + "score": 1.0, + "content": "we describe theoretical results quantifying the expressive power of existing GNNs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 354, + 446, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 449, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 449, + 367 + ], + "score": 1.0, + "content": "Higher-order GNNs. We recall the following families of higher-order GNN models:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "- Invariant (resp., equivariant) graph networks: Invariant (resp., equivariant) graph networks", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 114, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "(Maron et al., 2019b) represent graphs as a tensor where node adjacency is directly encoded, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 114, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "implicitly pass information between nodes through invariant (resp., equivariant) computational", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 113, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 113, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "blocks. Hence, these models are themselves invariant (resp., equivariant), a desirable property", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 114, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 114, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "for computations on graphs. Following intermediate blocks, higher-order tensors are typically", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 114, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "returned, and the order of these tensors correlates directly with the expressive power of the overall", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 441, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 114, + 441, + 505, + 451 + ], + "score": 1.0, + "content": "model. Indeed, invariant networks (Maron et al., 2019c), and later equivariant networks (Keriven", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 113, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 113, + 450, + 391, + 464 + ], + "score": 1.0, + "content": "& Peyre, 2019), are shown to be universal, but with tensor orders of ´", + "type": "text" + }, + { + "bbox": [ + 391, + 450, + 425, + 463 + ], + "score": 0.93, + "content": "\\dot { O } ( | V | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 450, + 456, + 464 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 457, + 451, + 471, + 463 + ], + "score": 0.89, + "content": "| V |", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 114, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "the number of graph nodes. Furthermore, invariant (resp., equivariant) networks with intermediate", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 114, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 114, + 472, + 165, + 486 + ], + "score": 1.0, + "content": "tensor order", + "type": "text" + }, + { + "bbox": [ + 165, + 473, + 172, + 483 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 472, + 330, + 486 + ], + "score": 1.0, + "content": "are shown to be equivalent in power to", + "type": "text" + }, + { + "bbox": [ + 330, + 473, + 359, + 484 + ], + "score": 0.83, + "content": "\\left( k - 1 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "-WL (Maron et al., 2019a), which is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 114, + 484, + 224, + 496 + ], + "score": 1.0, + "content": "strictly more expressive as", + "type": "text" + }, + { + "bbox": [ + 225, + 484, + 231, + 493 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "increases (Cai et al., 1992). Therefore, such universal higher-order", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 496, + 386, + 507 + ], + "spans": [ + { + "bbox": [ + 115, + 496, + 386, + 507 + ], + "score": 1.0, + "content": "models require intractably-sized intermediate tensors in practice.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 230, + 520 + ], + "score": 1.0, + "content": "- Higher-order MPNNs: The", + "type": "text" + }, + { + "bbox": [ + 230, + 509, + 259, + 519 + ], + "score": 0.59, + "content": "k { \\mathrm { - } } \\mathbf { W } \\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "hierarchy has been directly emulated in GNNs, such that these", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 114, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 114, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "models learn embeddings for tuples of nodes, and perform messaging passing between them, as", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 113, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 113, + 530, + 409, + 543 + ], + "score": 1.0, + "content": "opposed to individual nodes. This approach has yielded models such as", + "type": "text" + }, + { + "bbox": [ + 409, + 531, + 416, + 541 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "-GNNs (Morris et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 113, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 113, + 540, + 143, + 556 + ], + "score": 1.0, + "content": "2019).", + "type": "text" + }, + { + "bbox": [ + 144, + 542, + 150, + 552 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 540, + 201, + 556 + ], + "score": 1.0, + "content": "-GNNs have", + "type": "text" + }, + { + "bbox": [ + 202, + 542, + 230, + 554 + ], + "score": 0.86, + "content": "\\left( k - 1 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 540, + 363, + 556 + ], + "score": 1.0, + "content": "-WL expressive power,2 but need", + "type": "text" + }, + { + "bbox": [ + 363, + 541, + 398, + 554 + ], + "score": 0.93, + "content": "O ( | V | ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "memory to run, leading to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 114, + 554, + 254, + 565 + ], + "spans": [ + { + "bbox": [ + 114, + 554, + 254, + 565 + ], + "score": 1.0, + "content": "excessive memory requirements.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 581 + ], + "score": 1.0, + "content": "- Provably powerful graph networks (PPGNs): PPGN is an invariant GNN (Maron et al., 2019a),", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 114, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 114, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "based on “blocks” of multilayer perceptrons (MLPs) and matrix multiplication, which theoreti-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 113, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 113, + 588, + 362, + 602 + ], + "score": 1.0, + "content": "cally has 2-WL expressive power, and only requires memory", + "type": "text" + }, + { + "bbox": [ + 362, + 588, + 397, + 601 + ], + "score": 0.93, + "content": "O ( | V | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 588, + 454, + 602 + ], + "score": 1.0, + "content": "(compared to", + "type": "text" + }, + { + "bbox": [ + 455, + 588, + 489, + 601 + ], + "score": 0.93, + "content": "O ( | V | ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 113, + 598, + 507, + 614 + ], + "spans": [ + { + "bbox": [ + 113, + 598, + 507, + 614 + ], + "score": 1.0, + "content": "3-GNNs). However, PPGN theoretically requires exponentially many samples in the number of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 114, + 611, + 459, + 624 + ], + "spans": [ + { + "bbox": [ + 114, + 611, + 459, + 624 + ], + "score": 1.0, + "content": "graph nodes to learn necessary functions for 2-WL expressiveness (Puny et al., 2020).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 503, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "score": 1.0, + "content": "GNNs with random node initialization. MPNNs have been enhanced with random node initial-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "ization (Sato et al., 2020), such that the model trains and runs with partially randomized initial", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 682, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 679, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 119, + 680, + 132, + 691 + ], + "score": 0.28, + "content": "^ 2 \\mathrm { I n }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 679, + 506, + 694 + ], + "score": 1.0, + "content": "the literature, one can find different (though equally expressive) versions of the Weisfeiler Leman al-", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 360, + 703 + ], + "score": 1.0, + "content": "gorithm leading to inconsistent dimension counts. For example, the", + "type": "text" + }, + { + "bbox": [ + 360, + 691, + 389, + 702 + ], + "score": 0.85, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 691, + 438, + 703 + ], + "score": 1.0, + "content": "-WL and the", + "type": "text" + }, + { + "bbox": [ + 439, + 691, + 467, + 702 + ], + "score": 0.88, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "-GNNs of", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 701, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 316, + 712 + ], + "score": 1.0, + "content": "Morris et al. (2019) are equivalent in expressiveness to the", + "type": "text" + }, + { + "bbox": [ + 316, + 702, + 322, + 711 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 701, + 505, + 712 + ], + "score": 1.0, + "content": "-WL of Cai et al. (1992); Grohe (2017). We follow", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "the version of Cai et al. (1992), as it has been adopted as the standard in the literature on graph isomorphism", + "type": "text" + } + ] + }, + { + "bbox": [ + 104, + 720, + 136, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 136, + 734 + ], + "score": 1.0, + "content": "testing.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 109, + 660, + 504, + 681 + ], + "lines": [ + { + "bbox": [ + 118, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 118, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "1In the terminology of Maron et al. (2019c), such global readouts are tensors of order 1 that are invariant", + "type": "text" + } + ] + }, + { + "bbox": [ + 107, + 670, + 333, + 682 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 333, + 682 + ], + "score": 1.0, + "content": "under the symmetric group of the vertex set of the input graph.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 328, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 329, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 187, + 94 + ], + "score": 1.0, + "content": "Consider the graphs", + "type": "text" + }, + { + "bbox": [ + 188, + 83, + 197, + 93 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 82, + 214, + 94 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 83, + 225, + 92 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 82, + 329, + 94 + ], + "score": 1.0, + "content": "shown in Figure 1. 1-WL", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 329, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 290, + 106 + ], + "score": 1.0, + "content": "cannot distinguish any two nodes of the graph", + "type": "text" + }, + { + "bbox": [ + 291, + 94, + 300, + 104 + ], + "score": 0.73, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 93, + 329, + 106 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 329, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 250, + 116 + ], + "score": 1.0, + "content": "for example, the invariant function", + "type": "text" + }, + { + "bbox": [ + 250, + 105, + 309, + 116 + ], + "score": 0.93, + "content": "f : V ( G ) \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 104, + 329, + 116 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 329, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 235, + 126 + ], + "score": 1.0, + "content": "maps all nodes in the 4-cycle of", + "type": "text" + }, + { + "bbox": [ + 236, + 116, + 245, + 125 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 115, + 329, + 126 + ], + "score": 1.0, + "content": "to 1 and all nodes in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 328, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 328, + 139 + ], + "score": 1.0, + "content": "the triangle to 0 is not expressible (or approximable) by", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 329, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 329, + 149 + ], + "score": 1.0, + "content": "a GNN. Moreover, 1-WL cannot distinguish the graphs", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 148, + 328, + 160 + ], + "spans": [ + { + "bbox": [ + 107, + 149, + 129, + 160 + ], + "score": 0.89, + "content": "G , H", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 148, + 328, + 160 + ], + "score": 1.0, + "content": ", even though they are obviously non-isomorphic,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 329, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 329, + 171 + ], + "score": 1.0, + "content": "so no classifier based on the node embeddings com-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 313, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 313, + 183 + ], + "score": 1.0, + "content": "puted by an MPNN can distinguish the two graphs.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 329, + 183 + ] + }, + { + "type": "image", + "bbox": [ + 347, + 84, + 497, + 150 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 347, + 84, + 497, + 150 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 347, + 84, + 497, + 150 + ], + "spans": [ + { + "bbox": [ + 347, + 84, + 497, + 150 + ], + "score": 0.963, + "type": "image", + "image_path": "aa77d9397242894aa279436cb0ba0d6163929d466561f5dcda8966db0ef5a676.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 347, + 84, + 497, + 117.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 347, + 117.0, + 497, + 150.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 336, + 157, + 504, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 335, + 157, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 335, + 157, + 378, + 170 + ], + "score": 1.0, + "content": "Figure 1:", + "type": "text" + }, + { + "bbox": [ + 378, + 158, + 387, + 168 + ], + "score": 0.72, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 157, + 406, + 170 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 158, + 417, + 167 + ], + "score": 0.73, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 157, + 505, + 170 + ], + "score": 1.0, + "content": "are indistinguishable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 335, + 169, + 493, + 181 + ], + "spans": [ + { + "bbox": [ + 335, + 169, + 493, + 181 + ], + "score": 1.0, + "content": "by 1-WL and hence by (1-WL) GNNs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "A somewhat trivial limitation in the expressiveness of MPNNs is that information is only propagated", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "along edges, and hence can never be shared between distinct connected components of a graph", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "(Barcelo et al., 2020; Xu et al., 2019). An easy way to overcome this limitation is by adding ´ global", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 220, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 504, + 232 + ], + "score": 1.0, + "content": "readouts, that is, permutation-invariant functions that aggregate the current states of all nodes1.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "Throughout the paper, we therefore focus on MPNNs with global readouts (also called aggregate-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 242, + 412, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 412, + 254 + ], + "score": 1.0, + "content": "combine GNNs with global readout, i.e., ACR-GNNs (Barcelo et al., 2020)). ´", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 187, + 505, + 254 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 268, + 293, + 281 + ], + "lines": [ + { + "bbox": [ + 104, + 267, + 295, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 295, + 284 + ], + "score": 1.0, + "content": "3 RELATED WORK & MOTIVATION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "Developing more expressive GNNs is an active research area due to the prominence of GNNs for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "relational learning (Hamilton et al., 2017) and combinatorial optimization (Bengio et al., 2018). As", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "mentioned earlier, standard GNN models are at most as expressive as 1-WL (Morris et al., 2019; Xu", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "score": 1.0, + "content": "et al., 2019), and thus cannot distinguish between non-isomorphic input instances. In this section,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 337, + 438, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 438, + 350 + ], + "score": 1.0, + "content": "we describe theoretical results quantifying the expressive power of existing GNNs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 293, + 505, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 354, + 446, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 449, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 449, + 367 + ], + "score": 1.0, + "content": "Higher-order GNNs. We recall the following families of higher-order GNN models:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 353, + 449, + 367 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "- Invariant (resp., equivariant) graph networks: Invariant (resp., equivariant) graph networks", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 114, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "(Maron et al., 2019b) represent graphs as a tensor where node adjacency is directly encoded, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 114, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "implicitly pass information between nodes through invariant (resp., equivariant) computational", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 113, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 113, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "blocks. Hence, these models are themselves invariant (resp., equivariant), a desirable property", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 114, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 114, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "for computations on graphs. Following intermediate blocks, higher-order tensors are typically", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 114, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "returned, and the order of these tensors correlates directly with the expressive power of the overall", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 441, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 114, + 441, + 505, + 451 + ], + "score": 1.0, + "content": "model. Indeed, invariant networks (Maron et al., 2019c), and later equivariant networks (Keriven", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 113, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 113, + 450, + 391, + 464 + ], + "score": 1.0, + "content": "& Peyre, 2019), are shown to be universal, but with tensor orders of ´", + "type": "text" + }, + { + "bbox": [ + 391, + 450, + 425, + 463 + ], + "score": 0.93, + "content": "\\dot { O } ( | V | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 450, + 456, + 464 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 457, + 451, + 471, + 463 + ], + "score": 0.89, + "content": "| V |", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 114, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "the number of graph nodes. Furthermore, invariant (resp., equivariant) networks with intermediate", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 114, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 114, + 472, + 165, + 486 + ], + "score": 1.0, + "content": "tensor order", + "type": "text" + }, + { + "bbox": [ + 165, + 473, + 172, + 483 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 472, + 330, + 486 + ], + "score": 1.0, + "content": "are shown to be equivalent in power to", + "type": "text" + }, + { + "bbox": [ + 330, + 473, + 359, + 484 + ], + "score": 0.83, + "content": "\\left( k - 1 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "-WL (Maron et al., 2019a), which is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 114, + 484, + 224, + 496 + ], + "score": 1.0, + "content": "strictly more expressive as", + "type": "text" + }, + { + "bbox": [ + 225, + 484, + 231, + 493 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "increases (Cai et al., 1992). Therefore, such universal higher-order", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 496, + 386, + 507 + ], + "spans": [ + { + "bbox": [ + 115, + 496, + 386, + 507 + ], + "score": 1.0, + "content": "models require intractably-sized intermediate tensors in practice.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 373, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 230, + 520 + ], + "score": 1.0, + "content": "- Higher-order MPNNs: The", + "type": "text" + }, + { + "bbox": [ + 230, + 509, + 259, + 519 + ], + "score": 0.59, + "content": "k { \\mathrm { - } } \\mathbf { W } \\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "hierarchy has been directly emulated in GNNs, such that these", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 114, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 114, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "models learn embeddings for tuples of nodes, and perform messaging passing between them, as", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 113, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 113, + 530, + 409, + 543 + ], + "score": 1.0, + "content": "opposed to individual nodes. This approach has yielded models such as", + "type": "text" + }, + { + "bbox": [ + 409, + 531, + 416, + 541 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "-GNNs (Morris et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 113, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 113, + 540, + 143, + 556 + ], + "score": 1.0, + "content": "2019).", + "type": "text" + }, + { + "bbox": [ + 144, + 542, + 150, + 552 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 540, + 201, + 556 + ], + "score": 1.0, + "content": "-GNNs have", + "type": "text" + }, + { + "bbox": [ + 202, + 542, + 230, + 554 + ], + "score": 0.86, + "content": "\\left( k - 1 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 540, + 363, + 556 + ], + "score": 1.0, + "content": "-WL expressive power,2 but need", + "type": "text" + }, + { + "bbox": [ + 363, + 541, + 398, + 554 + ], + "score": 0.93, + "content": "O ( | V | ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "memory to run, leading to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 114, + 554, + 254, + 565 + ], + "spans": [ + { + "bbox": [ + 114, + 554, + 254, + 565 + ], + "score": 1.0, + "content": "excessive memory requirements.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 106, + 508, + 506, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 581 + ], + "score": 1.0, + "content": "- Provably powerful graph networks (PPGNs): PPGN is an invariant GNN (Maron et al., 2019a),", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 114, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 114, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "based on “blocks” of multilayer perceptrons (MLPs) and matrix multiplication, which theoreti-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 113, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 113, + 588, + 362, + 602 + ], + "score": 1.0, + "content": "cally has 2-WL expressive power, and only requires memory", + "type": "text" + }, + { + "bbox": [ + 362, + 588, + 397, + 601 + ], + "score": 0.93, + "content": "O ( | V | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 588, + 454, + 602 + ], + "score": 1.0, + "content": "(compared to", + "type": "text" + }, + { + "bbox": [ + 455, + 588, + 489, + 601 + ], + "score": 0.93, + "content": "O ( | V | ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 113, + 598, + 507, + 614 + ], + "spans": [ + { + "bbox": [ + 113, + 598, + 507, + 614 + ], + "score": 1.0, + "content": "3-GNNs). However, PPGN theoretically requires exponentially many samples in the number of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 114, + 611, + 459, + 624 + ], + "spans": [ + { + "bbox": [ + 114, + 611, + 459, + 624 + ], + "score": 1.0, + "content": "graph nodes to learn necessary functions for 2-WL expressiveness (Puny et al., 2020).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 565, + 507, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 503, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "score": 1.0, + "content": "GNNs with random node initialization. MPNNs have been enhanced with random node initial-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "ization (Sato et al., 2020), such that the model trains and runs with partially randomized initial", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "node features. These models, denoted rGNNs, are shown to near-optimally approximate solutions", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "to specific combinatorial optimization problems, and are able to distinguish between 1-WL indistin-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "guishable graph pairs. rGNNs can also detect characteristic sub-graphs in an input graph with high", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "probability. Nonetheless, it remains open as to how much expressive power is exactly gained through", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "RNI, and, in general, whether a GNN model that is universal, scalable, and structure-preserving, can", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "be developed. Our work strongly improves the theoretical result of Sato et al. (2020), as it shows", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "universality of MPNNs with RNI, and thus that arbitrary real-valued functions over graphs can be", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "learned by MPNNs with the help of RNI. On the empirical side, we highlight the power of RNI", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "in a significantly more challenging setting than rGNN, using a target function (SAT) beyond their", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "theoretical scope. Indeed, for SAT, approximation is known to be hard, and fixed local structures", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 222, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 222, + 203 + ], + "score": 1.0, + "content": "are not useful for prediction.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 48.5, + "bbox_fs": [ + 106, + 631, + 505, + 654 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "node features. These models, denoted rGNNs, are shown to near-optimally approximate solutions", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "to specific combinatorial optimization problems, and are able to distinguish between 1-WL indistin-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "guishable graph pairs. rGNNs can also detect characteristic sub-graphs in an input graph with high", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "probability. Nonetheless, it remains open as to how much expressive power is exactly gained through", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "RNI, and, in general, whether a GNN model that is universal, scalable, and structure-preserving, can", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "be developed. Our work strongly improves the theoretical result of Sato et al. 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On the empirical side, we highlight the power of RNI", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "in a significantly more challenging setting than rGNN, using a target function (SAT) beyond their", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "theoretical scope. Indeed, for SAT, approximation is known to be hard, and fixed local structures", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 222, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 222, + 203 + ], + "score": 1.0, + "content": "are not useful for prediction.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "Similar work to RNI has also been conducted in terms of randomly adding features from a pre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "determined set of colors (Dasoulas et al., 2020) to disambiguate between nodes. This model, known", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "as CLIP, is similar in spirit to RNI, in that it introduces randomness to node representations, but", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "explicitly makes graphs distinguishable by construction. By contrast, we study random features", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "produced by RNI, which (i) are not designed a priori to distinguish nodes, (ii) do not explicitly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "introduce a fixed underlying structure, and (iii) yield potentially infinitely many representations for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "a single graph. In this more general setting, we nonetheless show that RNI adds expressive power to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "distinguish between nodes with high probability, leads to a universality result, and performs strongly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 296, + 236, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 236, + 309 + ], + "score": 1.0, + "content": "in challenging problem settings.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 325, + 432, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 434, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 434, + 340 + ], + "score": 1.0, + "content": "4 RANDOM NODE INITIALIZATION MAKES GNNS UNIVERSAL", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 107, + 351, + 504, + 362 + ], + "spans": [ + { + "bbox": [ + 107, + 351, + 504, + 362 + ], + "score": 1.0, + "content": "We present the main result of the paper, showing that random node initialization significantly in-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "creases the expressiveness and makes MPNNs universal, in a natural sense. Our work is a first", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "positive result for the universality of MPNNs. This result is not based on a new model, but rather on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "random initialization of node features, which is widely used in practice, and in this respect, it also", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 462, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 462, + 407 + ], + "score": 1.0, + "content": "serves as a theoretical justification for models that are successfully employed in practice.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "It may appear somewhat surprising, and even counter-intuitive, that randomly initializing node fea-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "tures, on its own, would deliver such a gain in expressiveness. In fact, on the surface, random", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 445 + ], + "score": 1.0, + "content": "initialization no longer preserves the invariance of MPNNs, since the result of the computation of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "an MPNN with RNI not only depends on the structure (i.e., the isomorphism type) of the input", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "graph, but also on the random initialization. The broader picture is, however, rather subtle, as we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "score": 1.0, + "content": "can view such a model as computing a random variable (or as generating an output distribution),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "score": 1.0, + "content": "and this random variable would still be invariant. This means that the outcome of the computa-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "tion of an MPNN with RNI does still not depend on the specific representation of the input graph,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "which fundamentally maintains invariance. Indeed, random features vary around a mean which, in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 521 + ], + "score": 1.0, + "content": "expectation, will inform GNN predictions, and is identical across all nodes as randomization is i.i.d.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 520, + 504, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 504, + 532 + ], + "score": 1.0, + "content": "However, the variability between different samples, and the variability of a random sample relative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "score": 1.0, + "content": "to this mean, enable graph discrimination and improve expressiveness. Hence, in expectation, all", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "samples over training and evaluation fluctuate around a unique value, preserving invariance, whereas", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 554, + 355, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 355, + 566 + ], + "score": 1.0, + "content": "single-sample variance achieves the improved expressiveness.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 163, + 583 + ], + "score": 1.0, + "content": "Formally, let", + "type": "text" + }, + { + "bbox": [ + 163, + 571, + 176, + 582 + ], + "score": 0.89, + "content": "{ \\mathcal { G } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 570, + 258, + 583 + ], + "score": 1.0, + "content": "be the class of all", + "type": "text" + }, + { + "bbox": [ + 258, + 573, + 265, + 581 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 570, + 477, + 583 + ], + "score": 1.0, + "content": "-vertex graphs, i.e., graphs that consist of at most", + "type": "text" + }, + { + "bbox": [ + 477, + 573, + 484, + 581 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "ver-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 165, + 594 + ], + "score": 1.0, + "content": "tices, and let", + "type": "text" + }, + { + "bbox": [ + 165, + 582, + 211, + 592 + ], + "score": 0.9, + "content": "f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 581, + 275, + 594 + ], + "score": 1.0, + "content": ". We say that", + "type": "text" + }, + { + "bbox": [ + 276, + 582, + 283, + 593 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 581, + 444, + 594 + ], + "score": 1.0, + "content": "is invariant if for isomorphic graphs", + "type": "text" + }, + { + "bbox": [ + 445, + 582, + 494, + 593 + ], + "score": 0.92, + "content": "G , H \\ \\in \\ G _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "it", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 149, + 605 + ], + "score": 1.0, + "content": "holds that", + "type": "text" + }, + { + "bbox": [ + 150, + 592, + 209, + 604 + ], + "score": 0.93, + "content": "f ( G ) = f ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 591, + 361, + 605 + ], + "score": 1.0, + "content": ". We say that a randomized function", + "type": "text" + }, + { + "bbox": [ + 361, + 593, + 371, + 602 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "that associates with every graph", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 602, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 140, + 614 + ], + "score": 0.9, + "content": "G \\in \\mathcal G _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 602, + 219, + 616 + ], + "score": 1.0, + "content": "a random variable", + "type": "text" + }, + { + "bbox": [ + 219, + 603, + 246, + 615 + ], + "score": 0.92, + "content": "\\mathcal { X } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 602, + 271, + 616 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 272, + 603, + 294, + 615 + ], + "score": 0.92, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 602, + 370, + 616 + ], + "score": 1.0, + "content": "-approximation of", + "type": "text" + }, + { + "bbox": [ + 370, + 604, + 378, + 615 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 602, + 418, + 616 + ], + "score": 1.0, + "content": "if for all", + "type": "text" + }, + { + "bbox": [ + 418, + 604, + 451, + 615 + ], + "score": 0.92, + "content": "G \\in \\mathcal G _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 602, + 506, + 616 + ], + "score": 1.0, + "content": "it holds that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 613, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 236, + 628 + ], + "score": 0.91, + "content": "\\operatorname* { P r } \\left( | f ( G ) - \\mathcal { X } ( G ) | \\leq \\epsilon \\right) \\geq 1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 613, + 433, + 629 + ], + "score": 1.0, + "content": ". Note that MPNNs with RNI compute functions", + "type": "text" + }, + { + "bbox": [ + 433, + 616, + 443, + 626 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 613, + 506, + 629 + ], + "score": 1.0, + "content": "of this type. If", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 627, + 431, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 116, + 638 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 628, + 224, + 641 + ], + "score": 1.0, + "content": "is computed by an MPNN", + "type": "text" + }, + { + "bbox": [ + 224, + 628, + 235, + 638 + ], + "score": 0.63, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 628, + 325, + 641 + ], + "score": 1.0, + "content": "with RNI, we say that", + "type": "text" + }, + { + "bbox": [ + 326, + 627, + 361, + 640 + ], + "score": 0.55, + "content": "\\mathcal { N } \\left( \\epsilon , \\delta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 628, + 419, + 641 + ], + "score": 1.0, + "content": "-approximates", + "type": "text" + }, + { + "bbox": [ + 420, + 628, + 426, + 640 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 628, + 431, + 641 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 504, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 291, + 657 + ], + "score": 1.0, + "content": "Theorem 4.1 (Universal approximation). Let", + "type": "text" + }, + { + "bbox": [ + 292, + 644, + 315, + 654 + ], + "score": 0.88, + "content": "n \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 642, + 349, + 657 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 349, + 643, + 397, + 655 + ], + "score": 0.91, + "content": "f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "be invariant. Then, for all", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 654, + 358, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 136, + 666 + ], + "score": 0.91, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 654, + 266, + 667 + ], + "score": 1.0, + "content": ", there is a MPNN with RNI that", + "type": "text" + }, + { + "bbox": [ + 266, + 654, + 288, + 666 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 654, + 347, + 667 + ], + "score": 1.0, + "content": "-approximates", + "type": "text" + }, + { + "bbox": [ + 347, + 655, + 354, + 666 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 654, + 358, + 667 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 107, + 675, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 687 + ], + "score": 1.0, + "content": "For ease of presentation, we state the theorem only for real-valued functions, but it can be ex-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 685, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 457, + 700 + ], + "score": 1.0, + "content": "tended to equivariant functions defined on the nodes of a graph (that is, functions", + "type": "text" + }, + { + "bbox": [ + 457, + 687, + 465, + 698 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 685, + 506, + 700 + ], + "score": 1.0, + "content": "mapping", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 103, + 695, + 507, + 714 + ], + "spans": [ + { + "bbox": [ + 103, + 695, + 136, + 714 + ], + "score": 1.0, + "content": "graphs", + "type": "text" + }, + { + "bbox": [ + 137, + 699, + 146, + 709 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 695, + 191, + 714 + ], + "score": 1.0, + "content": "to vectors", + "type": "text" + }, + { + "bbox": [ + 191, + 697, + 235, + 709 + ], + "score": 0.92, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { V ( G ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 695, + 422, + 714 + ], + "score": 1.0, + "content": "with the property that for every permutation", + "type": "text" + }, + { + "bbox": [ + 422, + 700, + 430, + 709 + ], + "score": 0.73, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 695, + 443, + 714 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 444, + 699, + 469, + 711 + ], + "score": 0.92, + "content": "V ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 695, + 507, + 714 + ], + "score": 1.0, + "content": "it holds", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 124, + 722 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 709, + 195, + 722 + ], + "score": 0.92, + "content": "f ( G ^ { \\pi } ) = f ( G ) ^ { \\pi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "and even higher-order equivariant functions. The result can also be extended", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 460, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 272, + 733 + ], + "score": 1.0, + "content": "to weighted graphs, but then the function", + "type": "text" + }, + { + "bbox": [ + 272, + 721, + 280, + 732 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 721, + 460, + 733 + ], + "score": 1.0, + "content": "that we approximate needs to be continuous.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 105, + 82, + 506, + 203 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "Similar work to RNI has also been conducted in terms of randomly adding features from a pre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "determined set of colors (Dasoulas et al., 2020) to disambiguate between nodes. This model, known", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "as CLIP, is similar in spirit to RNI, in that it introduces randomness to node representations, but", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "explicitly makes graphs distinguishable by construction. By contrast, we study random features", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "produced by RNI, which (i) are not designed a priori to distinguish nodes, (ii) do not explicitly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "introduce a fixed underlying structure, and (iii) yield potentially infinitely many representations for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "a single graph. In this more general setting, we nonetheless show that RNI adds expressive power to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "distinguish between nodes with high probability, leads to a universality result, and performs strongly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 296, + 236, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 236, + 309 + ], + "score": 1.0, + "content": "in challenging problem settings.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 208, + 506, + 309 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 325, + 432, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 434, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 434, + 340 + ], + "score": 1.0, + "content": "4 RANDOM NODE INITIALIZATION MAKES GNNS UNIVERSAL", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 107, + 351, + 504, + 362 + ], + "spans": [ + { + "bbox": [ + 107, + 351, + 504, + 362 + ], + "score": 1.0, + "content": "We present the main result of the paper, showing that random node initialization significantly in-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "creases the expressiveness and makes MPNNs universal, in a natural sense. Our work is a first", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "positive result for the universality of MPNNs. This result is not based on a new model, but rather on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "random initialization of node features, which is widely used in practice, and in this respect, it also", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 462, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 462, + 407 + ], + "score": 1.0, + "content": "serves as a theoretical justification for models that are successfully employed in practice.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 351, + 506, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "It may appear somewhat surprising, and even counter-intuitive, that randomly initializing node fea-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "tures, on its own, would deliver such a gain in expressiveness. In fact, on the surface, random", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 445 + ], + "score": 1.0, + "content": "initialization no longer preserves the invariance of MPNNs, since the result of the computation of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "an MPNN with RNI not only depends on the structure (i.e., the isomorphism type) of the input", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "graph, but also on the random initialization. The broader picture is, however, rather subtle, as we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "score": 1.0, + "content": "can view such a model as computing a random variable (or as generating an output distribution),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "score": 1.0, + "content": "and this random variable would still be invariant. This means that the outcome of the computa-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "tion of an MPNN with RNI does still not depend on the specific representation of the input graph,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "which fundamentally maintains invariance. Indeed, random features vary around a mean which, in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 521 + ], + "score": 1.0, + "content": "expectation, will inform GNN predictions, and is identical across all nodes as randomization is i.i.d.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 520, + 504, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 504, + 532 + ], + "score": 1.0, + "content": "However, the variability between different samples, and the variability of a random sample relative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "score": 1.0, + "content": "to this mean, enable graph discrimination and improve expressiveness. Hence, in expectation, all", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "samples over training and evaluation fluctuate around a unique value, preserving invariance, whereas", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 554, + 355, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 355, + 566 + ], + "score": 1.0, + "content": "single-sample variance achieves the improved expressiveness.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 411, + 506, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 163, + 583 + ], + "score": 1.0, + "content": "Formally, let", + "type": "text" + }, + { + "bbox": [ + 163, + 571, + 176, + 582 + ], + "score": 0.89, + "content": "{ \\mathcal { G } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 570, + 258, + 583 + ], + "score": 1.0, + "content": "be the class of all", + "type": "text" + }, + { + "bbox": [ + 258, + 573, + 265, + 581 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 570, + 477, + 583 + ], + "score": 1.0, + "content": "-vertex graphs, i.e., graphs that consist of at most", + "type": "text" + }, + { + "bbox": [ + 477, + 573, + 484, + 581 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "ver-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 165, + 594 + ], + "score": 1.0, + "content": "tices, and let", + "type": "text" + }, + { + "bbox": [ + 165, + 582, + 211, + 592 + ], + "score": 0.9, + "content": "f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 581, + 275, + 594 + ], + "score": 1.0, + "content": ". We say that", + "type": "text" + }, + { + "bbox": [ + 276, + 582, + 283, + 593 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 581, + 444, + 594 + ], + "score": 1.0, + "content": "is invariant if for isomorphic graphs", + "type": "text" + }, + { + "bbox": [ + 445, + 582, + 494, + 593 + ], + "score": 0.92, + "content": "G , H \\ \\in \\ G _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "it", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 149, + 605 + ], + "score": 1.0, + "content": "holds that", + "type": "text" + }, + { + "bbox": [ + 150, + 592, + 209, + 604 + ], + "score": 0.93, + "content": "f ( G ) = f ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 591, + 361, + 605 + ], + "score": 1.0, + "content": ". We say that a randomized function", + "type": "text" + }, + { + "bbox": [ + 361, + 593, + 371, + 602 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "that associates with every graph", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 602, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 140, + 614 + ], + "score": 0.9, + "content": "G \\in \\mathcal G _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 602, + 219, + 616 + ], + "score": 1.0, + "content": "a random variable", + "type": "text" + }, + { + "bbox": [ + 219, + 603, + 246, + 615 + ], + "score": 0.92, + "content": "\\mathcal { X } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 602, + 271, + 616 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 272, + 603, + 294, + 615 + ], + "score": 0.92, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 602, + 370, + 616 + ], + "score": 1.0, + "content": "-approximation of", + "type": "text" + }, + { + "bbox": [ + 370, + 604, + 378, + 615 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 602, + 418, + 616 + ], + "score": 1.0, + "content": "if for all", + "type": "text" + }, + { + "bbox": [ + 418, + 604, + 451, + 615 + ], + "score": 0.92, + "content": "G \\in \\mathcal G _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 602, + 506, + 616 + ], + "score": 1.0, + "content": "it holds that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 613, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 236, + 628 + ], + "score": 0.91, + "content": "\\operatorname* { P r } \\left( | f ( G ) - \\mathcal { X } ( G ) | \\leq \\epsilon \\right) \\geq 1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 613, + 433, + 629 + ], + "score": 1.0, + "content": ". Note that MPNNs with RNI compute functions", + "type": "text" + }, + { + "bbox": [ + 433, + 616, + 443, + 626 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 613, + 506, + 629 + ], + "score": 1.0, + "content": "of this type. If", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 627, + 431, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 116, + 638 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 628, + 224, + 641 + ], + "score": 1.0, + "content": "is computed by an MPNN", + "type": "text" + }, + { + "bbox": [ + 224, + 628, + 235, + 638 + ], + "score": 0.63, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 628, + 325, + 641 + ], + "score": 1.0, + "content": "with RNI, we say that", + "type": "text" + }, + { + "bbox": [ + 326, + 627, + 361, + 640 + ], + "score": 0.55, + "content": "\\mathcal { N } \\left( \\epsilon , \\delta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 628, + 419, + 641 + ], + "score": 1.0, + "content": "-approximates", + "type": "text" + }, + { + "bbox": [ + 420, + 628, + 426, + 640 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 628, + 431, + 641 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 570, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 504, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 291, + 657 + ], + "score": 1.0, + "content": "Theorem 4.1 (Universal approximation). Let", + "type": "text" + }, + { + "bbox": [ + 292, + 644, + 315, + 654 + ], + "score": 0.88, + "content": "n \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 642, + 349, + 657 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 349, + 643, + 397, + 655 + ], + "score": 0.91, + "content": "f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "be invariant. Then, for all", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 654, + 358, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 136, + 666 + ], + "score": 0.91, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 654, + 266, + 667 + ], + "score": 1.0, + "content": ", there is a MPNN with RNI that", + "type": "text" + }, + { + "bbox": [ + 266, + 654, + 288, + 666 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 654, + 347, + 667 + ], + "score": 1.0, + "content": "-approximates", + "type": "text" + }, + { + "bbox": [ + 347, + 655, + 354, + 666 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 654, + 358, + 667 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 106, + 642, + 506, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 675, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 687 + ], + "score": 1.0, + "content": "For ease of presentation, we state the theorem only for real-valued functions, but it can be ex-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 685, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 457, + 700 + ], + "score": 1.0, + "content": "tended to equivariant functions defined on the nodes of a graph (that is, functions", + "type": "text" + }, + { + "bbox": [ + 457, + 687, + 465, + 698 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 685, + 506, + 700 + ], + "score": 1.0, + "content": "mapping", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 103, + 695, + 507, + 714 + ], + "spans": [ + { + "bbox": [ + 103, + 695, + 136, + 714 + ], + "score": 1.0, + "content": "graphs", + "type": "text" + }, + { + "bbox": [ + 137, + 699, + 146, + 709 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 695, + 191, + 714 + ], + "score": 1.0, + "content": "to vectors", + "type": "text" + }, + { + "bbox": [ + 191, + 697, + 235, + 709 + ], + "score": 0.92, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { V ( G ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 695, + 422, + 714 + ], + "score": 1.0, + "content": "with the property that for every permutation", + "type": "text" + }, + { + "bbox": [ + 422, + 700, + 430, + 709 + ], + "score": 0.73, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 695, + 443, + 714 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 444, + 699, + 469, + 711 + ], + "score": 0.92, + "content": "V ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 695, + 507, + 714 + ], + "score": 1.0, + "content": "it holds", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 124, + 722 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 709, + 195, + 722 + ], + "score": 0.92, + "content": "f ( G ^ { \\pi } ) = f ( G ) ^ { \\pi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "and even higher-order equivariant functions. The result can also be extended", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 460, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 272, + 733 + ], + "score": 1.0, + "content": "to weighted graphs, but then the function", + "type": "text" + }, + { + "bbox": [ + 272, + 721, + 280, + 732 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 721, + 460, + 733 + ], + "score": 1.0, + "content": "that we approximate needs to be continuous.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50, + "bbox_fs": [ + 103, + 676, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "To prove Theorem 4.1, we first show that MPNNs with RNI can capture arbitrary Boolean functions,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 474, + 108 + ], + "score": 1.0, + "content": "by building on the result of Barcelo et al. (2020), which states that any logical sentence in ´", + "type": "text" + }, + { + "bbox": [ + 475, + 94, + 487, + 105 + ], + "score": 0.87, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 338, + 118 + ], + "score": 1.0, + "content": "be captured by an MPNN with global readout. The logic", + "type": "text" + }, + { + "bbox": [ + 338, + 108, + 345, + 116 + ], + "score": 0.76, + "content": "\\complement", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 105, + 504, + 118 + ], + "score": 1.0, + "content": "is the extension of first-order predicate", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 281, + 131 + ], + "score": 1.0, + "content": "logic using counting quantifiers of the form", + "type": "text" + }, + { + "bbox": [ + 282, + 117, + 302, + 127 + ], + "score": 0.89, + "content": "\\exists ^ { \\geq k } x", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 115, + 318, + 131 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 318, + 119, + 339, + 127 + ], + "score": 0.88, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 115, + 370, + 131 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 371, + 118, + 412, + 129 + ], + "score": 0.93, + "content": "\\exists ^ { \\geq k } x \\varphi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 115, + 506, + 131 + ], + "score": 1.0, + "content": "means that there are at", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 507, + 144 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 127, + 144 + ], + "score": 1.0, + "content": "least", + "type": "text" + }, + { + "bbox": [ + 128, + 131, + 134, + 140 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 126, + 173, + 144 + ], + "score": 1.0, + "content": "elements", + "type": "text" + }, + { + "bbox": [ + 174, + 132, + 181, + 140 + ], + "score": 0.68, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 126, + 223, + 144 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 224, + 134, + 231, + 142 + ], + "score": 0.72, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 126, + 253, + 144 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 253, + 130, + 265, + 140 + ], + "score": 0.89, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 126, + 394, + 144 + ], + "score": 1.0, + "content": "is the two-variable fragment of", + "type": "text" + }, + { + "bbox": [ + 394, + 132, + 402, + 140 + ], + "score": 0.83, + "content": "\\mathrm { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 126, + 507, + 144 + ], + "score": 1.0, + "content": "(see appendix for further", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "details). We establish that any graph with identifying node features, which we call individualized", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 104, + 152, + 281, + 166 + ], + "score": 1.0, + "content": "graphs, can be represented by a sentence in", + "type": "text" + }, + { + "bbox": [ + 282, + 154, + 294, + 164 + ], + "score": 0.88, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 152, + 506, + 166 + ], + "score": 1.0, + "content": ". Then, we extend this result to sets of individualized", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "graphs, and thus to Boolean functions mapping these sets to True, by showing that these functions", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 188, + 190 + ], + "score": 1.0, + "content": "are represented by a", + "type": "text" + }, + { + "bbox": [ + 189, + 176, + 201, + 187 + ], + "score": 0.87, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "sentence, namely the disjunction of all constituent graph sentences. Follow-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 504, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 504, + 200 + ], + "score": 1.0, + "content": "ing this, we provide a construction with node embeddings based on random node initialization, and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "show that, with high probability, RNI makes the input graph individualized. Thus, with high proba-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "bility, RNI makes that MPNNs learn a Boolean function over individualized graphs. Since all such", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 277, + 236 + ], + "score": 1.0, + "content": "functions can be captured by a sentence in", + "type": "text" + }, + { + "bbox": [ + 278, + 223, + 290, + 233 + ], + "score": 0.9, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 220, + 505, + 236 + ], + "score": 1.0, + "content": ", and an MPNN with a global readout can capture any", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "Boolean function by Barcelo et al. (2020), we conclude that MPNNs with RNI can capture arbitrary ´", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 243, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 506, + 259 + ], + "score": 1.0, + "content": "Boolean functions. Finally, the result is extended to real-valued functions via a natural mapping,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 255, + 193, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 193, + 268 + ], + "score": 1.0, + "content": "yielding universality.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 107, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 107, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "The concrete implications of Theorem 4.1 can be summarized as follows: First, MPNNs enhanced", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 504, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 504, + 296 + ], + "score": 1.0, + "content": "with RNI are able to distinguish individual graphs, already with an embedding dimensionality poly-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 284, + 307 + ], + "score": 1.0, + "content": "nomial in the inverse of desired confidence", + "type": "text" + }, + { + "bbox": [ + 285, + 297, + 290, + 304 + ], + "score": 0.85, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 293, + 327, + 307 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + }, + { + "bbox": [ + 327, + 296, + 369, + 307 + ], + "score": 0.93, + "content": "O ( n ^ { 2 } \\delta ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 293, + 401, + 307 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 401, + 298, + 408, + 304 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "is the number of graph", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "nodes. Second, our universality results holds also with partial RNI. More specifically, it already", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "holds with only one randomized dimension. Third, although Theorem 4.1 can potentially result in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "very large constructions, we note that it is very adaptive and tightly linked to the descriptive com-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "plexity of the function that we want to approximate. That is, for a more restricted class of functions,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "there may be more efficient constructions, and our proof does not rely on a particular construction.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "This deserves a more thorough investigation, which we leave for future work. Finally, our construc-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "tion provides a logical characterization of the power of MPNNs with RNI, and substantiates the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "score": 1.0, + "content": "means through which randomization yields expressiveness improvements. This construction there-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "fore also serves as a basis for a more logically grounded theoretical study of randomized MPNN", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 405, + 361, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 361, + 416 + ], + "score": 1.0, + "content": "models, based on particular architectural or parametric choices.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Similarly to other universality results, Theorem 4.1 can potentially result in very large constructions.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "This is a simple consequence of the general nature of such universality results: Theorem 4.1 applies", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "to families of functions, describing problems of arbitrary computational complexity, including prob-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "lems that are computationally hard (even to approximate). Thus, a practically more relevant aspect is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "to empirically verify the formal statement, and test the capacity of MPNNs with RNI, in comparison", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "to higher-order GNNs. Higher-order GNNs typically suffer from prohibitive space requirements,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "which is not the case for MPNNs with RNI, and this already makes them more viable in practice.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "As we discuss later in detail, our experiments demonstrate that MPNNs with RNI indeed combine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 509, + 276, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 276, + 522 + ], + "score": 1.0, + "content": "expressiveness with efficiency in practice.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 542, + 456, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 458, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 458, + 557 + ], + "score": 1.0, + "content": "5 DATASETS FOR EVALUATING THE EXPRESSIVE POWER OF GNNS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "GNN models are evaluated on prominent real-world datasets, such as IMDB, TU, and Proteins", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "(Kersting et al., 2016). These datasets are not tailored for evaluating the expressive power of GNNs,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "as they do not contain instances or edge cases requiring expressiveness beyond 1-WL. In fact, higher-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "order models only marginally outperform MPNNs on these datasets (Maron et al., 2019a; Dwivedi", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 452, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 452, + 630 + ], + "score": 1.0, + "content": "et al., 2020), which further highlights their unsuitability for expressiveness evaluation.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "We develop the datasets EXP and CEXP. EXP is designed to explicitly evaluate the expressiveness", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 332, + 657 + ], + "score": 1.0, + "content": "of GNN models, and consists of a set of graph instances", + "type": "text" + }, + { + "bbox": [ + 333, + 644, + 429, + 656 + ], + "score": 0.9, + "content": "\\left\\{ { G _ { 1 } \\dots { \\bar { G } } _ { n } , \\dot { H } _ { 1 } \\dots { \\cal H } _ { n } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ", such that each in-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "stance is a graph encoding of a propositional formula. The classification task is to determine whether", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 664, + 504, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 276, + 679 + ], + "score": 1.0, + "content": "the formula is satisfiable (SAT). Each pair", + "type": "text" + }, + { + "bbox": [ + 276, + 666, + 312, + 677 + ], + "score": 0.92, + "content": "( G _ { i } , H _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 664, + 460, + 679 + ], + "score": 1.0, + "content": "respects the following properties: (i)", + "type": "text" + }, + { + "bbox": [ + 461, + 666, + 473, + 677 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 664, + 491, + 679 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 491, + 666, + 504, + 677 + ], + "score": 0.9, + "content": "H _ { i }", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 205, + 689 + ], + "score": 1.0, + "content": "are non-isomorphic, (ii)", + "type": "text" + }, + { + "bbox": [ + 206, + 677, + 218, + 688 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 677, + 237, + 689 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 677, + 250, + 688 + ], + "score": 0.88, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 677, + 406, + 689 + ], + "score": 1.0, + "content": "have different SAT outcomes, that is,", + "type": "text" + }, + { + "bbox": [ + 406, + 677, + 418, + 688 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "encodes a satisfiable", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 168, + 702 + ], + "score": 1.0, + "content": "formula, while", + "type": "text" + }, + { + "bbox": [ + 168, + 688, + 181, + 699 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 686, + 335, + 702 + ], + "score": 1.0, + "content": "encodes an unsatisfiable formula, (iii)", + "type": "text" + }, + { + "bbox": [ + 335, + 688, + 347, + 699 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 686, + 365, + 702 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 688, + 378, + 699 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "are 1-WL indistinguishable, so", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 419, + 712 + ], + "score": 1.0, + "content": "are guaranteed to be classified in the same way by standard MPNNs, and (iv)", + "type": "text" + }, + { + "bbox": [ + 419, + 699, + 431, + 710 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 699, + 450, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 699, + 462, + 710 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "are 2-WL", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "distinguishable, so can be classified differently by higher-order GNNs. Thanks to these properties,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 486, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 486, + 732 + ], + "score": 1.0, + "content": "we can explicitly compare the performance of MPNNs with RNI to these higher-order models.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "To prove Theorem 4.1, we first show that MPNNs with RNI can capture arbitrary Boolean functions,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 474, + 108 + ], + "score": 1.0, + "content": "by building on the result of Barcelo et al. (2020), which states that any logical sentence in ´", + "type": "text" + }, + { + "bbox": [ + 475, + 94, + 487, + 105 + ], + "score": 0.87, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 338, + 118 + ], + "score": 1.0, + "content": "be captured by an MPNN with global readout. The logic", + "type": "text" + }, + { + "bbox": [ + 338, + 108, + 345, + 116 + ], + "score": 0.76, + "content": "\\complement", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 105, + 504, + 118 + ], + "score": 1.0, + "content": "is the extension of first-order predicate", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 281, + 131 + ], + "score": 1.0, + "content": "logic using counting quantifiers of the form", + "type": "text" + }, + { + "bbox": [ + 282, + 117, + 302, + 127 + ], + "score": 0.89, + "content": "\\exists ^ { \\geq k } x", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 115, + 318, + 131 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 318, + 119, + 339, + 127 + ], + "score": 0.88, + "content": "k \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 115, + 370, + 131 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 371, + 118, + 412, + 129 + ], + "score": 0.93, + "content": "\\exists ^ { \\geq k } x \\varphi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 115, + 506, + 131 + ], + "score": 1.0, + "content": "means that there are at", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 507, + 144 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 127, + 144 + ], + "score": 1.0, + "content": "least", + "type": "text" + }, + { + "bbox": [ + 128, + 131, + 134, + 140 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 126, + 173, + 144 + ], + "score": 1.0, + "content": "elements", + "type": "text" + }, + { + "bbox": [ + 174, + 132, + 181, + 140 + ], + "score": 0.68, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 126, + 223, + 144 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 224, + 134, + 231, + 142 + ], + "score": 0.72, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 126, + 253, + 144 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 253, + 130, + 265, + 140 + ], + "score": 0.89, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 126, + 394, + 144 + ], + "score": 1.0, + "content": "is the two-variable fragment of", + "type": "text" + }, + { + "bbox": [ + 394, + 132, + 402, + 140 + ], + "score": 0.83, + "content": "\\mathrm { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 126, + 507, + 144 + ], + "score": 1.0, + "content": "(see appendix for further", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "details). We establish that any graph with identifying node features, which we call individualized", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 104, + 152, + 281, + 166 + ], + "score": 1.0, + "content": "graphs, can be represented by a sentence in", + "type": "text" + }, + { + "bbox": [ + 282, + 154, + 294, + 164 + ], + "score": 0.88, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 152, + 506, + 166 + ], + "score": 1.0, + "content": ". Then, we extend this result to sets of individualized", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "graphs, and thus to Boolean functions mapping these sets to True, by showing that these functions", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 188, + 190 + ], + "score": 1.0, + "content": "are represented by a", + "type": "text" + }, + { + "bbox": [ + 189, + 176, + 201, + 187 + ], + "score": 0.87, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "sentence, namely the disjunction of all constituent graph sentences. Follow-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 504, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 504, + 200 + ], + "score": 1.0, + "content": "ing this, we provide a construction with node embeddings based on random node initialization, and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "show that, with high probability, RNI makes the input graph individualized. Thus, with high proba-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "bility, RNI makes that MPNNs learn a Boolean function over individualized graphs. Since all such", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 277, + 236 + ], + "score": 1.0, + "content": "functions can be captured by a sentence in", + "type": "text" + }, + { + "bbox": [ + 278, + 223, + 290, + 233 + ], + "score": 0.9, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 220, + 505, + 236 + ], + "score": 1.0, + "content": ", and an MPNN with a global readout can capture any", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "Boolean function by Barcelo et al. (2020), we conclude that MPNNs with RNI can capture arbitrary ´", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 243, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 506, + 259 + ], + "score": 1.0, + "content": "Boolean functions. Finally, the result is extended to real-valued functions via a natural mapping,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 255, + 193, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 193, + 268 + ], + "score": 1.0, + "content": "yielding universality.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 82, + 507, + 268 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 107, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 107, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "The concrete implications of Theorem 4.1 can be summarized as follows: First, MPNNs enhanced", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 504, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 504, + 296 + ], + "score": 1.0, + "content": "with RNI are able to distinguish individual graphs, already with an embedding dimensionality poly-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 284, + 307 + ], + "score": 1.0, + "content": "nomial in the inverse of desired confidence", + "type": "text" + }, + { + "bbox": [ + 285, + 297, + 290, + 304 + ], + "score": 0.85, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 293, + 327, + 307 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + }, + { + "bbox": [ + 327, + 296, + 369, + 307 + ], + "score": 0.93, + "content": "O ( n ^ { 2 } \\delta ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 293, + 401, + 307 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 401, + 298, + 408, + 304 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "is the number of graph", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "nodes. Second, our universality results holds also with partial RNI. More specifically, it already", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "holds with only one randomized dimension. Third, although Theorem 4.1 can potentially result in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "very large constructions, we note that it is very adaptive and tightly linked to the descriptive com-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "plexity of the function that we want to approximate. That is, for a more restricted class of functions,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "there may be more efficient constructions, and our proof does not rely on a particular construction.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "This deserves a more thorough investigation, which we leave for future work. Finally, our construc-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "tion provides a logical characterization of the power of MPNNs with RNI, and substantiates the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "score": 1.0, + "content": "means through which randomization yields expressiveness improvements. This construction there-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "fore also serves as a basis for a more logically grounded theoretical study of randomized MPNN", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 405, + 361, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 361, + 416 + ], + "score": 1.0, + "content": "models, based on particular architectural or parametric choices.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 272, + 505, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Similarly to other universality results, Theorem 4.1 can potentially result in very large constructions.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "This is a simple consequence of the general nature of such universality results: Theorem 4.1 applies", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "to families of functions, describing problems of arbitrary computational complexity, including prob-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "lems that are computationally hard (even to approximate). Thus, a practically more relevant aspect is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "to empirically verify the formal statement, and test the capacity of MPNNs with RNI, in comparison", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "to higher-order GNNs. Higher-order GNNs typically suffer from prohibitive space requirements,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "which is not the case for MPNNs with RNI, and this already makes them more viable in practice.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "As we discuss later in detail, our experiments demonstrate that MPNNs with RNI indeed combine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 509, + 276, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 276, + 522 + ], + "score": 1.0, + "content": "expressiveness with efficiency in practice.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 421, + 505, + 522 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 542, + 456, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 458, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 458, + 557 + ], + "score": 1.0, + "content": "5 DATASETS FOR EVALUATING THE EXPRESSIVE POWER OF GNNS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "GNN models are evaluated on prominent real-world datasets, such as IMDB, TU, and Proteins", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "(Kersting et al., 2016). These datasets are not tailored for evaluating the expressive power of GNNs,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "as they do not contain instances or edge cases requiring expressiveness beyond 1-WL. In fact, higher-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "order models only marginally outperform MPNNs on these datasets (Maron et al., 2019a; Dwivedi", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 452, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 452, + 630 + ], + "score": 1.0, + "content": "et al., 2020), which further highlights their unsuitability for expressiveness evaluation.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 571, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "We develop the datasets EXP and CEXP. EXP is designed to explicitly evaluate the expressiveness", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 332, + 657 + ], + "score": 1.0, + "content": "of GNN models, and consists of a set of graph instances", + "type": "text" + }, + { + "bbox": [ + 333, + 644, + 429, + 656 + ], + "score": 0.9, + "content": "\\left\\{ { G _ { 1 } \\dots { \\bar { G } } _ { n } , \\dot { H } _ { 1 } \\dots { \\cal H } _ { n } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ", such that each in-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "stance is a graph encoding of a propositional formula. The classification task is to determine whether", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 664, + 504, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 276, + 679 + ], + "score": 1.0, + "content": "the formula is satisfiable (SAT). Each pair", + "type": "text" + }, + { + "bbox": [ + 276, + 666, + 312, + 677 + ], + "score": 0.92, + "content": "( G _ { i } , H _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 664, + 460, + 679 + ], + "score": 1.0, + "content": "respects the following properties: (i)", + "type": "text" + }, + { + "bbox": [ + 461, + 666, + 473, + 677 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 664, + 491, + 679 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 491, + 666, + 504, + 677 + ], + "score": 0.9, + "content": "H _ { i }", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 205, + 689 + ], + "score": 1.0, + "content": "are non-isomorphic, (ii)", + "type": "text" + }, + { + "bbox": [ + 206, + 677, + 218, + 688 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 677, + 237, + 689 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 677, + 250, + 688 + ], + "score": 0.88, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 677, + 406, + 689 + ], + "score": 1.0, + "content": "have different SAT outcomes, that is,", + "type": "text" + }, + { + "bbox": [ + 406, + 677, + 418, + 688 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "encodes a satisfiable", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 168, + 702 + ], + "score": 1.0, + "content": "formula, while", + "type": "text" + }, + { + "bbox": [ + 168, + 688, + 181, + 699 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 686, + 335, + 702 + ], + "score": 1.0, + "content": "encodes an unsatisfiable formula, (iii)", + "type": "text" + }, + { + "bbox": [ + 335, + 688, + 347, + 699 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 686, + 365, + 702 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 688, + 378, + 699 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "are 1-WL indistinguishable, so", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 419, + 712 + ], + "score": 1.0, + "content": "are guaranteed to be classified in the same way by standard MPNNs, and (iv)", + "type": "text" + }, + { + "bbox": [ + 419, + 699, + 431, + 710 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 699, + 450, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 699, + 462, + 710 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "are 2-WL", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "distinguishable, so can be classified differently by higher-order GNNs. Thanks to these properties,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 486, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 486, + 732 + ], + "score": 1.0, + "content": "we can explicitly compare the performance of MPNNs with RNI to these higher-order models.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 632, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Ensuring these properties in EXP is highly non-trivial, and the construction of this dataset is cum-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 237, + 107 + ], + "score": 1.0, + "content": "bersome. Fundamentally, every", + "type": "text" + }, + { + "bbox": [ + 237, + 93, + 273, + 105 + ], + "score": 0.93, + "content": "( G _ { i } , H _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "is carefully constructed on top of a basic building block,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 280, + 117 + ], + "score": 1.0, + "content": "the core pair, such that the 2 cores underlie", + "type": "text" + }, + { + "bbox": [ + 281, + 105, + 293, + 115 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 104, + 311, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 105, + 323, + 115 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ", respectively. In this core pair, both cores are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "based on propositional clauses, such that one core is satisfiable and the other is not, and that these", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 299, + 140 + ], + "score": 1.0, + "content": "cores exclusively determine the satisfiability of", + "type": "text" + }, + { + "bbox": [ + 299, + 127, + 312, + 137 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 126, + 341, + 140 + ], + "score": 1.0, + "content": "(resp.,", + "type": "text" + }, + { + "bbox": [ + 342, + 127, + 355, + 137 + ], + "score": 0.82, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 126, + 505, + 140 + ], + "score": 1.0, + "content": ") and have graph encodings enabling", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "all aforementioned properties. Core pairs, and their resulting graph instances in EXP are planar and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "are also carefully constrained to ensure they are 2-WL distinguishable. Hence, core pairs are key", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 473, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 473, + 172 + ], + "score": 1.0, + "content": "substructures within EXP, and distinguishing these cores is essential for good performance.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "Building on EXP, CEXP includes instances with varying expressiveness requirements. Specifically,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 269, + 200 + ], + "score": 1.0, + "content": "CEXP is a standard EXP dataset where", + "type": "text" + }, + { + "bbox": [ + 270, + 187, + 290, + 198 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "of all satisfiable graph pairs are modified, such that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "they become 1-WL distinguishable from their unsatisfiable counterparts, only differing from these", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 365, + 221 + ], + "score": 1.0, + "content": "by a small number of added edges. Hence, CEXP consists of", + "type": "text" + }, + { + "bbox": [ + 365, + 209, + 385, + 220 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 210, + 505, + 221 + ], + "score": 1.0, + "content": "“corrupted” data, which can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "be distinguished and learned by a standard MPNN model (1-WL), which we label CORRUPT, and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 126, + 242 + ], + "score": 0.84, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "unmodified data, generated analogously to EXP, and requiring expressive power beyond 1-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 214, + 255 + ], + "score": 1.0, + "content": "WL, which we refer to as", + "type": "text" + }, + { + "bbox": [ + 214, + 242, + 234, + 253 + ], + "score": 0.45, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 242, + 505, + 255 + ], + "score": 1.0, + "content": ". Thus, CEXP contains the same core structures as EXP, but these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 254, + 266 + ], + "score": 1.0, + "content": "now lead to different SAT values in", + "type": "text" + }, + { + "bbox": [ + 255, + 253, + 275, + 264 + ], + "score": 0.43, + "content": "\\overline { { \\mathrm { E x P } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "and CORRUPT, and this makes the overall learning task", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 233, + 279 + ], + "score": 1.0, + "content": "more challenging than learning", + "type": "text" + }, + { + "bbox": [ + 234, + 265, + 253, + 277 + ], + "score": 0.39, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "or CORRUPT in isolation. Complete details of the overall data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 277, + 372, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 372, + 290 + ], + "score": 1.0, + "content": "generation process for both datasets can be found in the appendix.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 304, + 277, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 279, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 279, + 319 + ], + "score": 1.0, + "content": "6 EXPERIMENTAL EVALUATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 327, + 504, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "In this section, we first evaluate the practical effect of RNI on MPNN expressiveness based on EXP,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "and compare MPNN-RNI against established higher-order GNNs. We then extend our empirical", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 427, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 427, + 364 + ], + "score": 1.0, + "content": "analysis to CEXP. Both experiments are conducted using the following models:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "- 1-WL GCN (1-GCN): A GCN with 8 distinct message passing iterations, ELU non-linearities", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 115, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "(Clevert et al., 2016), 64-dimensional embeddings, and deterministic learnable initial node em-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 114, + 391, + 480, + 405 + ], + "spans": [ + { + "bbox": [ + 114, + 391, + 385, + 405 + ], + "score": 1.0, + "content": "beddings indicating node type. This model is guaranteed to achieve", + "type": "text" + }, + { + "bbox": [ + 386, + 392, + 406, + 402 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 391, + 480, + 405 + ], + "score": 1.0, + "content": "accuracy on EXP.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "- GCN - Random node initialization (GCN-RNI): An analogous model to 1-GCN with an identi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 114, + 415, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 114, + 415, + 505, + 430 + ], + "score": 1.0, + "content": "cal architecture, enhanced with RNI. We evaluate this model with four initialization distributions,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 113, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 113, + 427, + 280, + 441 + ], + "score": 1.0, + "content": "namely the standard normal distribution", + "type": "text" + }, + { + "bbox": [ + 280, + 428, + 313, + 439 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 427, + 455, + 441 + ], + "score": 1.0, + "content": "(N), the uniform distribution over", + "type": "text" + }, + { + "bbox": [ + 456, + 428, + 484, + 439 + ], + "score": 0.88, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "(U),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 114, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "Xavier normal (XN), and the Xavier uniform distribution (XU) (Glorot & Bengio, 2010). We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 450, + 404, + 463 + ], + "spans": [ + { + "bbox": [ + 114, + 450, + 232, + 463 + ], + "score": 1.0, + "content": "denote the respective models", + "type": "text" + }, + { + "bbox": [ + 232, + 450, + 290, + 461 + ], + "score": 0.33, + "content": "\\mathbf { G C N - R N I } ( D )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 450, + 321, + 463 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 321, + 450, + 400, + 462 + ], + "score": 0.9, + "content": "D \\in \\{ \\mathrm { N , U , X N , X U } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 450, + 404, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 296, + 475 + ], + "score": 1.0, + "content": "- GCN - Partial Random node initialization", + "type": "text" + }, + { + "bbox": [ + 297, + 464, + 363, + 475 + ], + "score": 0.32, + "content": "( \\mathbf { G C N - } x \\% \\mathbf { R N I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 462, + 506, + 475 + ], + "score": 1.0, + "content": ": A GCN-RNI model, where only a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 113, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 113, + 474, + 161, + 487 + ], + "score": 1.0, + "content": "percentage", + "type": "text" + }, + { + "bbox": [ + 161, + 477, + 168, + 485 + ], + "score": 0.66, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 474, + 445, + 487 + ], + "score": 1.0, + "content": "of initial node embedding dimensions are randomized. That is, for", + "type": "text" + }, + { + "bbox": [ + 446, + 475, + 452, + 485 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 114, + 486, + 217, + 499 + ], + "score": 1.0, + "content": "node embeddings, GCN-", + "type": "text" + }, + { + "bbox": [ + 217, + 487, + 232, + 497 + ], + "score": 0.48, + "content": "x \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 486, + 300, + 499 + ], + "score": 1.0, + "content": "RNI randomizes", + "type": "text" + }, + { + "bbox": [ + 301, + 486, + 324, + 500 + ], + "score": 0.92, + "content": "\\lfloor \\frac { x d } { 1 0 0 } \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "dimensions, and sets the remaining dimen-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 114, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 114, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "sions deterministically from input features, namely, a one-hot representation of the two possible", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 114, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "node types (literal and disjunction) in the input graph representation (see appendix for more de-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 114, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 114, + 519, + 169, + 531 + ], + "score": 1.0, + "content": "tails). We set", + "type": "text" + }, + { + "bbox": [ + 170, + 522, + 177, + 529 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 519, + 290, + 531 + ], + "score": 1.0, + "content": "to the extreme values 0 and", + "type": "text" + }, + { + "bbox": [ + 291, + 520, + 315, + 530 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 519, + 318, + 531 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 318, + 520, + 338, + 530 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 519, + 460, + 531 + ], + "score": 1.0, + "content": ", as well as near-edge cases of", + "type": "text" + }, + { + "bbox": [ + 460, + 520, + 487, + 530 + ], + "score": 0.86, + "content": "8 7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 115, + 529, + 198, + 543 + ], + "spans": [ + { + "bbox": [ + 115, + 531, + 142, + 542 + ], + "score": 0.85, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 529, + 198, + 543 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "- Provably Powerful Graph Network (PPGN) (Maron et al., 2019a): A higher-order GNN with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 113, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "2-WL expressive power, and which requires quadratic memory relative to the number of graph", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 114, + 567, + 416, + 579 + ], + "spans": [ + { + "bbox": [ + 114, + 567, + 416, + 579 + ], + "score": 1.0, + "content": "nodes. We set up PPGN with eight 400-dimensional computational blocks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 581, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 592 + ], + "score": 1.0, + "content": "- 1-2-3-GCN-L: A higher-order GNN (Morris et al., 2019) emulating 2-WL on 3-tuples of nodes.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 115, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 115, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "1-2-3-GCN-L operates at increasingly coarse node granularities, starting with single nodes and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 114, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 114, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "rising to 3-tuples. The model first computes all possible 3-tuples of nodes, then represents them", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 114, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 114, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "as standard graph nodes. These nodes are connected to one another following the 2-WL neighbor-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 113, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 113, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "hood definition, i.e., tuples that exchange messages in 2-WL are connected by an edge in 3-GCN.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 113, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 113, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "3-GCN-L implements a connected relaxation of 2-WL, in that only 3-tuples forming a connected", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 114, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 114, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "graph are used, which comes at the cost of some theoretical guarantees. Nonetheless, the compu-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 114, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 114, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "tation and representation of all tuples still imposes a severe overhead relative to MPNNs. We set", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 114, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 114, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "up 1-2-3-GCN-L with 64-dimensional embeddings, 3 message passing iterations at level 1, 2 at", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 114, + 679, + 209, + 691 + ], + "spans": [ + { + "bbox": [ + 114, + 679, + 209, + 691 + ], + "score": 1.0, + "content": "level 2 and 8 at level 3.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 108, + 694, + 505, + 727 + ], + "lines": [ + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "score": 1.0, + "content": "- 3-GCN: A modification to 1-2-3-GCN-L, such that (i) only the 3rd level is used, and (ii) the full", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 113, + 703, + 506, + 717 + ], + "spans": [ + { + "bbox": [ + 113, + 703, + 506, + 717 + ], + "score": 1.0, + "content": "2-WL procedure is implemented, i.e., all 3-tuples are computed, as in standard 2-WL, rather than", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 114, + 715, + 216, + 728 + ], + "spans": [ + { + "bbox": [ + 114, + 715, + 216, + 728 + ], + "score": 1.0, + "content": "only the connected ones.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Ensuring these properties in EXP is highly non-trivial, and the construction of this dataset is cum-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 237, + 107 + ], + "score": 1.0, + "content": "bersome. Fundamentally, every", + "type": "text" + }, + { + "bbox": [ + 237, + 93, + 273, + 105 + ], + "score": 0.93, + "content": "( G _ { i } , H _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "is carefully constructed on top of a basic building block,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 280, + 117 + ], + "score": 1.0, + "content": "the core pair, such that the 2 cores underlie", + "type": "text" + }, + { + "bbox": [ + 281, + 105, + 293, + 115 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 104, + 311, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 105, + 323, + 115 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ", respectively. In this core pair, both cores are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "based on propositional clauses, such that one core is satisfiable and the other is not, and that these", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 299, + 140 + ], + "score": 1.0, + "content": "cores exclusively determine the satisfiability of", + "type": "text" + }, + { + "bbox": [ + 299, + 127, + 312, + 137 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 126, + 341, + 140 + ], + "score": 1.0, + "content": "(resp.,", + "type": "text" + }, + { + "bbox": [ + 342, + 127, + 355, + 137 + ], + "score": 0.82, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 126, + 505, + 140 + ], + "score": 1.0, + "content": ") and have graph encodings enabling", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "all aforementioned properties. Core pairs, and their resulting graph instances in EXP are planar and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "are also carefully constrained to ensure they are 2-WL distinguishable. Hence, core pairs are key", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 473, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 473, + 172 + ], + "score": 1.0, + "content": "substructures within EXP, and distinguishing these cores is essential for good performance.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 83, + 506, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "Building on EXP, CEXP includes instances with varying expressiveness requirements. Specifically,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 269, + 200 + ], + "score": 1.0, + "content": "CEXP is a standard EXP dataset where", + "type": "text" + }, + { + "bbox": [ + 270, + 187, + 290, + 198 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "of all satisfiable graph pairs are modified, such that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "they become 1-WL distinguishable from their unsatisfiable counterparts, only differing from these", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 365, + 221 + ], + "score": 1.0, + "content": "by a small number of added edges. Hence, CEXP consists of", + "type": "text" + }, + { + "bbox": [ + 365, + 209, + 385, + 220 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 210, + 505, + 221 + ], + "score": 1.0, + "content": "“corrupted” data, which can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "be distinguished and learned by a standard MPNN model (1-WL), which we label CORRUPT, and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 126, + 242 + ], + "score": 0.84, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "unmodified data, generated analogously to EXP, and requiring expressive power beyond 1-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 214, + 255 + ], + "score": 1.0, + "content": "WL, which we refer to as", + "type": "text" + }, + { + "bbox": [ + 214, + 242, + 234, + 253 + ], + "score": 0.45, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 242, + 505, + 255 + ], + "score": 1.0, + "content": ". Thus, CEXP contains the same core structures as EXP, but these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 254, + 266 + ], + "score": 1.0, + "content": "now lead to different SAT values in", + "type": "text" + }, + { + "bbox": [ + 255, + 253, + 275, + 264 + ], + "score": 0.43, + "content": "\\overline { { \\mathrm { E x P } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "and CORRUPT, and this makes the overall learning task", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 233, + 279 + ], + "score": 1.0, + "content": "more challenging than learning", + "type": "text" + }, + { + "bbox": [ + 234, + 265, + 253, + 277 + ], + "score": 0.39, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "or CORRUPT in isolation. Complete details of the overall data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 277, + 372, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 372, + 290 + ], + "score": 1.0, + "content": "generation process for both datasets can be found in the appendix.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 176, + 506, + 290 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 304, + 277, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 279, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 279, + 319 + ], + "score": 1.0, + "content": "6 EXPERIMENTAL EVALUATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 327, + 504, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "In this section, we first evaluate the practical effect of RNI on MPNN expressiveness based on EXP,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "and compare MPNN-RNI against established higher-order GNNs. We then extend our empirical", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 427, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 427, + 364 + ], + "score": 1.0, + "content": "analysis to CEXP. Both experiments are conducted using the following models:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 327, + 506, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "- 1-WL GCN (1-GCN): A GCN with 8 distinct message passing iterations, ELU non-linearities", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 115, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "(Clevert et al., 2016), 64-dimensional embeddings, and deterministic learnable initial node em-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 114, + 391, + 480, + 405 + ], + "spans": [ + { + "bbox": [ + 114, + 391, + 385, + 405 + ], + "score": 1.0, + "content": "beddings indicating node type. This model is guaranteed to achieve", + "type": "text" + }, + { + "bbox": [ + 386, + 392, + 406, + 402 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 391, + 480, + 405 + ], + "score": 1.0, + "content": "accuracy on EXP.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 369, + 505, + 405 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "- GCN - Random node initialization (GCN-RNI): An analogous model to 1-GCN with an identi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 114, + 415, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 114, + 415, + 505, + 430 + ], + "score": 1.0, + "content": "cal architecture, enhanced with RNI. We evaluate this model with four initialization distributions,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 113, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 113, + 427, + 280, + 441 + ], + "score": 1.0, + "content": "namely the standard normal distribution", + "type": "text" + }, + { + "bbox": [ + 280, + 428, + 313, + 439 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 427, + 455, + 441 + ], + "score": 1.0, + "content": "(N), the uniform distribution over", + "type": "text" + }, + { + "bbox": [ + 456, + 428, + 484, + 439 + ], + "score": 0.88, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "(U),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 114, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "Xavier normal (XN), and the Xavier uniform distribution (XU) (Glorot & Bengio, 2010). We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 450, + 404, + 463 + ], + "spans": [ + { + "bbox": [ + 114, + 450, + 232, + 463 + ], + "score": 1.0, + "content": "denote the respective models", + "type": "text" + }, + { + "bbox": [ + 232, + 450, + 290, + 461 + ], + "score": 0.33, + "content": "\\mathbf { G C N - R N I } ( D )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 450, + 321, + 463 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 321, + 450, + 400, + 462 + ], + "score": 0.9, + "content": "D \\in \\{ \\mathrm { N , U , X N , X U } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 450, + 404, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 405, + 505, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 296, + 475 + ], + "score": 1.0, + "content": "- GCN - Partial Random node initialization", + "type": "text" + }, + { + "bbox": [ + 297, + 464, + 363, + 475 + ], + "score": 0.32, + "content": "( \\mathbf { G C N - } x \\% \\mathbf { R N I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 462, + 506, + 475 + ], + "score": 1.0, + "content": ": A GCN-RNI model, where only a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 113, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 113, + 474, + 161, + 487 + ], + "score": 1.0, + "content": "percentage", + "type": "text" + }, + { + "bbox": [ + 161, + 477, + 168, + 485 + ], + "score": 0.66, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 474, + 445, + 487 + ], + "score": 1.0, + "content": "of initial node embedding dimensions are randomized. That is, for", + "type": "text" + }, + { + "bbox": [ + 446, + 475, + 452, + 485 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 114, + 486, + 217, + 499 + ], + "score": 1.0, + "content": "node embeddings, GCN-", + "type": "text" + }, + { + "bbox": [ + 217, + 487, + 232, + 497 + ], + "score": 0.48, + "content": "x \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 486, + 300, + 499 + ], + "score": 1.0, + "content": "RNI randomizes", + "type": "text" + }, + { + "bbox": [ + 301, + 486, + 324, + 500 + ], + "score": 0.92, + "content": "\\lfloor \\frac { x d } { 1 0 0 } \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "dimensions, and sets the remaining dimen-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 114, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 114, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "sions deterministically from input features, namely, a one-hot representation of the two possible", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 114, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "node types (literal and disjunction) in the input graph representation (see appendix for more de-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 114, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 114, + 519, + 169, + 531 + ], + "score": 1.0, + "content": "tails). We set", + "type": "text" + }, + { + "bbox": [ + 170, + 522, + 177, + 529 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 519, + 290, + 531 + ], + "score": 1.0, + "content": "to the extreme values 0 and", + "type": "text" + }, + { + "bbox": [ + 291, + 520, + 315, + 530 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 519, + 318, + 531 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 318, + 520, + 338, + 530 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 519, + 460, + 531 + ], + "score": 1.0, + "content": ", as well as near-edge cases of", + "type": "text" + }, + { + "bbox": [ + 460, + 520, + 487, + 530 + ], + "score": 0.86, + "content": "8 7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 115, + 529, + 198, + 543 + ], + "spans": [ + { + "bbox": [ + 115, + 531, + 142, + 542 + ], + "score": 0.85, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 529, + 198, + 543 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 462, + 506, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "- Provably Powerful Graph Network (PPGN) (Maron et al., 2019a): A higher-order GNN with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 113, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "2-WL expressive power, and which requires quadratic memory relative to the number of graph", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 114, + 567, + 416, + 579 + ], + "spans": [ + { + "bbox": [ + 114, + 567, + 416, + 579 + ], + "score": 1.0, + "content": "nodes. We set up PPGN with eight 400-dimensional computational blocks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 545, + 505, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 581, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 592 + ], + "score": 1.0, + "content": "- 1-2-3-GCN-L: A higher-order GNN (Morris et al., 2019) emulating 2-WL on 3-tuples of nodes.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 115, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 115, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "1-2-3-GCN-L operates at increasingly coarse node granularities, starting with single nodes and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 114, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 114, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "rising to 3-tuples. The model first computes all possible 3-tuples of nodes, then represents them", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 114, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 114, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "as standard graph nodes. These nodes are connected to one another following the 2-WL neighbor-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 113, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 113, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "hood definition, i.e., tuples that exchange messages in 2-WL are connected by an edge in 3-GCN.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 113, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 113, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "3-GCN-L implements a connected relaxation of 2-WL, in that only 3-tuples forming a connected", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 114, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 114, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "graph are used, which comes at the cost of some theoretical guarantees. Nonetheless, the compu-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 114, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 114, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "tation and representation of all tuples still imposes a severe overhead relative to MPNNs. We set", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 114, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 114, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "up 1-2-3-GCN-L with 64-dimensional embeddings, 3 message passing iterations at level 1, 2 at", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 114, + 679, + 209, + 691 + ], + "spans": [ + { + "bbox": [ + 114, + 679, + 209, + 691 + ], + "score": 1.0, + "content": "level 2 and 8 at level 3.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44.5, + "bbox_fs": [ + 106, + 581, + 506, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 694, + 505, + 727 + ], + "lines": [ + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "score": 1.0, + "content": "- 3-GCN: A modification to 1-2-3-GCN-L, such that (i) only the 3rd level is used, and (ii) the full", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 113, + 703, + 506, + 717 + ], + "spans": [ + { + "bbox": [ + 113, + 703, + 506, + 717 + ], + "score": 1.0, + "content": "2-WL procedure is implemented, i.e., all 3-tuples are computed, as in standard 2-WL, rather than", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 114, + 715, + 216, + 728 + ], + "spans": [ + { + "bbox": [ + 114, + 715, + 216, + 728 + ], + "score": 1.0, + "content": "only the connected ones.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 692, + 506, + 728 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 426, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 428, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 428, + 95 + ], + "score": 1.0, + "content": "6.1 EXPERIMENT 1: HOW DOES RNI IMPROVE MPNN EXPRESSIVENESS?", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "In this experiment, we evaluate GCNs using different RNI settings on EXP, and compare with stan-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "dard GNNs and higher-order models. Specifically, we generate an EXP dataset consisting of 600", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "graph pairs, and discuss this generation in more detail in the appendix. Then, we evaluate all mod-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "els on EXP using 10-fold cross-validation. We train 3-GCN for 100 epochs per fold, and all other", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 495, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 495, + 161 + ], + "score": 1.0, + "content": "systems for 500 epochs. Mean test accuracy across all validation folds is measured and reported.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 166, + 336, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 166, + 336, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 336, + 177 + ], + "score": 1.0, + "content": "Full test accuracy results for all models are reported", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 176, + 336, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 336, + 189 + ], + "score": 1.0, + "content": "in Table 1, and model convergence for 3-GCN and all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 336, + 200 + ], + "score": 1.0, + "content": "GCN-RNI models are shown in Figure 2. In line with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 336, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 336, + 210 + ], + "score": 1.0, + "content": "Theorem 4.1, GCN-RNI achieves a near-perfect perfor-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 336, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 275, + 221 + ], + "score": 1.0, + "content": "mance on EXP, substantially surpassing", + "type": "text" + }, + { + "bbox": [ + 276, + 210, + 295, + 220 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 210, + 336, + 221 + ], + "score": 1.0, + "content": ". Indeed,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 336, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 336, + 232 + ], + "score": 1.0, + "content": "all fully randomized GCN-RNI models achieve a perfor-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 232, + 336, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 162, + 243 + ], + "score": 1.0, + "content": "mance above", + "type": "text" + }, + { + "bbox": [ + 162, + 232, + 182, + 242 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 232, + 336, + 243 + ], + "score": 1.0, + "content": "with all four RNI distributions. This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 337, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 337, + 255 + ], + "score": 1.0, + "content": "finding supports observations made in related studies on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 337, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 337, + 265 + ], + "score": 1.0, + "content": "RNI (Sato et al., 2020), which suggest that RNI enables", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 264, + 337, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 337, + 277 + ], + "score": 1.0, + "content": "(sub)structure detection beyond the theoretical limits of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 275, + 336, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 336, + 287 + ], + "score": 1.0, + "content": "1-WL. Empirically, we observed that GCN-RNI is highly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 286, + 336, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 336, + 299 + ], + "score": 1.0, + "content": "sensitive to changes in learning rate, activation function,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 297, + 336, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 336, + 309 + ], + "score": 1.0, + "content": "and/or randomization distribution, and required delicate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 263, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 263, + 321 + ], + "score": 1.0, + "content": "tuning to achieve its best performance.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5 + }, + { + "type": "table", + "bbox": [ + 350, + 199, + 495, + 316 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 350, + 199, + 495, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 350, + 199, + 495, + 316 + ], + "spans": [ + { + "bbox": [ + 350, + 199, + 495, + 316 + ], + "score": 0.977, + "html": "
ModelTest Accuracy (%)
GCN-RNI(U)97.3 ± 2.55
GCN-RNI(N) GCN-RNI(XU)98.0 ± 1.85 97.0 ± 1.43
GCN-RNI(XN)96.6 ± 2.20
PPGN50.0
1-2-3-GCN-L50.0
3-GCN99.7 ± 0.004
", + "type": "table", + "image_path": "8e39b157b9b1c4e26382f2283262d684738bb36f6ae20784dfe694cbe3ee2387.jpg" + } + ] + } + ], + "index": 20.0, + "virtual_lines": [ + { + "bbox": [ + 350, + 199, + 495, + 257.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 350, + 257.5, + 495, + 316.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 355, + 179, + 493, + 191 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 354, + 178, + 494, + 192 + ], + "spans": [ + { + "bbox": [ + 354, + 178, + 494, + 192 + ], + "score": 1.0, + "content": "Table 1: Accuracy results on EXP.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 343, + 338 + ], + "score": 1.0, + "content": "Surprisingly, PPGN does not achieve performance above", + "type": "text" + }, + { + "bbox": [ + 343, + 325, + 362, + 336 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 325, + 505, + 338 + ], + "score": 1.0, + "content": ", despite being theoretically 2-WL", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "expressive. Essentially, PPGN learns an approximation of 2-WL, based on power-sum multi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 346, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 361 + ], + "score": 1.0, + "content": "symmetric polynomials (PMP), but fails to distinguish EXP graph pairs, despite extensive training.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "This suggests that PPGNs struggle to learn the required PMPs, and we could not improve these", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "results, both for training and testing, with hyperparameter tuning. Furthermore, as mentioned in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "Section 3, PPGN requires exponentially many data samples in the size of the input graph (Puny", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "et al., 2020) for learning. Hence, PPGN is likely struggling to discern between EXP graph pairs", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 402, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 484, + 413 + ], + "score": 1.0, + "content": "due to the smaller sample size and variability of the dataset. 1-2-3-GCN-L also only achieves", + "type": "text" + }, + { + "bbox": [ + 484, + 402, + 504, + 412 + ], + "score": 0.84, + "content": "50 \\%", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 413, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 424 + ], + "score": 1.0, + "content": "accuracy, which can be attributed to theoretical model limitations. Indeed, the local and connected", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "algorithm drops necessary information to distinguish graph pairs, as it only considers 3-tuples of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "nodes that form a connected sub-graph. Thus, 1-2-3-GCN-L discards disconnected 3-tuples that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "crucially are where the difference between the EXP cores lies. This further highlights the difficulty", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "of EXP instances, as even a relaxation of 2-WL costs the model the ability to achieve above-random", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 468, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 504, + 479 + ], + "score": 1.0, + "content": "performance. Note that 3-GCN achieves near-perfect performance, as it explicitly has the suffi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "cient theoretical power needed for the task, irrespective of learning constraints, and must only learn", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 490, + 451, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 451, + 502 + ], + "score": 1.0, + "content": "appropriate injective aggregation functions for neighbor aggregation (Xu et al., 2019).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 30.5 + }, + { + "type": "image", + "bbox": [ + 114, + 521, + 267, + 655 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 521, + 267, + 655 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 521, + 267, + 655 + ], + "spans": [ + { + "bbox": [ + 114, + 521, + 267, + 655 + ], + "score": 0.963, + "type": "image", + "image_path": "bc876dd422d8f56b42cd4c08770bee5dbcdab007b0466e2888f1f7d72dd04bae.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 114, + 521, + 267, + 588.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 114, + 588.0, + 267, + 655.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 123, + 666, + 263, + 678 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 664, + 264, + 680 + ], + "spans": [ + { + "bbox": [ + 121, + 664, + 264, + 680 + ], + "score": 1.0, + "content": "Figure 2: Learning curves on EXP.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + } + ], + "index": 45.75 + }, + { + "type": "text", + "bbox": [ + 287, + 506, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 286, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 286, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "In terms of model convergence, we observe that 3-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 286, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 286, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "GCN converges significantly faster than all GCN-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 286, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 286, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "RNI models, for all randomization percentages. In-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 286, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 286, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "deed, 3-GCN only requires about 10 epochs to achieve", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 286, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 286, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "its optimal performance, whereas GCN-RNI models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 285, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 285, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "all require in excess of 100 epochs. The rp Intuitively,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 286, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 286, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "the slower convergence of GCN-RNI can be attributed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 286, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 286, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "to a significantly harder learning task compared to 3-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 286, + 594, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 286, + 594, + 505, + 605 + ], + "score": 1.0, + "content": "GCN: Whereas 3-GCN must learn from a determinis-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 286, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 286, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "tic set of node embeddings, and is naturally capable of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 286, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 286, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "discerning between dataset cores, GCN-RNI relies on", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 286, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 286, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "RNI to discern between data points in EXP, via an ar-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 286, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 286, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "tificial node ordering. This in turn implies that GCN-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 286, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 286, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "RNI must first leverage RNI to detect structure, then", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 286, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 286, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "subsequently learn robustness against the variability of", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 286, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 286, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "RNI, which makes the learning task for GCN-RNI es-", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 286, + 681, + 372, + 695 + ], + "spans": [ + { + "bbox": [ + 286, + 681, + 372, + 695 + ], + "score": 1.0, + "content": "pecially challenging.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Our findings suggest that RNI can practically improve the expressiveness of MPNNs, and make them", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 709, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 504, + 723 + ], + "score": 1.0, + "content": "competitive with higher-order models, despite being significantly less demanding computationally.", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "Indeed, for a typical EXP instance with 50 nodes, GCN-RNI only requires 3200 parameters (using", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 60 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 426, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 428, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 428, + 95 + ], + "score": 1.0, + "content": "6.1 EXPERIMENT 1: HOW DOES RNI IMPROVE MPNN EXPRESSIVENESS?", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 428, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "In this experiment, we evaluate GCNs using different RNI settings on EXP, and compare with stan-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "dard GNNs and higher-order models. Specifically, we generate an EXP dataset consisting of 600", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "graph pairs, and discuss this generation in more detail in the appendix. Then, we evaluate all mod-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "els on EXP using 10-fold cross-validation. We train 3-GCN for 100 epochs per fold, and all other", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 495, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 495, + 161 + ], + "score": 1.0, + "content": "systems for 500 epochs. Mean test accuracy across all validation folds is measured and reported.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 105, + 505, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 166, + 336, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 166, + 336, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 336, + 177 + ], + "score": 1.0, + "content": "Full test accuracy results for all models are reported", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 176, + 336, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 336, + 189 + ], + "score": 1.0, + "content": "in Table 1, and model convergence for 3-GCN and all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 336, + 200 + ], + "score": 1.0, + "content": "GCN-RNI models are shown in Figure 2. In line with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 336, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 336, + 210 + ], + "score": 1.0, + "content": "Theorem 4.1, GCN-RNI achieves a near-perfect perfor-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 336, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 275, + 221 + ], + "score": 1.0, + "content": "mance on EXP, substantially surpassing", + "type": "text" + }, + { + "bbox": [ + 276, + 210, + 295, + 220 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 210, + 336, + 221 + ], + "score": 1.0, + "content": ". Indeed,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 336, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 336, + 232 + ], + "score": 1.0, + "content": "all fully randomized GCN-RNI models achieve a perfor-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 232, + 336, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 162, + 243 + ], + "score": 1.0, + "content": "mance above", + "type": "text" + }, + { + "bbox": [ + 162, + 232, + 182, + 242 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 232, + 336, + 243 + ], + "score": 1.0, + "content": "with all four RNI distributions. This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 337, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 337, + 255 + ], + "score": 1.0, + "content": "finding supports observations made in related studies on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 337, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 337, + 265 + ], + "score": 1.0, + "content": "RNI (Sato et al., 2020), which suggest that RNI enables", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 264, + 337, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 337, + 277 + ], + "score": 1.0, + "content": "(sub)structure detection beyond the theoretical limits of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 275, + 336, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 336, + 287 + ], + "score": 1.0, + "content": "1-WL. Empirically, we observed that GCN-RNI is highly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 286, + 336, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 336, + 299 + ], + "score": 1.0, + "content": "sensitive to changes in learning rate, activation function,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 297, + 336, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 336, + 309 + ], + "score": 1.0, + "content": "and/or randomization distribution, and required delicate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 263, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 263, + 321 + ], + "score": 1.0, + "content": "tuning to achieve its best performance.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 166, + 337, + 321 + ] + }, + { + "type": "table", + "bbox": [ + 350, + 199, + 495, + 316 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 350, + 199, + 495, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 350, + 199, + 495, + 316 + ], + "spans": [ + { + "bbox": [ + 350, + 199, + 495, + 316 + ], + "score": 0.977, + "html": "
ModelTest Accuracy (%)
GCN-RNI(U)97.3 ± 2.55
GCN-RNI(N) GCN-RNI(XU)98.0 ± 1.85 97.0 ± 1.43
GCN-RNI(XN)96.6 ± 2.20
PPGN50.0
1-2-3-GCN-L50.0
3-GCN99.7 ± 0.004
", + "type": "table", + "image_path": "8e39b157b9b1c4e26382f2283262d684738bb36f6ae20784dfe694cbe3ee2387.jpg" + } + ] + } + ], + "index": 20.0, + "virtual_lines": [ + { + "bbox": [ + 350, + 199, + 495, + 257.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 350, + 257.5, + 495, + 316.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 355, + 179, + 493, + 191 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 354, + 178, + 494, + 192 + ], + "spans": [ + { + "bbox": [ + 354, + 178, + 494, + 192 + ], + "score": 1.0, + "content": "Table 1: Accuracy results on EXP.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 343, + 338 + ], + "score": 1.0, + "content": "Surprisingly, PPGN does not achieve performance above", + "type": "text" + }, + { + "bbox": [ + 343, + 325, + 362, + 336 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 325, + 505, + 338 + ], + "score": 1.0, + "content": ", despite being theoretically 2-WL", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "expressive. Essentially, PPGN learns an approximation of 2-WL, based on power-sum multi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 346, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 361 + ], + "score": 1.0, + "content": "symmetric polynomials (PMP), but fails to distinguish EXP graph pairs, despite extensive training.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "This suggests that PPGNs struggle to learn the required PMPs, and we could not improve these", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "results, both for training and testing, with hyperparameter tuning. Furthermore, as mentioned in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "Section 3, PPGN requires exponentially many data samples in the size of the input graph (Puny", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "et al., 2020) for learning. Hence, PPGN is likely struggling to discern between EXP graph pairs", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 402, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 484, + 413 + ], + "score": 1.0, + "content": "due to the smaller sample size and variability of the dataset. 1-2-3-GCN-L also only achieves", + "type": "text" + }, + { + "bbox": [ + 484, + 402, + 504, + 412 + ], + "score": 0.84, + "content": "50 \\%", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 413, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 424 + ], + "score": 1.0, + "content": "accuracy, which can be attributed to theoretical model limitations. Indeed, the local and connected", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "algorithm drops necessary information to distinguish graph pairs, as it only considers 3-tuples of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "nodes that form a connected sub-graph. Thus, 1-2-3-GCN-L discards disconnected 3-tuples that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "crucially are where the difference between the EXP cores lies. This further highlights the difficulty", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "of EXP instances, as even a relaxation of 2-WL costs the model the ability to achieve above-random", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 468, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 504, + 479 + ], + "score": 1.0, + "content": "performance. Note that 3-GCN achieves near-perfect performance, as it explicitly has the suffi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "cient theoretical power needed for the task, irrespective of learning constraints, and must only learn", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 490, + 451, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 451, + 502 + ], + "score": 1.0, + "content": "appropriate injective aggregation functions for neighbor aggregation (Xu et al., 2019).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 325, + 506, + 502 + ] + }, + { + "type": "image", + "bbox": [ + 114, + 521, + 267, + 655 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 521, + 267, + 655 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 521, + 267, + 655 + ], + "spans": [ + { + "bbox": [ + 114, + 521, + 267, + 655 + ], + "score": 0.963, + "type": "image", + "image_path": "bc876dd422d8f56b42cd4c08770bee5dbcdab007b0466e2888f1f7d72dd04bae.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 114, + 521, + 267, + 588.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 114, + 588.0, + 267, + 655.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 123, + 666, + 263, + 678 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 664, + 264, + 680 + ], + "spans": [ + { + "bbox": [ + 121, + 664, + 264, + 680 + ], + "score": 1.0, + "content": "Figure 2: Learning curves on EXP.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + } + ], + "index": 45.75 + }, + { + "type": "text", + "bbox": [ + 287, + 506, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 286, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 286, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "In terms of model convergence, we observe that 3-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 286, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 286, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "GCN converges significantly faster than all GCN-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 286, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 286, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "RNI models, for all randomization percentages. In-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 286, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 286, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "deed, 3-GCN only requires about 10 epochs to achieve", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 286, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 286, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "its optimal performance, whereas GCN-RNI models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 285, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 285, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "all require in excess of 100 epochs. The rp Intuitively,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 286, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 286, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "the slower convergence of GCN-RNI can be attributed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 286, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 286, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "to a significantly harder learning task compared to 3-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 286, + 594, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 286, + 594, + 505, + 605 + ], + "score": 1.0, + "content": "GCN: Whereas 3-GCN must learn from a determinis-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 286, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 286, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "tic set of node embeddings, and is naturally capable of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 286, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 286, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "discerning between dataset cores, GCN-RNI relies on", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 286, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 286, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "RNI to discern between data points in EXP, via an ar-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 286, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 286, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "tificial node ordering. This in turn implies that GCN-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 286, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 286, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "RNI must first leverage RNI to detect structure, then", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 286, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 286, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "subsequently learn robustness against the variability of", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 286, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 286, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "RNI, which makes the learning task for GCN-RNI es-", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 286, + 681, + 372, + 695 + ], + "spans": [ + { + "bbox": [ + 286, + 681, + 372, + 695 + ], + "score": 1.0, + "content": "pecially challenging.", + "type": "text" + } + ], + "index": 58 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Nonetheless, GCN-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "RNI performs comparably to 3-GCN, and, unlike the latter model, can easily scale to larger instances", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "exceeding the range used in our datasets. This increase in expressive power, however, comes at the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "cost of slower convergence. Even so, RNI proves to be a promising direction for building scalable", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 352, + 197, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 197, + 363 + ], + "score": 1.0, + "content": "yet powerful MPNNs.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 376, + 477, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 478, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 478, + 389 + ], + "score": 1.0, + "content": "6.2 EXPERIMENT 2: HOW DOES RNI AFFECT MPNN ON MORE VARIABLE DATASETS?", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 504, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "In Experiment 1, we observed that RNI practically improves the expressive power of GCNs over", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "EXP. However, EXP is solely designed for expressiveness evaluation, and this leaves multiple ques-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "tions open: How does RNI impact learning when data contains instances with varying expressive-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "ness requirements, and how does RNI affect model generalization on more variable datasets? We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 442, + 350, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 350, + 455 + ], + "score": 1.0, + "content": "experiment with CEXP to explicitly address these questions.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "Analogously to Experiment 1, we generate an EXP dataset with 600 pairs of graphs. Then, we create", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "score": 1.0, + "content": "CEXP by selecting 300 graph pairs and modifying their satisfiable graph, yielding CORRUPT. CEXP", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "is well-suited for evaluating the efficacy of RNI more holistically, as it allows (i) the evaluation of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "the contribution of RNI on EXP conjointly with a second learning task on CORRUPT involving very", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "similar core structures, and (ii) a study of the effect of different degrees of randomization on overall", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 514, + 267, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 267, + 527 + ], + "score": 1.0, + "content": "and subset-specific model performance.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "In this experiment, we train GCN-RNI (with varying randomization degrees) and 3-GCN on CEXP,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "and compare their test accuracy across all cross-validation splits. For GCN-RNI models, we observe", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 273, + 567 + ], + "score": 1.0, + "content": "the effect of RNI on specifically learning", + "type": "text" + }, + { + "bbox": [ + 274, + 553, + 293, + 565 + ], + "score": 0.42, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "and CORRUPT, and the interplay between these two", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 563, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 580 + ], + "score": 1.0, + "content": "tasks. In all experiments, we exclusively use normal distribution initialization, given its strong", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 577, + 229, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 229, + 588 + ], + "score": 1.0, + "content": "performance in Experiment 1.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "The learning curves of all GCN-RNI and 3-GCN on CEXP are shown in Figure 3a, and the same", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "curves for the EXP and CORRUPT subsets are shown in Figure 3b. As on EXP, we observe that 3-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 268, + 629 + ], + "score": 1.0, + "content": "GCN converges very quickly, exceeding", + "type": "text" + }, + { + "bbox": [ + 268, + 616, + 288, + 627 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "test accuracy within 25 epochs on CEXP. By contrast,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "GCN-RNI, for all randomization levels, converges much slower, around after 200 epochs, despite the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 214, + 651 + ], + "score": 1.0, + "content": "small size of input graphs (", + "type": "text" + }, + { + "bbox": [ + 214, + 638, + 231, + 649 + ], + "score": 0.73, + "content": "\\mathord { \\sim } 7 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "nodes at most). Furthermore, fully randomized GCN-RNI performs", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "worse than partly randomized GCN-RNI models, particularly on CEXP, due to its weak performance", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 660, + 257, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 257, + 672 + ], + "score": 1.0, + "content": "on CORRUPT, as shown in Figure 3b.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 54 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "First, we observe that partial randomization can significantly improve model performance. This can", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 375, + 700 + ], + "score": 1.0, + "content": "clearly be seen on CEXP, in Figure 3a and Figure 3b, where GCN-", + "type": "text" + }, + { + "bbox": [ + 376, + 688, + 399, + 698 + ], + "score": 0.3, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "RNI and GCN-87.5%RNI", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "achieve the best performance, by far outperforming GCN-RNI, which struggles on CORRUPT. This", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "can be attributed to having a better inductive bias than a fully randomized model. Indeed, GCN-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 107, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 720, + 151, + 731 + ], + "score": 0.49, + "content": "1 2 . 5 \\% \\mathrm { R N I }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "has mostly deterministic node embeddings, which simplifies learning over CORRUPT.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 60 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 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GCN-RNI models and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 112, + 254, + 180, + 264 + ], + "spans": [ + { + "bbox": [ + 112, + 254, + 180, + 264 + ], + "score": 1.0, + "content": "3-GCN on CEXP.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 112, + 243, + 298, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "64-dimensional embeddings), whereas 3-GCN requires 1,254,400 parameters. Nonetheless, GCN-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "RNI performs comparably to 3-GCN, and, unlike the latter model, can easily scale to larger instances", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "exceeding the range used in our datasets. This increase in expressive power, however, comes at the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "cost of slower convergence. Even so, RNI proves to be a promising direction for building scalable", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 352, + 197, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 197, + 363 + ], + "score": 1.0, + "content": "yet powerful MPNNs.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 307, + 505, + 363 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 376, + 477, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 478, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 478, + 389 + ], + "score": 1.0, + "content": "6.2 EXPERIMENT 2: HOW DOES RNI AFFECT MPNN ON MORE VARIABLE DATASETS?", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 504, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "In Experiment 1, we observed that RNI practically improves the expressive power of GCNs over", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "EXP. However, EXP is solely designed for expressiveness evaluation, and this leaves multiple ques-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "tions open: How does RNI impact learning when data contains instances with varying expressive-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "ness requirements, and how does RNI affect model generalization on more variable datasets? We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 442, + 350, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 350, + 455 + ], + "score": 1.0, + "content": "experiment with CEXP to explicitly address these questions.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 396, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "Analogously to Experiment 1, we generate an EXP dataset with 600 pairs of graphs. Then, we create", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "score": 1.0, + "content": "CEXP by selecting 300 graph pairs and modifying their satisfiable graph, yielding CORRUPT. CEXP", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "is well-suited for evaluating the efficacy of RNI more holistically, as it allows (i) the evaluation of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "the contribution of RNI on EXP conjointly with a second learning task on CORRUPT involving very", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "similar core structures, and (ii) a study of the effect of different degrees of randomization on overall", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 514, + 267, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 267, + 527 + ], + "score": 1.0, + "content": "and subset-specific model performance.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 458, + 505, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "In this experiment, we train GCN-RNI (with varying randomization degrees) and 3-GCN on CEXP,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "and compare their test accuracy across all cross-validation splits. For GCN-RNI models, we observe", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 273, + 567 + ], + "score": 1.0, + "content": "the effect of RNI on specifically learning", + "type": "text" + }, + { + "bbox": [ + 274, + 553, + 293, + 565 + ], + "score": 0.42, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "and CORRUPT, and the interplay between these two", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 563, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 580 + ], + "score": 1.0, + "content": "tasks. In all experiments, we exclusively use normal distribution initialization, given its strong", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 577, + 229, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 229, + 588 + ], + "score": 1.0, + "content": "performance in Experiment 1.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 531, + 506, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "The learning curves of all GCN-RNI and 3-GCN on CEXP are shown in Figure 3a, and the same", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "curves for the EXP and CORRUPT subsets are shown in Figure 3b. As on EXP, we observe that 3-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 268, + 629 + ], + "score": 1.0, + "content": "GCN converges very quickly, exceeding", + "type": "text" + }, + { + "bbox": [ + 268, + 616, + 288, + 627 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "test accuracy within 25 epochs on CEXP. By contrast,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "GCN-RNI, for all randomization levels, converges much slower, around after 200 epochs, despite the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 214, + 651 + ], + "score": 1.0, + "content": "small size of input graphs (", + "type": "text" + }, + { + "bbox": [ + 214, + 638, + 231, + 649 + ], + "score": 0.73, + "content": "\\mathord { \\sim } 7 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "nodes at most). Furthermore, fully randomized GCN-RNI performs", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "worse than partly randomized GCN-RNI models, particularly on CEXP, due to its weak performance", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 660, + 257, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 257, + 672 + ], + "score": 1.0, + "content": "on CORRUPT, as shown in Figure 3b.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 54, + "bbox_fs": [ + 105, + 593, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "First, we observe that partial randomization can significantly improve model performance. This can", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 375, + 700 + ], + "score": 1.0, + "content": "clearly be seen on CEXP, in Figure 3a and Figure 3b, where GCN-", + "type": "text" + }, + { + "bbox": [ + 376, + 688, + 399, + 698 + ], + "score": 0.3, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "RNI and GCN-87.5%RNI", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "achieve the best performance, by far outperforming GCN-RNI, which struggles on CORRUPT. This", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "can be attributed to having a better inductive bias than a fully randomized model. Indeed, GCN-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 107, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 720, + 151, + 731 + ], + "score": 0.49, + "content": "1 2 . 5 \\% \\mathrm { R N I }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "has mostly deterministic node embeddings, which simplifies learning over CORRUPT.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 60, + "bbox_fs": [ + 105, + 677, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 186, + 94 + ], + "score": 1.0, + "content": "This also applies to", + "type": "text" + }, + { + "bbox": [ + 186, + 82, + 249, + 93 + ], + "score": 0.68, + "content": "\\mathrm { G C N - } 8 7 . 5 \\% \\mathrm { R N }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "I, where the number of deterministic dimensions, though small,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "remains sufficient for learning over CORRUPT. Both models also benefit from randomization to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 188, + 118 + ], + "score": 1.0, + "content": "perform strongly on", + "type": "text" + }, + { + "bbox": [ + 188, + 105, + 208, + 116 + ], + "score": 0.53, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 105, + 505, + 118 + ], + "score": 1.0, + "content": ", and have sufficient randomization to perform similarly to a fully random-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 179, + 129 + ], + "score": 1.0, + "content": "ized GCN. GCN-", + "type": "text" + }, + { + "bbox": [ + 180, + 117, + 223, + 127 + ], + "score": 0.28, + "content": "1 2 . 5 \\% \\mathrm { R N I }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 116, + 243, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 116, + 302, + 127 + ], + "score": 0.39, + "content": "\\mathrm { G C N - } 8 7 . 5 \\% \\mathrm { R l }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "NI effectively achieve the best of both worlds on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "CEXP, leveraging inductive bias from deterministic node embeddings, while harnessing the power", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 297, + 153 + ], + "score": 1.0, + "content": "of random embeddings to perform strongly on", + "type": "text" + }, + { + "bbox": [ + 297, + 138, + 316, + 150 + ], + "score": 0.4, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 140, + 505, + 153 + ], + "score": 1.0, + "content": ". This is best shown in Figure 3b, where stan-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 207, + 164 + ], + "score": 1.0, + "content": "dard GCN fails to learn", + "type": "text" + }, + { + "bbox": [ + 208, + 150, + 227, + 162 + ], + "score": 0.43, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 151, + 505, + 164 + ], + "score": 1.0, + "content": ", fully randomized GCN-RNI struggles to learn CORRUPT, and the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 162, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 179, + 175 + ], + "score": 1.0, + "content": "semi-randomized", + "type": "text" + }, + { + "bbox": [ + 180, + 163, + 230, + 173 + ], + "score": 0.36, + "content": "G C N { - } 5 0 \\% \\mathrm { R l }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 162, + 506, + 175 + ], + "score": 1.0, + "content": "NI achieves perfect performance on both subsets. Overall, this is a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "surprising finding, as it suggests that MPNNs can perform significantly better with partial, and even", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 185, + 242, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 242, + 197 + ], + "score": 1.0, + "content": "small, amounts of randomization.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "Second, we observe that the fully randomized GCN-RNI performs substantially worse than its", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "partially randomized counterparts. Whereas fully randomized GCN-RNI only performs marginally", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "worse on EXP (cf. Figure 2) than partially randomized models, this gap is very large on CEXP,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "primarily due to CORRUPT. This observation concurs with the earlier idea of inductive bias: Fully", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "randomized GCN-RNI loses all node type information, which is valuable for making robust and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "consistent decisions, and therefore struggles to match 3-GCN and partially randomized models.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 267, + 280 + ], + "score": 1.0, + "content": "Indeed, the model fails to achieve even", + "type": "text" + }, + { + "bbox": [ + 268, + 267, + 288, + 277 + ], + "score": 0.85, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "accuracy on CORRUPT, where other models are near", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 279, + 498, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 265, + 292 + ], + "score": 1.0, + "content": "perfect, and also relatively struggles on", + "type": "text" + }, + { + "bbox": [ + 265, + 279, + 284, + 290 + ], + "score": 0.3, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 279, + 344, + 292 + ], + "score": 1.0, + "content": ", only reaching", + "type": "text" + }, + { + "bbox": [ + 345, + 279, + 365, + 290 + ], + "score": 0.85, + "content": "91 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 279, + 498, + 292 + ], + "score": 1.0, + "content": "accuracy and converging slower.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Third, all GCN-RNI models, at all randomization levels, converge significantly slower on both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "datasets than 3-GCN, similarly to Experiment 1. However, an interesting phenomenon can be seen", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 293, + 330 + ], + "score": 1.0, + "content": "on CEXP: All GCN-RNI models hover around", + "type": "text" + }, + { + "bbox": [ + 293, + 318, + 313, + 329 + ], + "score": 0.87, + "content": "55 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "accuracy within the first 100 epochs over CEXP", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 398, + 343 + ], + "score": 1.0, + "content": "(cf. Figure 3a), suggesting a struggle jointly fitting both CORRUPT and", + "type": "text" + }, + { + "bbox": [ + 398, + 329, + 417, + 340 + ], + "score": 0.29, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 330, + 505, + 343 + ], + "score": 1.0, + "content": ", before these models", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "ultimately improve. This, however, is not observed with 3-GCN. Unlike on EXP, randomness is not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 364 + ], + "score": 1.0, + "content": "necessarily beneficial on CEXP, as it can hurt performance on CORRUPT. Hence, RNI-enhanced", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "models must additionally learn to isolate deterministic dimensions for CORRUPT, and randomized", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "dimensions for EXP. These findings consolidate the earlier observations made on EXP on the impact", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "of RNI on MPNN learning behavior, and highlight that the variability and slower learning for RNI", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 397, + 368, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 368, + 409 + ], + "score": 1.0, + "content": "also hinges on the variability and complexity of the input dataset.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "Finally, we observe that both fully randomized GCN-RNI, and, surprisingly, 1-GCN, struggle to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "learn CORRUPT relative to partially randomized GCN-RNI. We can also observe that 1-GCN does", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "not present a “struggle” phase, and begins improving consistently from the start of training. These", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "observations can be attributed to key conceptual , but very distinct hindrances impeding both models.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 472 + ], + "score": 1.0, + "content": "In the case of 1-GCN, the model is jointly trying to learn both EXP and CORRUPT, when it is proven", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "that it cannot fit the former. This joint optimization severely hinders CORRUPT learning, as data pairs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "from both subsets are highly similar, and share identically generated UNSAT graphs (cf. Appendix).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "Hence, 1-GCN, in attempting to fit SAT graphs from both subsets, knowing it cannot distinguish", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "EXP pairs, struggles to learn the simpler difference in CORRUPT pairs. For GCN-RNI, the model", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "score": 1.0, + "content": "discards key type information, so must only rely on structural differences to learn CORRUPT, which", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "impedes its convergence. All in all, this further consolidates the promise of partial RNI as a means", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 537, + 378, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 378, + 548 + ], + "score": 1.0, + "content": "to combine the strengths of both deterministic and random features.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "Further to the earlier two experiments, we also conducted analogous experiments using sparser", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "analogs of the datasets EXP and CEXP. In these cases, we observed similar behavior, albeit with", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 575, + 480, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 480, + 589 + ], + "score": 1.0, + "content": "slower convergence overall. More details on these experiments can be found in the appendix.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "title", + "bbox": [ + 109, + 606, + 260, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 263, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 263, + 621 + ], + "score": 1.0, + "content": "7 SUMMARY AND OUTLOOK", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "We studied the expressive power of MPNNs with RNI, and showed that these are universal models.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "score": 1.0, + "content": "We empirically evaluated this model on carefully designed datasets, and observed that RNI prac-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "tically improves the learning abilities of MPNNs for challenging data, though it does slow down", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "model convergence owing to the need to learn robustness against random variability. Our work", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "score": 1.0, + "content": "delivers a strong theoretical result, supported by empirical evaluation and practical insights, to rig-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "orously quantify the effect of RNI on GNNs. Somewhat surprisingly, our experiments suggest that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "partial randomization may be the best strategy in most practical scenarios. An important direction", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "for future work is to theoretically study the sensitivity of RNI to model architectures and initializa-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 493, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 493, + 733 + ], + "score": 1.0, + "content": "tion distributions, to yield a more complete understanding of the benefits and limitations of RNI.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 186, + 94 + ], + "score": 1.0, + "content": "This also applies to", + "type": "text" + }, + { + "bbox": [ + 186, + 82, + 249, + 93 + ], + "score": 0.68, + "content": "\\mathrm { G C N - } 8 7 . 5 \\% \\mathrm { R N }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "I, where the number of deterministic dimensions, though small,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "remains sufficient for learning over CORRUPT. Both models also benefit from randomization to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 188, + 118 + ], + "score": 1.0, + "content": "perform strongly on", + "type": "text" + }, + { + "bbox": [ + 188, + 105, + 208, + 116 + ], + "score": 0.53, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 105, + 505, + 118 + ], + "score": 1.0, + "content": ", and have sufficient randomization to perform similarly to a fully random-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 179, + 129 + ], + "score": 1.0, + "content": "ized GCN. GCN-", + "type": "text" + }, + { + "bbox": [ + 180, + 117, + 223, + 127 + ], + "score": 0.28, + "content": "1 2 . 5 \\% \\mathrm { R N I }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 116, + 243, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 116, + 302, + 127 + ], + "score": 0.39, + "content": "\\mathrm { G C N - } 8 7 . 5 \\% \\mathrm { R l }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "NI effectively achieve the best of both worlds on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "CEXP, leveraging inductive bias from deterministic node embeddings, while harnessing the power", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 297, + 153 + ], + "score": 1.0, + "content": "of random embeddings to perform strongly on", + "type": "text" + }, + { + "bbox": [ + 297, + 138, + 316, + 150 + ], + "score": 0.4, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 140, + 505, + 153 + ], + "score": 1.0, + "content": ". This is best shown in Figure 3b, where stan-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 207, + 164 + ], + "score": 1.0, + "content": "dard GCN fails to learn", + "type": "text" + }, + { + "bbox": [ + 208, + 150, + 227, + 162 + ], + "score": 0.43, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 151, + 505, + 164 + ], + "score": 1.0, + "content": ", fully randomized GCN-RNI struggles to learn CORRUPT, and the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 162, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 179, + 175 + ], + "score": 1.0, + "content": "semi-randomized", + "type": "text" + }, + { + "bbox": [ + 180, + 163, + 230, + 173 + ], + "score": 0.36, + "content": "G C N { - } 5 0 \\% \\mathrm { R l }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 162, + 506, + 175 + ], + "score": 1.0, + "content": "NI achieves perfect performance on both subsets. Overall, this is a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "surprising finding, as it suggests that MPNNs can perform significantly better with partial, and even", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 185, + 242, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 242, + 197 + ], + "score": 1.0, + "content": "small, amounts of randomization.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 197 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "Second, we observe that the fully randomized GCN-RNI performs substantially worse than its", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "partially randomized counterparts. Whereas fully randomized GCN-RNI only performs marginally", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "worse on EXP (cf. Figure 2) than partially randomized models, this gap is very large on CEXP,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "primarily due to CORRUPT. This observation concurs with the earlier idea of inductive bias: Fully", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "randomized GCN-RNI loses all node type information, which is valuable for making robust and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "consistent decisions, and therefore struggles to match 3-GCN and partially randomized models.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 267, + 280 + ], + "score": 1.0, + "content": "Indeed, the model fails to achieve even", + "type": "text" + }, + { + "bbox": [ + 268, + 267, + 288, + 277 + ], + "score": 0.85, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "accuracy on CORRUPT, where other models are near", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 279, + 498, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 265, + 292 + ], + "score": 1.0, + "content": "perfect, and also relatively struggles on", + "type": "text" + }, + { + "bbox": [ + 265, + 279, + 284, + 290 + ], + "score": 0.3, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 279, + 344, + 292 + ], + "score": 1.0, + "content": ", only reaching", + "type": "text" + }, + { + "bbox": [ + 345, + 279, + 365, + 290 + ], + "score": 0.85, + "content": "91 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 279, + 498, + 292 + ], + "score": 1.0, + "content": "accuracy and converging slower.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 201, + 506, + 292 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Third, all GCN-RNI models, at all randomization levels, converge significantly slower on both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "datasets than 3-GCN, similarly to Experiment 1. However, an interesting phenomenon can be seen", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 293, + 330 + ], + "score": 1.0, + "content": "on CEXP: All GCN-RNI models hover around", + "type": "text" + }, + { + "bbox": [ + 293, + 318, + 313, + 329 + ], + "score": 0.87, + "content": "55 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "accuracy within the first 100 epochs over CEXP", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 398, + 343 + ], + "score": 1.0, + "content": "(cf. Figure 3a), suggesting a struggle jointly fitting both CORRUPT and", + "type": "text" + }, + { + "bbox": [ + 398, + 329, + 417, + 340 + ], + "score": 0.29, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 330, + 505, + 343 + ], + "score": 1.0, + "content": ", before these models", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "ultimately improve. This, however, is not observed with 3-GCN. Unlike on EXP, randomness is not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 364 + ], + "score": 1.0, + "content": "necessarily beneficial on CEXP, as it can hurt performance on CORRUPT. Hence, RNI-enhanced", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "models must additionally learn to isolate deterministic dimensions for CORRUPT, and randomized", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "dimensions for EXP. These findings consolidate the earlier observations made on EXP on the impact", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "of RNI on MPNN learning behavior, and highlight that the variability and slower learning for RNI", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 397, + 368, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 368, + 409 + ], + "score": 1.0, + "content": "also hinges on the variability and complexity of the input dataset.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 295, + 506, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "Finally, we observe that both fully randomized GCN-RNI, and, surprisingly, 1-GCN, struggle to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "learn CORRUPT relative to partially randomized GCN-RNI. We can also observe that 1-GCN does", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "not present a “struggle” phase, and begins improving consistently from the start of training. These", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "observations can be attributed to key conceptual , but very distinct hindrances impeding both models.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 472 + ], + "score": 1.0, + "content": "In the case of 1-GCN, the model is jointly trying to learn both EXP and CORRUPT, when it is proven", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "that it cannot fit the former. This joint optimization severely hinders CORRUPT learning, as data pairs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "from both subsets are highly similar, and share identically generated UNSAT graphs (cf. Appendix).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "Hence, 1-GCN, in attempting to fit SAT graphs from both subsets, knowing it cannot distinguish", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "EXP pairs, struggles to learn the simpler difference in CORRUPT pairs. For GCN-RNI, the model", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 528 + ], + "score": 1.0, + "content": "discards key type information, so must only rely on structural differences to learn CORRUPT, which", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "impedes its convergence. All in all, this further consolidates the promise of partial RNI as a means", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 537, + 378, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 378, + 548 + ], + "score": 1.0, + "content": "to combine the strengths of both deterministic and random features.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 414, + 506, + 548 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "Further to the earlier two experiments, we also conducted analogous experiments using sparser", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "analogs of the datasets EXP and CEXP. In these cases, we observed similar behavior, albeit with", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 575, + 480, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 480, + 589 + ], + "score": 1.0, + "content": "slower convergence overall. More details on these experiments can be found in the appendix.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 553, + 505, + 589 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 606, + 260, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 263, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 263, + 621 + ], + "score": 1.0, + "content": "7 SUMMARY AND OUTLOOK", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "We studied the expressive power of MPNNs with RNI, and showed that these are universal models.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "score": 1.0, + "content": "We empirically evaluated this model on carefully designed datasets, and observed that RNI prac-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "tically improves the learning abilities of MPNNs for challenging data, though it does slow down", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "model convergence owing to the need to learn robustness against random variability. Our work", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "score": 1.0, + "content": "delivers a strong theoretical result, supported by empirical evaluation and practical insights, to rig-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "orously quantify the effect of RNI on GNNs. Somewhat surprisingly, our experiments suggest that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "partial randomization may be the best strategy in most practical scenarios. 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Let", + "type": "text" + }, + { + "bbox": [ + 182, + 587, + 259, + 599 + ], + "score": 0.91, + "content": "h : { \\mathcal { G } } _ { n , k } \\to \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 585, + 487, + 599 + ], + "score": 1.0, + "content": "be an invariant Boolean function. Then there exists a", + "type": "text" + }, + { + "bbox": [ + 487, + 585, + 500, + 597 + ], + "score": 0.51, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 585, + 504, + 599 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 597, + 384, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 143, + 611 + ], + "score": 1.0, + "content": "sentence", + "type": "text" + }, + { + "bbox": [ + 144, + 599, + 156, + 609 + ], + "score": 0.89, + "content": "\\psi _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 597, + 223, + 611 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 223, + 599, + 259, + 610 + ], + "score": 0.92, + "content": "G \\in \\mathcal { G } _ { n , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 597, + 311, + 611 + ], + "score": 1.0, + "content": "it holds that", + "type": "text" + }, + { + "bbox": [ + 312, + 598, + 381, + 610 + ], + "score": 0.93, + "content": "[ [ \\psi _ { h } ] ] ( G ) = h ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 597, + 384, + 611 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 105, + 621, + 445, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 446, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 152, + 635 + ], + "score": 1.0, + "content": "Proof. 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Then there exists a", + "type": "text" + }, + { + "bbox": [ + 487, + 585, + 500, + 597 + ], + "score": 0.51, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 585, + 504, + 599 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 597, + 384, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 143, + 611 + ], + "score": 1.0, + "content": "sentence", + "type": "text" + }, + { + "bbox": [ + 144, + 599, + 156, + 609 + ], + "score": 0.89, + "content": "\\psi _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 597, + 223, + 611 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 223, + 599, + 259, + 610 + ], + "score": 0.92, + "content": "G \\in \\mathcal { G } _ { n , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 597, + 311, + 611 + ], + "score": 1.0, + "content": "it holds that", + "type": "text" + }, + { + "bbox": [ + 312, + 598, + 381, + 610 + ], + "score": 0.93, + "content": "[ [ \\psi _ { h } ] ] ( G ) = h ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 597, + 384, + 611 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 585, + 504, + 611 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 621, + 445, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 446, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 152, + 635 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 622, + 190, + 634 + ], + "score": 0.92, + "content": "{ \\mathcal { H } } \\subseteq { \\mathcal { G } } _ { n , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 621, + 340, + 635 + ], + "score": 1.0, + "content": "be the subset consisting of all graphs", + "type": "text" + }, + { + "bbox": [ + 341, + 622, + 351, + 632 + ], + "score": 0.85, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 621, + 372, + 635 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 372, + 622, + 412, + 634 + ], + "score": 0.92, + "content": "h ( H ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 621, + 446, + 635 + ], + "score": 1.0, + "content": ". We let", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 621, + 446, + 635 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 275, + 637, + 336, + 659 + ], + "lines": [ + { + "bbox": [ + 275, + 637, + 336, + 659 + ], + "spans": [ + { + "bbox": [ + 275, + 637, + 336, + 659 + ], + "score": 0.93, + "content": "\\psi _ { h } : = \\bigvee _ { H \\in \\mathcal { H } } \\chi _ { H } .", + "type": "interline_equation", + "image_path": "49874527490f55ab52810e93895774282e09833140c26bcfe589949691871a8f.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 275, + 637, + 336, + 659 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 106, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 430, + 676 + ], + "score": 1.0, + "content": "We eliminate duplicates in the disjunction. 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Let", + "type": "text" + }, + { + "bbox": [ + 191, + 82, + 260, + 95 + ], + "score": 0.92, + "content": "f : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 80, + 487, + 95 + ], + "score": 1.0, + "content": "be an invariant Boolean function. 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We let", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 143, + 363, + 169 + ], + "lines": [ + { + "bbox": [ + 248, + 143, + 363, + 169 + ], + "spans": [ + { + "bbox": [ + 248, + 143, + 363, + 169 + ], + "score": 0.93, + "content": "c : = \\left\\lceil { \\frac { 2 } { \\delta } } \\right\\rceil \\quad { \\mathrm { a n d } } \\quad k : = c ^ { 2 } \\cdot n ^ { 3 }", + "type": "interline_equation", + "image_path": "ba7e9ba578d79f745c90f449340b738ae19ffb6bc9a251db3c8484f2bf543f85.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 248, + 143, + 363, + 169 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "The technical details of the proof of Lemma A.1 and Theorem 4.1 depend on the exact choice of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "random initialization and the activation functions used in the neural networks, but the idea is always", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 365, + 209 + ], + "score": 1.0, + "content": "the same. 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Let", + "type": "text" + }, + { + "bbox": [ + 180, + 245, + 221, + 254 + ], + "score": 0.87, + "content": "r _ { 1 } , \\ldots , r _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 241, + 478, + 256 + ], + "score": 1.0, + "content": "be chosen independently uniformly at random from the interval", + "type": "text" + }, + { + "bbox": [ + 479, + 243, + 501, + 255 + ], + "score": 0.31, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 241, + 504, + 256 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 252, + 248, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 123, + 267 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 254, + 159, + 264 + ], + "score": 0.9, + "content": "1 \\leq i \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 252, + 178, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 254, + 230, + 265 + ], + "score": 0.91, + "content": "1 \\leq j \\leq c \\cdot n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 252, + 248, + 267 + ], + "score": 1.0, + "content": ", let", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 270, + 363, + 295 + ], + "lines": [ + { + "bbox": [ + 247, + 270, + 363, + 295 + ], + "spans": [ + { + "bbox": [ + 247, + 270, + 363, + 295 + ], + "score": 0.93, + "content": "s _ { i j } : = k \\cdot r _ { i } - \\left( j - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } .", + "type": "interline_equation", + "image_path": "0258962bac425b6802425ddc9855ec6eef3641e804a5d99f7b6570262f6d3558.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 247, + 270, + 363, + 295 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 299, + 422, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 423, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 246, + 313 + ], + "score": 1.0, + "content": "Then with probability greater than", + "type": "text" + }, + { + "bbox": [ + 247, + 300, + 267, + 310 + ], + "score": 0.61, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 298, + 423, + 313 + ], + "score": 1.0, + "content": ", the following conditions are satisfied.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 124, + 339, + 503, + 353 + ], + "lines": [ + { + "bbox": [ + 123, + 337, + 501, + 356 + ], + "spans": [ + { + "bbox": [ + 123, + 337, + 203, + 356 + ], + "score": 1.0, + "content": "(ii) For all distinct", + "type": "text" + }, + { + "bbox": [ + 204, + 340, + 269, + 353 + ], + "score": 0.93, + "content": "i , i ^ { \\prime } \\in \\{ 1 , . . . , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 337, + 325, + 356 + ], + "score": 1.0, + "content": "there exists a", + "type": "text" + }, + { + "bbox": [ + 325, + 340, + 395, + 353 + ], + "score": 0.93, + "content": "j \\in \\left\\{ 1 , \\dots , c \\cdot n ^ { 2 } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 337, + 434, + 356 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 434, + 340, + 501, + 353 + ], + "score": 0.92, + "content": "\\sigma ( s _ { i j } ) \\neq \\sigma ( s _ { i ^ { \\prime } j } )", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 177, + 378 + ], + "score": 1.0, + "content": "Proof. For every", + "type": "text" + }, + { + "bbox": [ + 178, + 366, + 182, + 375 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 364, + 199, + 378 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 200, + 365, + 251, + 377 + ], + "score": 0.93, + "content": "p _ { i } : = \\lfloor r _ { i } \\cdot k \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 364, + 282, + 378 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 282, + 366, + 304, + 376 + ], + "score": 0.88, + "content": "k \\cdot r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 364, + 462, + 378 + ], + "score": 1.0, + "content": "is uniformly random from the interval", + "type": "text" + }, + { + "bbox": [ + 463, + 365, + 486, + 377 + ], + "score": 0.91, + "content": "[ 0 , k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 364, + 505, + 378 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 136, + 389 + ], + "score": 1.0, + "content": "integer", + "type": "text" + }, + { + "bbox": [ + 137, + 378, + 146, + 387 + ], + "score": 0.84, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 375, + 257, + 389 + ], + "score": 1.0, + "content": "is uniformly random from", + "type": "text" + }, + { + "bbox": [ + 257, + 376, + 314, + 388 + ], + "score": 0.92, + "content": "\\{ 0 , \\ldots , k - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 375, + 374, + 389 + ], + "score": 1.0, + "content": ". Observe that", + "type": "text" + }, + { + "bbox": [ + 374, + 376, + 435, + 388 + ], + "score": 0.93, + "content": "0 < \\sigma ( s _ { i j } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 375, + 466, + 389 + ], + "score": 1.0, + "content": "only if", + "type": "text" + }, + { + "bbox": [ + 466, + 376, + 505, + 388 + ], + "score": 0.9, + "content": "p _ { i } - ( j -", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 383, + 501, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 154, + 401 + ], + "score": 0.91, + "content": "\\textstyle 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 383, + 257, + 407 + ], + "score": 1.0, + "content": "(here we use the fact that", + "type": "text" + }, + { + "bbox": [ + 257, + 389, + 263, + 398 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 383, + 322, + 407 + ], + "score": 1.0, + "content": "is divisible by", + "type": "text" + }, + { + "bbox": [ + 322, + 388, + 344, + 399 + ], + "score": 0.87, + "content": "c \\cdot n ^ { 2 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 383, + 493, + 407 + ], + "score": 1.0, + "content": "). The probability that this happens is", + "type": "text" + }, + { + "bbox": [ + 494, + 388, + 501, + 402 + ], + "score": 0.86, + "content": "\\frac { 1 } { k }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 399, + 218, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 218, + 412 + ], + "score": 1.0, + "content": "⋅Thus, by the Union Bound,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 416, + 379, + 442 + ], + "lines": [ + { + "bbox": [ + 233, + 416, + 379, + 442 + ], + "spans": [ + { + "bbox": [ + 233, + 416, + 379, + 442 + ], + "score": 0.93, + "content": "\\operatorname* { P r } \\left( \\exists i , j : 0 < \\sigma ( s _ { i j } ) < 1 \\right) \\leq \\frac { c \\cdot n ^ { 3 } } { k } .", + "type": "interline_equation", + "image_path": "bd4afacfa46bc02b6c3ee163416d9fae134482461bb962ca484f5fed1aa068ed.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 233, + 416, + 379, + 442 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 447, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 142, + 462 + ], + "score": 1.0, + "content": "Now let", + "type": "text" + }, + { + "bbox": [ + 142, + 447, + 157, + 459 + ], + "score": 0.89, + "content": "i , i ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 445, + 275, + 462 + ], + "score": 1.0, + "content": "be distinct and suppose that", + "type": "text" + }, + { + "bbox": [ + 275, + 447, + 344, + 460 + ], + "score": 0.92, + "content": "\\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 445, + 374, + 462 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 374, + 448, + 380, + 459 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 445, + 437, + 462 + ], + "score": 1.0, + "content": ". Then for all", + "type": "text" + }, + { + "bbox": [ + 438, + 448, + 444, + 459 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 445, + 482, + 462 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 482, + 449, + 505, + 460 + ], + "score": 0.88, + "content": "s _ { i j } \\leq", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 458, + 384, + 471 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 170, + 471 + ], + "score": 0.91, + "content": "0 \\iff s _ { i ^ { \\prime } j } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 458, + 226, + 471 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 227, + 459, + 327, + 471 + ], + "score": 0.92, + "content": "\\lfloor s _ { i j } \\rfloor \\le 0 \\iff \\lfloor \\stackrel { \\sim } { s } _ { i ^ { \\prime } j } \\rfloor \\le 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 458, + 384, + 471 + ], + "score": 1.0, + "content": ". This implies", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 475, + 447, + 500 + ], + "lines": [ + { + "bbox": [ + 164, + 475, + 447, + 500 + ], + "spans": [ + { + "bbox": [ + 164, + 475, + 447, + 500 + ], + "score": 0.94, + "content": "\\forall j \\in \\{ 1 , \\ldots , c \\cdot n ^ { 2 } \\} : \\quad p _ { i } \\leq ( j - 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } \\Longleftrightarrow p _ { i ^ { \\prime } } \\leq ( j - 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } .", + "type": "interline_equation", + "image_path": "cb2e1e4fdb6329bb8691bc30d59a92c0ea0650ad0c4d973f46995b676839b4f6.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 164, + 475, + 447, + 500 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 506, + 571 + ], + "lines": [ + { + "bbox": [ + 101, + 500, + 509, + 529 + ], + "spans": [ + { + "bbox": [ + 101, + 500, + 123, + 529 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 509, + 203, + 522 + ], + "score": 0.92, + "content": "j ^ { * } \\in \\{ 1 , \\ldots , c \\cdot n ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 500, + 244, + 529 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 244, + 506, + 402, + 525 + ], + "score": 0.93, + "content": "p _ { i } \\in \\left\\{ \\left( j ^ { * } - 1 \\right) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } , \\ldots , j ^ { * } \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } - 1 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 500, + 509, + 529 + ], + "score": 1.0, + "content": ". Then by (A.4) we have", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 525, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 262, + 544 + ], + "score": 0.91, + "content": "\\begin{array} { r } { p _ { i } ^ { \\prime } \\in \\left\\{ \\left( j ^ { * } - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } , \\ldots , j ^ { * } \\cdot \\frac { k } { c \\cdot n ^ { 2 } } - 1 \\right\\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 525, + 279, + 542 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 280, + 530, + 291, + 540 + ], + "score": 0.83, + "content": "p _ { i ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 525, + 364, + 542 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 365, + 530, + 374, + 540 + ], + "score": 0.85, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 525, + 429, + 542 + ], + "score": 1.0, + "content": "and hence of", + "type": "text" + }, + { + "bbox": [ + 429, + 529, + 439, + 540 + ], + "score": 0.88, + "content": "j ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 525, + 505, + 542 + ], + "score": 1.0, + "content": ", the probability", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 542, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 219, + 558 + ], + "score": 1.0, + "content": "that this happens is at most", + "type": "text" + }, + { + "bbox": [ + 219, + 543, + 277, + 558 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { 1 } { k } \\cdot \\frac { k } { c \\cdot n ^ { 2 } } = \\frac { 1 } { c \\cdot n ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 542, + 409, + 559 + ], + "score": 1.0, + "content": ". This proves that for all distinct", + "type": "text" + }, + { + "bbox": [ + 410, + 545, + 424, + 556 + ], + "score": 0.91, + "content": "i , i ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 542, + 507, + 559 + ], + "score": 1.0, + "content": "the probability that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 219, + 550, + 228, + 565 + ], + "spans": [ + { + "bbox": [ + 219, + 550, + 228, + 565 + ], + "score": 1.0, + "content": "k1", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 556, + 380, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 173, + 570 + ], + "score": 0.93, + "content": "\\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 556, + 216, + 572 + ], + "score": 1.0, + "content": "is at most", + "type": "text" + }, + { + "bbox": [ + 216, + 558, + 233, + 571 + ], + "score": 0.9, + "content": "\\textstyle { \\frac { 1 } { c \\cdot n ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 556, + 380, + 572 + ], + "score": 1.0, + "content": ". Hence, again by the Union Bound,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 576, + 384, + 600 + ], + "lines": [ + { + "bbox": [ + 227, + 576, + 384, + 600 + ], + "spans": [ + { + "bbox": [ + 227, + 576, + 384, + 600 + ], + "score": 0.93, + "content": "\\operatorname* { P r } ( \\exists i \\neq i ^ { \\prime } \\forall j : \\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } ) ) \\leq \\frac { 1 } { c } .", + "type": "interline_equation", + "image_path": "dac60406468b443641503a34a679b80846b803faea6fccbc9b5b1bc8a71abf25.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 227, + 576, + 384, + 600 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 439, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 441, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 441, + 618 + ], + "score": 1.0, + "content": "(A.3) and (A.5) imply that the probability that either (i) or (ii) is violated is at most", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 621, + 344, + 648 + ], + "lines": [ + { + "bbox": [ + 267, + 621, + 344, + 648 + ], + "spans": [ + { + "bbox": [ + 267, + 621, + 344, + 648 + ], + "score": 0.94, + "content": "\\frac { c \\cdot n ^ { 3 } } { k } + \\frac { 1 } { c } \\leq \\frac { 2 } { c } \\leq \\delta .", + "type": "interline_equation", + "image_path": "69eac52e7ccaccfbf44b2a2ca33d0511400584990b8b4dbbfa3a5384a4218f2a.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 267, + 621, + 344, + 648 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 273, + 673 + ], + "score": 1.0, + "content": "Proof of Lemma A.1. For given function", + "type": "text" + }, + { + "bbox": [ + 274, + 659, + 343, + 671 + ], + "score": 0.95, + "content": "f : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 657, + 448, + 673 + ], + "score": 1.0, + "content": ", we choose the sentence", + "type": "text" + }, + { + "bbox": [ + 448, + 659, + 461, + 671 + ], + "score": 0.85, + "content": "\\psi _ { f } ^ { \\wedge }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 657, + 505, + 673 + ], + "score": 1.0, + "content": "according", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 354, + 683 + ], + "score": 1.0, + "content": "to Corollary A.1. Applying Lemma A.2 to this sentence and", + "type": "text" + }, + { + "bbox": [ + 354, + 673, + 360, + 681 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 671, + 451, + 683 + ], + "score": 1.0, + "content": ", we obtain an MPNN", + "type": "text" + }, + { + "bbox": [ + 451, + 671, + 465, + 683 + ], + "score": 0.84, + "content": "\\ddot { \\mathcal { N } } _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "that on a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 681, + 366, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 164, + 695 + ], + "score": 1.0, + "content": "colored graph", + "type": "text" + }, + { + "bbox": [ + 164, + 682, + 200, + 694 + ], + "score": 0.92, + "content": "G \\in \\mathcal { G } _ { n , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 681, + 254, + 695 + ], + "score": 1.0, + "content": "computes an", + "type": "text" + }, + { + "bbox": [ + 255, + 684, + 259, + 691 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 681, + 333, + 695 + ], + "score": 1.0, + "content": "-approximation of", + "type": "text" + }, + { + "bbox": [ + 333, + 682, + 362, + 694 + ], + "score": 0.92, + "content": "f ( G ^ { \\vee } )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 681, + 366, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Without loss of generality, we assume that the vertex set of the input graph to our MPNN is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 150, + 722 + ], + "score": 0.92, + "content": "\\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 710, + 204, + 721 + ], + "score": 1.0, + "content": ". We choose", + "type": "text" + }, + { + "bbox": [ + 204, + 710, + 210, + 720 + ], + "score": 0.73, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 710, + 438, + 721 + ], + "score": 1.0, + "content": "(the dimension of the state vectors) in such a way that", + "type": "text" + }, + { + "bbox": [ + 438, + 709, + 478, + 720 + ], + "score": 0.92, + "content": "\\ell \\geq c \\cdot n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 710, + 498, + 721 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 498, + 710, + 504, + 720 + ], + "score": 0.77, + "content": "\\ell", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 351, + 733 + ], + "score": 1.0, + "content": "is at least as large as the dimension of the state vectors of", + "type": "text" + }, + { + "bbox": [ + 351, + 721, + 365, + 733 + ], + "score": 0.9, + "content": "\\mathcal { N } _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 191, + 82, + 260, + 95 + ], + "score": 0.92, + "content": "f : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 80, + 487, + 95 + ], + "score": 1.0, + "content": "be an invariant Boolean function. Then there exists a", + "type": "text" + }, + { + "bbox": [ + 487, + 81, + 500, + 93 + ], + "score": 0.75, + "content": "\\mathsf { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 80, + 505, + 95 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 143, + 107 + ], + "score": 1.0, + "content": "sentence", + "type": "text" + }, + { + "bbox": [ + 144, + 94, + 156, + 107 + ], + "score": 0.89, + "content": "\\varphi _ { f } ^ { \\wedge }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 92, + 369, + 107 + ], + "score": 1.0, + "content": "(in the language of colored graphs) such that for all", + "type": "text" + }, + { + "bbox": [ + 369, + 94, + 406, + 106 + ], + "score": 0.92, + "content": "G \\in { \\mathcal { G } } _ { n , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 92, + 458, + 107 + ], + "score": 1.0, + "content": "it holds that", + "type": "text" + }, + { + "bbox": [ + 458, + 93, + 505, + 107 + ], + "score": 0.92, + "content": "\\mathbb { [ } \\psi _ { f } ^ { \\wedge } ] ( G ) =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 140, + 120 + ], + "spans": [ + { + "bbox": [ + 107, + 106, + 135, + 118 + ], + "score": 0.9, + "content": "f ( G ^ { \\vee } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 104, + 140, + 120 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 80, + 505, + 120 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 126, + 377, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 378, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 267, + 140 + ], + "score": 1.0, + "content": "Towards proving Lemma A.1, we fix an", + "type": "text" + }, + { + "bbox": [ + 268, + 128, + 290, + 137 + ], + "score": 0.9, + "content": "n \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 126, + 315, + 140 + ], + "score": 1.0, + "content": "and a", + "type": "text" + }, + { + "bbox": [ + 315, + 127, + 345, + 138 + ], + "score": 0.91, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 126, + 378, + 140 + ], + "score": 1.0, + "content": ". We let", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 126, + 378, + 140 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 143, + 363, + 169 + ], + "lines": [ + { + "bbox": [ + 248, + 143, + 363, + 169 + ], + "spans": [ + { + "bbox": [ + 248, + 143, + 363, + 169 + ], + "score": 0.93, + "content": "c : = \\left\\lceil { \\frac { 2 } { \\delta } } \\right\\rceil \\quad { \\mathrm { a n d } } \\quad k : = c ^ { 2 } \\cdot n ^ { 3 }", + "type": "interline_equation", + "image_path": "ba7e9ba578d79f745c90f449340b738ae19ffb6bc9a251db3c8484f2bf543f85.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 248, + 143, + 363, + 169 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "The technical details of the proof of Lemma A.1 and Theorem 4.1 depend on the exact choice of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "random initialization and the activation functions used in the neural networks, but the idea is always", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 365, + 209 + ], + "score": 1.0, + "content": "the same. 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Let", + "type": "text" + }, + { + "bbox": [ + 180, + 245, + 221, + 254 + ], + "score": 0.87, + "content": "r _ { 1 } , \\ldots , r _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 241, + 478, + 256 + ], + "score": 1.0, + "content": "be chosen independently uniformly at random from the interval", + "type": "text" + }, + { + "bbox": [ + 479, + 243, + 501, + 255 + ], + "score": 0.31, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 241, + 504, + 256 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 252, + 248, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 123, + 267 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 254, + 159, + 264 + ], + "score": 0.9, + "content": "1 \\leq i \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 252, + 178, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 254, + 230, + 265 + ], + "score": 0.91, + "content": "1 \\leq j \\leq c \\cdot n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 252, + 248, + 267 + ], + "score": 1.0, + "content": ", let", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 241, + 504, + 267 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 270, + 363, + 295 + ], + "lines": [ + { + "bbox": [ + 247, + 270, + 363, + 295 + ], + "spans": [ + { + "bbox": [ + 247, + 270, + 363, + 295 + ], + "score": 0.93, + "content": "s _ { i j } : = k \\cdot r _ { i } - \\left( j - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } .", + "type": "interline_equation", + "image_path": "0258962bac425b6802425ddc9855ec6eef3641e804a5d99f7b6570262f6d3558.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 247, + 270, + 363, + 295 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 299, + 422, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 423, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 246, + 313 + ], + "score": 1.0, + "content": "Then with probability greater than", + "type": "text" + }, + { + "bbox": [ + 247, + 300, + 267, + 310 + ], + "score": 0.61, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 298, + 423, + 313 + ], + "score": 1.0, + "content": ", the following conditions are satisfied.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 298, + 423, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 124, + 339, + 503, + 353 + ], + "lines": [ + { + "bbox": [ + 123, + 337, + 501, + 356 + ], + "spans": [ + { + "bbox": [ + 123, + 337, + 203, + 356 + ], + "score": 1.0, + "content": "(ii) For all distinct", + "type": "text" + }, + { + "bbox": [ + 204, + 340, + 269, + 353 + ], + "score": 0.93, + "content": "i , i ^ { \\prime } \\in \\{ 1 , . . . , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 337, + 325, + 356 + ], + "score": 1.0, + "content": "there exists a", + "type": "text" + }, + { + "bbox": [ + 325, + 340, + 395, + 353 + ], + "score": 0.93, + "content": "j \\in \\left\\{ 1 , \\dots , c \\cdot n ^ { 2 } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 337, + 434, + 356 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 434, + 340, + 501, + 353 + ], + "score": 0.92, + "content": "\\sigma ( s _ { i j } ) \\neq \\sigma ( s _ { i ^ { \\prime } j } )", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 123, + 337, + 501, + 356 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 177, + 378 + ], + "score": 1.0, + "content": "Proof. 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Since", + "type": "text" + }, + { + "bbox": [ + 282, + 366, + 304, + 376 + ], + "score": 0.88, + "content": "k \\cdot r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 364, + 462, + 378 + ], + "score": 1.0, + "content": "is uniformly random from the interval", + "type": "text" + }, + { + "bbox": [ + 463, + 365, + 486, + 377 + ], + "score": 0.91, + "content": "[ 0 , k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 364, + 505, + 378 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 136, + 389 + ], + "score": 1.0, + "content": "integer", + "type": "text" + }, + { + "bbox": [ + 137, + 378, + 146, + 387 + ], + "score": 0.84, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 375, + 257, + 389 + ], + "score": 1.0, + "content": "is uniformly random from", + "type": "text" + }, + { + "bbox": [ + 257, + 376, + 314, + 388 + ], + "score": 0.92, + "content": "\\{ 0 , \\ldots , k - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 375, + 374, + 389 + ], + "score": 1.0, + "content": ". Observe that", + "type": "text" + }, + { + "bbox": [ + 374, + 376, + 435, + 388 + ], + "score": 0.93, + "content": "0 < \\sigma ( s _ { i j } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 375, + 466, + 389 + ], + "score": 1.0, + "content": "only if", + "type": "text" + }, + { + "bbox": [ + 466, + 376, + 505, + 388 + ], + "score": 0.9, + "content": "p _ { i } - ( j -", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 383, + 501, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 154, + 401 + ], + "score": 0.91, + "content": "\\textstyle 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 383, + 257, + 407 + ], + "score": 1.0, + "content": "(here we use the fact that", + "type": "text" + }, + { + "bbox": [ + 257, + 389, + 263, + 398 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 383, + 322, + 407 + ], + "score": 1.0, + "content": "is divisible by", + "type": "text" + }, + { + "bbox": [ + 322, + 388, + 344, + 399 + ], + "score": 0.87, + "content": "c \\cdot n ^ { 2 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 383, + 493, + 407 + ], + "score": 1.0, + "content": "). The probability that this happens is", + "type": "text" + }, + { + "bbox": [ + 494, + 388, + 501, + 402 + ], + "score": 0.86, + "content": "\\frac { 1 } { k }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 399, + 218, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 218, + 412 + ], + "score": 1.0, + "content": "⋅Thus, by the Union Bound,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 364, + 505, + 412 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 416, + 379, + 442 + ], + "lines": [ + { + "bbox": [ + 233, + 416, + 379, + 442 + ], + "spans": [ + { + "bbox": [ + 233, + 416, + 379, + 442 + ], + "score": 0.93, + "content": "\\operatorname* { P r } \\left( \\exists i , j : 0 < \\sigma ( s _ { i j } ) < 1 \\right) \\leq \\frac { c \\cdot n ^ { 3 } } { k } .", + "type": "interline_equation", + "image_path": "bd4afacfa46bc02b6c3ee163416d9fae134482461bb962ca484f5fed1aa068ed.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 233, + 416, + 379, + 442 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 447, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 142, + 462 + ], + "score": 1.0, + "content": "Now let", + "type": "text" + }, + { + "bbox": [ + 142, + 447, + 157, + 459 + ], + "score": 0.89, + "content": "i , i ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 445, + 275, + 462 + ], + "score": 1.0, + "content": "be distinct and suppose that", + "type": "text" + }, + { + "bbox": [ + 275, + 447, + 344, + 460 + ], + "score": 0.92, + "content": "\\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 445, + 374, + 462 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 374, + 448, + 380, + 459 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 445, + 437, + 462 + ], + "score": 1.0, + "content": ". Then for all", + "type": "text" + }, + { + "bbox": [ + 438, + 448, + 444, + 459 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 445, + 482, + 462 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 482, + 449, + 505, + 460 + ], + "score": 0.88, + "content": "s _ { i j } \\leq", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 458, + 384, + 471 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 170, + 471 + ], + "score": 0.91, + "content": "0 \\iff s _ { i ^ { \\prime } j } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 458, + 226, + 471 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 227, + 459, + 327, + 471 + ], + "score": 0.92, + "content": "\\lfloor s _ { i j } \\rfloor \\le 0 \\iff \\lfloor \\stackrel { \\sim } { s } _ { i ^ { \\prime } j } \\rfloor \\le 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 458, + 384, + 471 + ], + "score": 1.0, + "content": ". This implies", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 445, + 505, + 471 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 475, + 447, + 500 + ], + "lines": [ + { + "bbox": [ + 164, + 475, + 447, + 500 + ], + "spans": [ + { + "bbox": [ + 164, + 475, + 447, + 500 + ], + "score": 0.94, + "content": "\\forall j \\in \\{ 1 , \\ldots , c \\cdot n ^ { 2 } \\} : \\quad p _ { i } \\leq ( j - 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } \\Longleftrightarrow p _ { i ^ { \\prime } } \\leq ( j - 1 ) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } .", + "type": "interline_equation", + "image_path": "cb2e1e4fdb6329bb8691bc30d59a92c0ea0650ad0c4d973f46995b676839b4f6.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 164, + 475, + 447, + 500 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "list", + "bbox": [ + 106, + 506, + 506, + 571 + ], + "lines": [ + { + "bbox": [ + 101, + 500, + 509, + 529 + ], + "spans": [ + { + "bbox": [ + 101, + 500, + 123, + 529 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 509, + 203, + 522 + ], + "score": 0.92, + "content": "j ^ { * } \\in \\{ 1 , \\ldots , c \\cdot n ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 500, + 244, + 529 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 244, + 506, + 402, + 525 + ], + "score": 0.93, + "content": "p _ { i } \\in \\left\\{ \\left( j ^ { * } - 1 \\right) \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } , \\ldots , j ^ { * } \\cdot { \\frac { k } { c \\cdot n ^ { 2 } } } - 1 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 500, + 509, + 529 + ], + "score": 1.0, + "content": ". Then by (A.4) we have", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 525, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 262, + 544 + ], + "score": 0.91, + "content": "\\begin{array} { r } { p _ { i } ^ { \\prime } \\in \\left\\{ \\left( j ^ { * } - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } , \\ldots , j ^ { * } \\cdot \\frac { k } { c \\cdot n ^ { 2 } } - 1 \\right\\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 525, + 279, + 542 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 280, + 530, + 291, + 540 + ], + "score": 0.83, + "content": "p _ { i ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 525, + 364, + 542 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 365, + 530, + 374, + 540 + ], + "score": 0.85, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 525, + 429, + 542 + ], + "score": 1.0, + "content": "and hence of", + "type": "text" + }, + { + "bbox": [ + 429, + 529, + 439, + 540 + ], + "score": 0.88, + "content": "j ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 525, + 505, + 542 + ], + "score": 1.0, + "content": ", the probability", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 542, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 219, + 558 + ], + "score": 1.0, + "content": "that this happens is at most", + "type": "text" + }, + { + "bbox": [ + 219, + 543, + 277, + 558 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { 1 } { k } \\cdot \\frac { k } { c \\cdot n ^ { 2 } } = \\frac { 1 } { c \\cdot n ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 542, + 409, + 559 + ], + "score": 1.0, + "content": ". This proves that for all distinct", + "type": "text" + }, + { + "bbox": [ + 410, + 545, + 424, + 556 + ], + "score": 0.91, + "content": "i , i ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 542, + 507, + 559 + ], + "score": 1.0, + "content": "the probability that", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 219, + 550, + 228, + 565 + ], + "spans": [ + { + "bbox": [ + 219, + 550, + 228, + 565 + ], + "score": 1.0, + "content": "k1", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 107, + 556, + 380, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 173, + 570 + ], + "score": 0.93, + "content": "\\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 556, + 216, + 572 + ], + "score": 1.0, + "content": "is at most", + "type": "text" + }, + { + "bbox": [ + 216, + 558, + 233, + 571 + ], + "score": 0.9, + "content": "\\textstyle { \\frac { 1 } { c \\cdot n ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 556, + 380, + 572 + ], + "score": 1.0, + "content": ". Hence, again by the Union Bound,", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 26, + "bbox_fs": [ + 101, + 500, + 509, + 572 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 576, + 384, + 600 + ], + "lines": [ + { + "bbox": [ + 227, + 576, + 384, + 600 + ], + "spans": [ + { + "bbox": [ + 227, + 576, + 384, + 600 + ], + "score": 0.93, + "content": "\\operatorname* { P r } ( \\exists i \\neq i ^ { \\prime } \\forall j : \\sigma ( s _ { i j } ) = \\sigma ( s _ { i ^ { \\prime } j } ) ) \\leq \\frac { 1 } { c } .", + "type": "interline_equation", + "image_path": "dac60406468b443641503a34a679b80846b803faea6fccbc9b5b1bc8a71abf25.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 227, + 576, + 384, + 600 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 439, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 441, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 441, + 618 + ], + "score": 1.0, + "content": "(A.3) and (A.5) imply that the probability that either (i) or (ii) is violated is at most", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 604, + 441, + 618 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 621, + 344, + 648 + ], + "lines": [ + { + "bbox": [ + 267, + 621, + 344, + 648 + ], + "spans": [ + { + "bbox": [ + 267, + 621, + 344, + 648 + ], + "score": 0.94, + "content": "\\frac { c \\cdot n ^ { 3 } } { k } + \\frac { 1 } { c } \\leq \\frac { 2 } { c } \\leq \\delta .", + "type": "interline_equation", + "image_path": "69eac52e7ccaccfbf44b2a2ca33d0511400584990b8b4dbbfa3a5384a4218f2a.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 267, + 621, + 344, + 648 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 273, + 673 + ], + "score": 1.0, + "content": "Proof of Lemma A.1. For given function", + "type": "text" + }, + { + "bbox": [ + 274, + 659, + 343, + 671 + ], + "score": 0.95, + "content": "f : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 657, + 448, + 673 + ], + "score": 1.0, + "content": ", we choose the sentence", + "type": "text" + }, + { + "bbox": [ + 448, + 659, + 461, + 671 + ], + "score": 0.85, + "content": "\\psi _ { f } ^ { \\wedge }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 657, + 505, + 673 + ], + "score": 1.0, + "content": "according", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 354, + 683 + ], + "score": 1.0, + "content": "to Corollary A.1. Applying Lemma A.2 to this sentence and", + "type": "text" + }, + { + "bbox": [ + 354, + 673, + 360, + 681 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 671, + 451, + 683 + ], + "score": 1.0, + "content": ", we obtain an MPNN", + "type": "text" + }, + { + "bbox": [ + 451, + 671, + 465, + 683 + ], + "score": 0.84, + "content": "\\ddot { \\mathcal { N } } _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "that on a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 681, + 366, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 164, + 695 + ], + "score": 1.0, + "content": "colored graph", + "type": "text" + }, + { + "bbox": [ + 164, + 682, + 200, + 694 + ], + "score": 0.92, + "content": "G \\in \\mathcal { G } _ { n , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 681, + 254, + 695 + ], + "score": 1.0, + "content": "computes an", + "type": "text" + }, + { + "bbox": [ + 255, + 684, + 259, + 691 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 681, + 333, + 695 + ], + "score": 1.0, + "content": "-approximation of", + "type": "text" + }, + { + "bbox": [ + 333, + 682, + 362, + 694 + ], + "score": 0.92, + "content": "f ( G ^ { \\vee } )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 681, + 366, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 657, + 505, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Without loss of generality, we assume that the vertex set of the input graph to our MPNN is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 150, + 722 + ], + "score": 0.92, + "content": "\\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 710, + 204, + 721 + ], + "score": 1.0, + "content": ". We choose", + "type": "text" + }, + { + "bbox": [ + 204, + 710, + 210, + 720 + ], + "score": 0.73, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 710, + 438, + 721 + ], + "score": 1.0, + "content": "(the dimension of the state vectors) in such a way that", + "type": "text" + }, + { + "bbox": [ + 438, + 709, + 478, + 720 + ], + "score": 0.92, + "content": "\\ell \\geq c \\cdot n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 710, + 498, + 721 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 498, + 710, + 504, + 720 + ], + "score": 0.77, + "content": "\\ell", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 351, + 733 + ], + "score": 1.0, + "content": "is at least as large as the dimension of the state vectors of", + "type": "text" + }, + { + "bbox": [ + 351, + 721, + 365, + 733 + ], + "score": 0.9, + "content": "\\mathcal { N } _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 216, + 292, + 267, + 303 + ], + "score": 0.91, + "content": "f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 290, + 350, + 305 + ], + "score": 1.0, + "content": "be invariant. 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By Lemma A.1, for every", + "type": "text" + }, + { + "bbox": [ + 371, + 390, + 430, + 403 + ], + "score": 0.94, + "content": "i \\in \\{ 1 , \\ldots , N \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 385, + 509, + 409 + ], + "score": 1.0, + "content": "there is an MPNN", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 126, + 416 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 403, + 159, + 414 + ], + "score": 0.27, + "content": "\\boldsymbol { \\mathrm { R N I } } \\mathcal { N } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 402, + 177, + 416 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 178, + 403, + 203, + 415 + ], + "score": 0.93, + "content": "( \\epsilon ^ { \\prime } , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 402, + 261, + 416 + ], + "score": 1.0, + "content": "-approximates", + "type": "text" + }, + { + "bbox": [ + 261, + 405, + 270, + 414 + ], + "score": 0.84, + "content": "g _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 402, + 333, + 416 + ], + "score": 1.0, + "content": ". Putting all the", + "type": "text" + }, + { + "bbox": [ + 333, + 403, + 345, + 414 + ], + "score": 0.88, + "content": "{ \\mathcal { N } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "together, we obtain an invariant MPNN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 414, + 486, + 428 + ], + "spans": [ + { + "bbox": [ + 107, + 415, + 117, + 425 + ], + "score": 0.79, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 414, + 219, + 428 + ], + "score": 1.0, + "content": "that computes a function", + "type": "text" + }, + { + "bbox": [ + 220, + 414, + 289, + 427 + ], + "score": 0.95, + "content": "g : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\} ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 414, + 486, + 428 + ], + "score": 1.0, + "content": ". We only need to apply the linear transformation", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 432, + 334, + 462 + ], + "lines": [ + { + "bbox": [ + 276, + 432, + 334, + 462 + ], + "spans": [ + { + "bbox": [ + 276, + 432, + 334, + 462 + ], + "score": 0.94, + "content": "\\pmb { x } \\mapsto \\sum _ { i = 1 } ^ { N } x _ { i } \\cdot y _ { i }", + "type": "interline_equation", + "image_path": "d3d99d70fdddcbd9c1bdb870b842520c2da7540f0984497e7388c45560c7696b.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 276, + 432, + 334, + 447.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 276, + 447.0, + 334, + 462.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 347, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 348, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 170, + 481 + ], + "score": 1.0, + "content": "to the output of", + "type": "text" + }, + { + "bbox": [ + 171, + 467, + 181, + 477 + ], + "score": 0.84, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 465, + 337, + 481 + ], + "score": 1.0, + "content": "to obtain the desired approximation of", + "type": "text" + }, + { + "bbox": [ + 338, + 468, + 344, + 479 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 465, + 348, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "Remark 1. Obviously, our construction yields MPNNs with a prohibitively large state space. In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "particular, this is the case for the brute force step from Boolean to general functions. We doubt that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "there are much more efficient approximators, after all we make no assumption whatsoever on the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 517, + 155, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 142, + 531 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 519, + 149, + 530 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 517, + 155, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "The approximation of Boolean functions is more interesting. It may still happen that the GNNs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 211, + 559 + ], + "score": 1.0, + "content": "get exponentially large in", + "type": "text" + }, + { + "bbox": [ + 212, + 548, + 219, + 556 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "; this seems unavoidable. However, the nice thing here is that our con-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "struction is very adaptive and tightly linked to the descriptive complexity of the function we want to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 569, + 473, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 473, + 581 + ], + "score": 1.0, + "content": "approximate. This deserves a more thorough investigation, which we leave for future work.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "As opposed to other universality results for GNNs, our construction needs no higher-order tensors", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "defined on tuples of nodes, with practically infeasible space requirements on all but very small", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "graphs. Instead, the complexity of our construction goes entirely into the dimension of the state", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "space. The advantage of this is that we can treat this dimension as a hyperparameter that we can eas-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "ily adapt and that gives us more fine-grained control over the space requirements. Our experiments", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 640, + 455, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 455, + 652 + ], + "score": 1.0, + "content": "show that usually in practice a small dimension already yields very powerful networks.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "Remark 2. In our experiments, we found that a partial random initialization, which only assigns", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "random values to a fraction of all node embedding vectors, often yields very good results, sometimes", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 504, + 690 + ], + "score": 1.0, + "content": "better than a full random initialization. There is plausibility to this from a theoretical perspective.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "For most graphs, we do not lose much by only initializing a small fraction of vertex embeddings,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "because in a few message-passing rounds GNNs can propagate the randomness and, referring our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "construction above, individualize the full input graph. On the other hand, we reduce the amount of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 378, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 378, + 733 + ], + "score": 1.0, + "content": "noise our models have to handle when we only randomize partially.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 307, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 467, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 496, + 469, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 496, + 469, + 504, + 477 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 80, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 78, + 506, + 99 + ], + "spans": [ + { + "bbox": [ + 104, + 78, + 174, + 99 + ], + "score": 1.0, + "content": "initialized as x(0i", + "type": "text" + }, + { + "bbox": [ + 160, + 80, + 241, + 95 + ], + "score": 0.92, + "content": "\\pmb { x } _ { i } ^ { ( 0 ) } = ( r _ { i } , 0 , \\ldots , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 79, + 284, + 98 + ], + "score": 1.0, + "content": "for values", + "type": "text" + }, + { + "bbox": [ + 284, + 84, + 293, + 94 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 79, + 506, + 98 + ], + "score": 1.0, + "content": "chosen independently uniformly at random from the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 167, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 139, + 108 + ], + "score": 1.0, + "content": "interval", + "type": "text" + }, + { + "bbox": [ + 140, + 94, + 162, + 106 + ], + "score": 0.74, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 93, + 167, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 78, + 506, + 108 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 110, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 101, + 110, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 101, + 118, + 206, + 145 + ], + "score": 1.0, + "content": "passed) that maps x(0)i t", + "type": "text" + }, + { + "bbox": [ + 193, + 118, + 238, + 144 + ], + "score": 1.0, + "content": "o x( 1 )i =", + "type": "text" + }, + { + "bbox": [ + 211, + 122, + 303, + 137 + ], + "score": 0.93, + "content": "\\pmb { x } _ { i } ^ { ( 1 ) } = \\bar { ( x _ { i 1 } ^ { ( 1 ) } , \\dots , x _ { i \\ell } ^ { ( 1 ) } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 110, + 505, + 144 + ], + "score": 1.0, + "content": "ly local transformation (no messages need to be with", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 101, + 110, + 505, + 145 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 142, + 417, + 176 + ], + "lines": [ + { + "bbox": [ + 192, + 142, + 417, + 176 + ], + "spans": [ + { + "bbox": [ + 192, + 142, + 417, + 176 + ], + "score": 0.92, + "content": "\\begin{array} { r } { x _ { i j } ^ { ( 1 ) } = \\left\\{ \\begin{array} { l l } { \\sigma \\Big ( \\boldsymbol { k } \\cdot \\boldsymbol { r _ { i } } - \\left( j - 1 \\right) \\cdot \\frac { \\boldsymbol { k } } { c \\cdot n ^ { 2 } } \\Big ) } & { \\mathrm { f o r ~ } 1 \\leq j \\leq c \\cdot n ^ { 2 } , } \\\\ { 0 } & { \\mathrm { f o r ~ } c \\cdot n ^ { 2 } + 1 \\leq j \\leq \\ell . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "f491397e8a04115d9493e751e43fe8d3424c020ba3e5659d872ec83ef1a90ff4.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 192, + 142, + 417, + 159.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 192, + 159.0, + 417, + 176.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 502, + 211 + ], + "lines": [ + { + "bbox": [ + 102, + 177, + 507, + 202 + ], + "spans": [ + { + "bbox": [ + 102, + 177, + 166, + 202 + ], + "score": 1.0, + "content": "Since we treat", + "type": "text" + }, + { + "bbox": [ + 167, + 184, + 192, + 195 + ], + "score": 0.91, + "content": "k , c , n", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 177, + 300, + 202 + ], + "score": 1.0, + "content": "as constants, the mapping", + "type": "text" + }, + { + "bbox": [ + 300, + 182, + 406, + 197 + ], + "score": 0.92, + "content": "\\begin{array} { r } { r _ { i } \\mapsto k \\cdot r _ { i } - \\left( j - 1 \\right) \\cdot \\frac { k } { c \\cdot n ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 177, + 507, + 202 + ], + "score": 1.0, + "content": "is just a linear mapping", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 194, + 190, + 216 + ], + "spans": [ + { + "bbox": [ + 104, + 194, + 148, + 216 + ], + "score": 1.0, + "content": "applied to", + "type": "text" + }, + { + "bbox": [ + 149, + 196, + 184, + 212 + ], + "score": 0.92, + "content": "r _ { i } = x _ { i 1 } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 194, + 190, + 216 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 102, + 177, + 507, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 103, + 216, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 103, + 217, + 270, + 234 + ], + "score": 1.0, + "content": "By Lemma A.5, with probability at least", + "type": "text" + }, + { + "bbox": [ + 270, + 219, + 290, + 229 + ], + "score": 0.78, + "content": "1 - 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Let", + "type": "text" + }, + { + "bbox": [ + 478, + 230, + 491, + 240 + ], + "score": 0.89, + "content": "G ^ { \\wedge }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 230, + 506, + 242 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 241, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 294, + 257 + ], + "score": 1.0, + "content": "the resulting colored graph. Since the vectors", + "type": "text" + }, + { + "bbox": [ + 294, + 241, + 313, + 255 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { x } } _ { i } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 242, + 403, + 256 + ], + "score": 1.0, + "content": "are mutually distinct,", + "type": "text" + }, + { + "bbox": [ + 403, + 243, + 417, + 254 + ], + "score": 0.88, + "content": "G ^ { \\wedge }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 242, + 506, + 256 + ], + "score": 1.0, + "content": "is individualized and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 253, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 171, + 268 + ], + "score": 1.0, + "content": "thus in the class", + "type": "text" + }, + { + "bbox": [ + 171, + 255, + 191, + 267 + ], + "score": 0.91, + "content": "\\mathcal { G } _ { n , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 253, + 299, + 268 + ], + "score": 1.0, + "content": ". We now apply the MPNN", + "type": "text" + }, + { + "bbox": [ + 300, + 255, + 314, + 267 + ], + "score": 0.79, + "content": "\\mathcal { N } _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 253, + 412, + 268 + ], + "score": 1.0, + "content": ", and it computes a value", + "type": "text" + }, + { + "bbox": [ + 413, + 257, + 418, + 264 + ], + "score": 0.75, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 253, + 453, + 268 + ], + "score": 1.0, + "content": "-close to", + "type": "text" + }, + { + "bbox": [ + 453, + 254, + 505, + 268 + ], + "score": 0.91, + "content": "\\mathbb { } [ \\psi _ { f } ^ { \\wedge } ] ( G ^ { \\wedge } ) =", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 267, + 182, + 279 + ], + "score": 0.92, + "content": "f ( ( G ^ { \\wedge } ) ^ { \\vee } ) = f ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 267, + 186, + 280 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 268, + 505, + 277 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 103, + 216, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 504, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 215, + 305 + ], + "score": 1.0, + "content": "Proof of Theorem 4.1. Let", + "type": "text" + }, + { + "bbox": [ + 216, + 292, + 267, + 303 + ], + "score": 0.91, + "content": "f : { \\mathcal { G } } _ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 290, + 350, + 305 + ], + "score": 1.0, + "content": "be invariant. Since", + "type": "text" + }, + { + "bbox": [ + 351, + 292, + 363, + 303 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 290, + 443, + 305 + ], + "score": 1.0, + "content": "is finite, the range", + "type": "text" + }, + { + "bbox": [ + 443, + 291, + 493, + 304 + ], + "score": 0.93, + "content": "Y : = f ( { \\mathcal { G } } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 290, + 506, + 305 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 303, + 318, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 225, + 317 + ], + "score": 1.0, + "content": "finite. To be precise, we have", + "type": "text" + }, + { + "bbox": [ + 226, + 303, + 314, + 317 + ], + "score": 0.92, + "content": "N : = | Y | \\leq | \\mathcal { G } _ { n } | = 2 ^ { { \\binom { n } { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 303, + 318, + 317 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 290, + 506, + 317 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 321, + 494, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 495, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 126, + 336 + ], + "score": 1.0, + "content": "Say,", + "type": "text" + }, + { + "bbox": [ + 126, + 321, + 198, + 334 + ], + "score": 0.94, + "content": "Y = \\{ y _ { 1 } , \\dots , y _ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 320, + 219, + 336 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 219, + 322, + 269, + 333 + ], + "score": 0.92, + "content": "i = 1 , \\ldots , N", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 320, + 285, + 336 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 286, + 322, + 350, + 334 + ], + "score": 0.93, + "content": "g _ { i } : { \\mathcal { G } } _ { n } \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 320, + 495, + 336 + ], + "score": 1.0, + "content": "be the Boolean function defined by", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 320, + 495, + 336 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 339, + 360, + 368 + ], + "lines": [ + { + "bbox": [ + 249, + 339, + 360, + 368 + ], + "spans": [ + { + "bbox": [ + 249, + 339, + 360, + 368 + ], + "score": 0.94, + "content": "g _ { i } ( G ) = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } f ( G ) = y _ { i } , } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "f8d35281f86440cd78ab828e31e4926334bf5f34e161f63493791264032c0240.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 249, + 339, + 360, + 353.5 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 249, + 353.5, + 360, + 368.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 204, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 373, + 205, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 145, + 386 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 376, + 155, + 385 + ], + "score": 0.84, + "content": "g _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 373, + 205, + 386 + ], + "score": 1.0, + "content": "is invariant.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 373, + 205, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 427 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 509, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 122, + 402 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 391, + 152, + 402 + ], + "score": 0.9, + "content": "\\epsilon , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 390, + 169, + 402 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 390, + 214, + 403 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\epsilon ^ { \\prime } : = \\frac { \\epsilon } { \\operatorname* { m a x } Y } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 385, + 232, + 409 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 389, + 263, + 404 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta ^ { \\prime } : = \\frac { \\delta } { N } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 385, + 371, + 409 + ], + "score": 1.0, + "content": ". By Lemma A.1, for every", + "type": "text" + }, + { + "bbox": [ + 371, + 390, + 430, + 403 + ], + "score": 0.94, + "content": "i \\in \\{ 1 , \\ldots , N \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 385, + 509, + 409 + ], + "score": 1.0, + "content": "there is an MPNN", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 126, + 416 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 403, + 159, + 414 + ], + "score": 0.27, + "content": "\\boldsymbol { \\mathrm { R N I } } \\mathcal { N } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 402, + 177, + 416 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 178, + 403, + 203, + 415 + ], + "score": 0.93, + "content": "( \\epsilon ^ { \\prime } , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 402, + 261, + 416 + ], + "score": 1.0, + "content": "-approximates", + "type": "text" + }, + { + "bbox": [ + 261, + 405, + 270, + 414 + ], + "score": 0.84, + "content": "g _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 402, + 333, + 416 + ], + "score": 1.0, + "content": ". Putting all the", + "type": "text" + }, + { + "bbox": [ + 333, + 403, + 345, + 414 + ], + "score": 0.88, + "content": "{ \\mathcal { N } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "together, we obtain an invariant MPNN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 414, + 486, + 428 + ], + "spans": [ + { + "bbox": [ + 107, + 415, + 117, + 425 + ], + "score": 0.79, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 414, + 219, + 428 + ], + "score": 1.0, + "content": "that computes a function", + "type": "text" + }, + { + "bbox": [ + 220, + 414, + 289, + 427 + ], + "score": 0.95, + "content": "g : { \\mathcal { G } } _ { n } \\to \\{ 0 , 1 \\} ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 414, + 486, + 428 + ], + "score": 1.0, + "content": ". We only need to apply the linear transformation", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 385, + 509, + 428 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 432, + 334, + 462 + ], + "lines": [ + { + "bbox": [ + 276, + 432, + 334, + 462 + ], + "spans": [ + { + "bbox": [ + 276, + 432, + 334, + 462 + ], + "score": 0.94, + "content": "\\pmb { x } \\mapsto \\sum _ { i = 1 } ^ { N } x _ { i } \\cdot y _ { i }", + "type": "interline_equation", + "image_path": "d3d99d70fdddcbd9c1bdb870b842520c2da7540f0984497e7388c45560c7696b.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 276, + 432, + 334, + 447.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 276, + 447.0, + 334, + 462.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 347, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 348, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 170, + 481 + ], + "score": 1.0, + "content": "to the output of", + "type": "text" + }, + { + "bbox": [ + 171, + 467, + 181, + 477 + ], + "score": 0.84, + "content": "\\mathcal { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 465, + 337, + 481 + ], + "score": 1.0, + "content": "to obtain the desired approximation of", + "type": "text" + }, + { + "bbox": [ + 338, + 468, + 344, + 479 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 465, + 348, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 465, + 348, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "Remark 1. Obviously, our construction yields MPNNs with a prohibitively large state space. In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "particular, this is the case for the brute force step from Boolean to general functions. We doubt that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "there are much more efficient approximators, after all we make no assumption whatsoever on the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 517, + 155, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 142, + 531 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 519, + 149, + 530 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 517, + 155, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 485, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "The approximation of Boolean functions is more interesting. It may still happen that the GNNs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 211, + 559 + ], + "score": 1.0, + "content": "get exponentially large in", + "type": "text" + }, + { + "bbox": [ + 212, + 548, + 219, + 556 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "; this seems unavoidable. However, the nice thing here is that our con-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "struction is very adaptive and tightly linked to the descriptive complexity of the function we want to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 569, + 473, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 473, + 581 + ], + "score": 1.0, + "content": "approximate. This deserves a more thorough investigation, which we leave for future work.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 535, + 506, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "As opposed to other universality results for GNNs, our construction needs no higher-order tensors", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "defined on tuples of nodes, with practically infeasible space requirements on all but very small", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "graphs. Instead, the complexity of our construction goes entirely into the dimension of the state", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "space. The advantage of this is that we can treat this dimension as a hyperparameter that we can eas-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "ily adapt and that gives us more fine-grained control over the space requirements. Our experiments", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 640, + 455, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 455, + 652 + ], + "score": 1.0, + "content": "show that usually in practice a small dimension already yields very powerful networks.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 585, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "Remark 2. In our experiments, we found that a partial random initialization, which only assigns", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "random values to a fraction of all node embedding vectors, often yields very good results, sometimes", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 504, + 690 + ], + "score": 1.0, + "content": "better than a full random initialization. There is plausibility to this from a theoretical perspective.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "For most graphs, we do not lose much by only initializing a small fraction of vertex embeddings,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "because in a few message-passing rounds GNNs can propagate the randomness and, referring our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "construction above, individualize the full input graph. 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This, however, does not imply that these", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "GNNs will solve EXP in practice, but only that an appropriate approximation function exists and can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 608, + 204, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 204, + 620 + ], + "score": 1.0, + "content": "theoretically be learned.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 564, + 506, + 620 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 641, + 245, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 246, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 246, + 653 + ], + "score": 1.0, + "content": "A.3.1 CONSTRUCTION OF EXP", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 663, + 505, + 730 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 675 + ], + "score": 1.0, + "content": "We now explain the construction and composition of EXP. Fundamentally, EXP consists of two", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "main components, (i) a pair of cores, which are non-isomorphic, planar, 1-WL indistinguishable,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "2-WL distinguishable, and decide the satisfiability of every instance, and (ii) an additional randomly", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 696, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 505, + 709 + ], + "score": 1.0, + "content": "generated and satisfiable planar component, identically added to the core pair, to add variability", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 708, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 720 + ], + "score": 1.0, + "content": "to EXP and make learning more challenging. We first present both components, and then provide", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 717, + 351, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 351, + 732 + ], + "score": 1.0, + "content": "further details about graph encoding and planar embeddings.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 664, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 362, + 95 + ], + "score": 1.0, + "content": "Core pair. In EXP, a core pair consists of two CNF formulas", + "type": "text" + }, + { + "bbox": [ + 362, + 84, + 389, + 94 + ], + "score": 0.9, + "content": "\\varphi _ { 1 } , \\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 82, + 470, + 95 + ], + "score": 1.0, + "content": ", both defined using", + "type": "text" + }, + { + "bbox": [ + 470, + 83, + 483, + 93 + ], + "score": 0.77, + "content": "2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "vari-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 132, + 107 + ], + "score": 1.0, + "content": "ables,", + "type": "text" + }, + { + "bbox": [ + 132, + 93, + 160, + 104 + ], + "score": 0.91, + "content": "n \\in \\mathbb { N } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 92, + 202, + 107 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 203, + 95, + 214, + 105 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 92, + 294, + 107 + ], + "score": 1.0, + "content": "is unsatisfiable and", + "type": "text" + }, + { + "bbox": [ + 294, + 95, + 306, + 105 + ], + "score": 0.87, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "is satisfiable, and such that their graph encodings", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 268, + 117 + ], + "score": 1.0, + "content": "are 1-WL indistinguishable and planar.", + "type": "text" + }, + { + "bbox": [ + 269, + 106, + 281, + 116 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 105, + 300, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 301, + 106, + 313, + 116 + ], + "score": 0.86, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "are constructed using two structures which we", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 349, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 349, + 128 + ], + "score": 1.0, + "content": "refer to as variable chains and variable bridges respectively.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 180, + 145 + ], + "score": 1.0, + "content": "A variable chain", + "type": "text" + }, + { + "bbox": [ + 180, + 134, + 209, + 144 + ], + "score": 0.9, + "content": "\\varphi _ { c h a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 131, + 311, + 145 + ], + "score": 1.0, + "content": "is defined over a set of", + "type": "text" + }, + { + "bbox": [ + 311, + 133, + 338, + 143 + ], + "score": 0.89, + "content": "n \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "Boolean variables, and imposes that all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "variables be equally set. The variable chain can be defined in increasing or decreasing order over", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 349, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 307, + 168 + ], + "score": 1.0, + "content": "these variables. More specifically, given variables", + "type": "text" + }, + { + "bbox": [ + 307, + 156, + 343, + 167 + ], + "score": 0.91, + "content": "x _ { i } , . . . , x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 153, + 349, + 168 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 173, + 406, + 233 + ], + "lines": [ + { + "bbox": [ + 206, + 173, + 406, + 233 + ], + "spans": [ + { + "bbox": [ + 206, + 173, + 406, + 233 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\mathrm { C h a i n } _ { \\mathrm { I n c } } ( i , j ) = \\displaystyle \\bigwedge _ { k = i } ^ { j - 1 } ( \\bar { x _ { k } } \\vee x _ { i + ( k + 1 ) } \\% ( j - i + 1 ) ) , \\mathrm { ~ a n d } } \\\\ & { \\mathrm { C h a i n } _ { \\mathrm { I n e c } } ( i , j ) = \\displaystyle \\bigwedge _ { k = i } ^ { j - 1 } ( x _ { k } \\vee \\bar { x } _ { i + ( k + 1 ) } \\% ( j - i + 1 ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "5a443b8e1d851aa053f60a9652603c292767f2223ecc0cf224db16d61bf1de93.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 173, + 406, + 188.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 206, + 188.0, + 406, + 203.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 206, + 203.0, + 406, + 218.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 206, + 218.0, + 406, + 233.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 238, + 473, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 475, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 407, + 252 + ], + "score": 1.0, + "content": "Additionally, a variable bridge is defined over an even number of variables", + "type": "text" + }, + { + "bbox": [ + 408, + 241, + 460, + 250 + ], + "score": 0.89, + "content": "x _ { 0 } , . . . , x _ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 237, + 475, + 252 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 256, + 403, + 285 + ], + "lines": [ + { + "bbox": [ + 207, + 256, + 403, + 285 + ], + "spans": [ + { + "bbox": [ + 207, + 256, + 403, + 285 + ], + "score": 0.93, + "content": "\\varphi _ { b r i d g e } = \\bigwedge _ { i = 0 } ^ { n - 1 } { \\big ( } ( x _ { i } \\vee x _ { 2 n - 1 - i } ) \\wedge ( { \\bar { x } } _ { i } \\vee { \\bar { x } } _ { 2 n - 1 - i } ) { \\big ) } .", + "type": "interline_equation", + "image_path": "3d2890cb3545abacee0aed89cd8d925528982926abce8d5f6f2af3d238b2937d.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 256, + 403, + 270.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 207, + 270.5, + 403, + 285.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 504, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 454, + 305 + ], + "score": 1.0, + "content": "A variable bridge makes the variables it connects forcibly have opposite values, e.g.,", + "type": "text" + }, + { + "bbox": [ + 455, + 293, + 489, + 302 + ], + "score": 0.91, + "content": "x _ { 0 } = \\bar { x _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 302, + 383, + 315 + ], + "spans": [ + { + "bbox": [ + 107, + 303, + 129, + 312 + ], + "score": 0.87, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 302, + 267, + 315 + ], + "score": 1.0, + "content": ". We denote a variable bridge over", + "type": "text" + }, + { + "bbox": [ + 267, + 304, + 319, + 314 + ], + "score": 0.88, + "content": "x _ { 0 } , . . . , x _ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 302, + 359, + 315 + ], + "score": 1.0, + "content": "as Bridge", + "type": "text" + }, + { + "bbox": [ + 359, + 303, + 379, + 315 + ], + "score": 0.7, + "content": "( 2 n )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 302, + 383, + 315 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 134, + 333 + ], + "score": 1.0, + "content": "To get", + "type": "text" + }, + { + "bbox": [ + 134, + 321, + 146, + 331 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 317, + 164, + 333 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 165, + 321, + 177, + 331 + ], + "score": 0.86, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 317, + 222, + 333 + ], + "score": 1.0, + "content": ", we define", + "type": "text" + }, + { + "bbox": [ + 222, + 321, + 234, + 331 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 317, + 506, + 333 + ], + "score": 1.0, + "content": "as a variable chain and bridge on all variables, yielding contrasting", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 264, + 342 + ], + "score": 1.0, + "content": "and unsatisfiable constraints. To define", + "type": "text" + }, + { + "bbox": [ + 264, + 332, + 276, + 342 + ], + "score": 0.86, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 330, + 458, + 342 + ], + "score": 1.0, + "content": ", we “cut” the chain in half, such that the first", + "type": "text" + }, + { + "bbox": [ + 459, + 332, + 466, + 340 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "variables", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 207, + 354 + ], + "score": 1.0, + "content": "can differ from the latter", + "type": "text" + }, + { + "bbox": [ + 207, + 344, + 214, + 352 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 340, + 505, + 354 + ], + "score": 1.0, + "content": ", satisfying the bridge. The second half of the “cut” chain is then flipped", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 352, + 365 + ], + "score": 1.0, + "content": "to a decrementing order, which preserves the satisfiability of", + "type": "text" + }, + { + "bbox": [ + 353, + 354, + 365, + 364 + ], + "score": 0.87, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ", but maintains the planarity of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 362, + 292, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 292, + 376 + ], + "score": 1.0, + "content": "resulting graph. More specifically, this yields:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 380, + 414, + 410 + ], + "lines": [ + { + "bbox": [ + 196, + 380, + 414, + 410 + ], + "spans": [ + { + "bbox": [ + 196, + 380, + 414, + 410 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\varphi _ { 1 } = \\mathbf { C h a i n } _ { \\mathrm { I n c } } ( 0 , 2 n ) \\wedge \\mathbf { B r i d g e } ( 2 n ) , \\mathrm { a n d } } \\\\ & { \\varphi _ { 2 } = \\mathbf { C h a i n } _ { \\mathrm { I n c } } ( 0 , n ) \\wedge \\mathbf { C h a i n } _ { \\mathrm { D e c } } ( n , 2 n ) \\wedge \\mathbf { B r i d g e } ( 2 n ) . } \\end{array}", + "type": "interline_equation", + "image_path": "a2e58547bb169db7ae266da9d0b79cb00beba7b954b1d7d862a337bd8fff6e3c.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 380, + 414, + 395.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 196, + 395.0, + 414, + 410.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 316, + 434 + ], + "score": 1.0, + "content": "Planar component. Following the generation of", + "type": "text" + }, + { + "bbox": [ + 316, + 423, + 328, + 433 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 421, + 348, + 434 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 423, + 361, + 433 + ], + "score": 0.86, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 421, + 505, + 434 + ], + "score": 1.0, + "content": ", a disjoint satisfiable planar graph", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 431, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 153, + 446 + ], + "score": 1.0, + "content": "component", + "type": "text" + }, + { + "bbox": [ + 154, + 433, + 179, + 444 + ], + "score": 0.9, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 431, + 218, + 446 + ], + "score": 1.0, + "content": "is added.", + "type": "text" + }, + { + "bbox": [ + 218, + 434, + 244, + 444 + ], + "score": 0.89, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 431, + 505, + 446 + ], + "score": 1.0, + "content": "shares no variables or disjunctions with the cores, so is primarily", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 370, + 457 + ], + "score": 1.0, + "content": "introduced to create noise and make learning more challenging.", + "type": "text" + }, + { + "bbox": [ + 370, + 444, + 396, + 456 + ], + "score": 0.88, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "is generated starting from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "random 2-connected (i.e., at least 2 edges must be removed to disconnect a component within the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "graph) bipartite planar graphs from the Plantri tool (Brinkmann et al., 2007), such that (i) the larger", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 475, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 504, + 489 + ], + "score": 1.0, + "content": "set of nodes in the graph is the variable set3, (ii) highly-connected disjunctions are split in a planarity-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "preserving fashion to maintain disjunction widths not exceeding 5, (iii) literal signs for variables are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 495, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 478, + 513 + ], + "score": 1.0, + "content": "uniformly randomly assigned, and (iv) redundant disjunctions, if any, are removed. If this", + "type": "text" + }, + { + "bbox": [ + 478, + 500, + 504, + 510 + ], + "score": 0.52, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 507, + 468, + 523 + ], + "score": 1.0, + "content": "is satisfiable, then it is accepted and used. Otherwise, the formula is discarded and a new", + "type": "text" + }, + { + "bbox": [ + 469, + 511, + 495, + 521 + ], + "score": 0.89, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 354, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 354, + 533 + ], + "score": 1.0, + "content": "analogously generated until a satisfiable formula is produced.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 504, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 199, + 551 + ], + "score": 1.0, + "content": "Since the core pair and", + "type": "text" + }, + { + "bbox": [ + 199, + 539, + 225, + 549 + ], + "score": 0.91, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 536, + 459, + 551 + ], + "score": 1.0, + "content": "are disjoint, it is easy to deduce that the graph encoding of", + "type": "text" + }, + { + "bbox": [ + 459, + 539, + 504, + 549 + ], + "score": 0.88, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 1 }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 123, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 549, + 169, + 560 + ], + "score": 0.83, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 547, + 404, + 561 + ], + "score": 1.0, + "content": "are both planar and 1-WL indistinguishable. Furthermore,", + "type": "text" + }, + { + "bbox": [ + 404, + 550, + 450, + 560 + ], + "score": 0.85, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "is satisfiable,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 123, + 572 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 561, + 170, + 571 + ], + "score": 0.88, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 559, + 306, + 572 + ], + "score": 1.0, + "content": "is not. Hence, the introduction of", + "type": "text" + }, + { + "bbox": [ + 307, + 560, + 333, + 571 + ], + "score": 0.9, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "maintains all the desirable core properties,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 568, + 358, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 358, + 583 + ], + "score": 1.0, + "content": "all while making any generated EXP dataset more challenging.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 586, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 504, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 504, + 598 + ], + "score": 1.0, + "content": "The structural properties of the cores, combined with the combinatorial difficulty of SAT, make", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "EXP a challenging dataset. For example, even minor formula changes, such as flipping a literal,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "can lead to a change in the SAT outcome, which enables the creation of near-identical, yet seman-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "tically different instances. Moreover, SAT is NP-complete (Cook, 1971), and remains so on planar", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "instances (Hunt III et al., 1998). Hence, EXP is cast to be challenging, both from an expressiveness", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 641, + 233, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 233, + 654 + ], + "score": 1.0, + "content": "and computational perspective.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 205, + 679 + ], + "score": 1.0, + "content": "Remark 3. Intuitively,", + "type": "text" + }, + { + "bbox": [ + 206, + 668, + 218, + 678 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 666, + 235, + 679 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 236, + 668, + 248, + 678 + ], + "score": 0.85, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 666, + 505, + 679 + ], + "score": 1.0, + "content": ", generated as described, can be distinguished by 2-WL, as 2-WL", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "can detect the break in cycles resulting from the aforementioned “cut”. In other words, 2-WL can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "identify that the chain has been broken in between these two formulas, and thus will return distinct", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 363, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 184, + 712 + ], + "score": 1.0, + "content": "colourings. Hence,", + "type": "text" + }, + { + "bbox": [ + 185, + 700, + 196, + 711 + ], + "score": 0.85, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 699, + 215, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 701, + 227, + 711 + ], + "score": 0.85, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 699, + 363, + 712 + ], + "score": 1.0, + "content": "can be distinguished by 3-GCNs.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 334, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 335, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 335, + 734 + ], + "score": 1.0, + "content": "3Ties are broken arbitrarily if the two sets are equally sized.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 362, + 95 + ], + "score": 1.0, + "content": "Core pair. 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The variable chain can be defined in increasing or decreasing order over", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 349, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 307, + 168 + ], + "score": 1.0, + "content": "these variables. More specifically, given variables", + "type": "text" + }, + { + "bbox": [ + 307, + 156, + 343, + 167 + ], + "score": 0.91, + "content": "x _ { i } , . . . , x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 153, + 349, + 168 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 131, + 505, + 168 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 173, + 406, + 233 + ], + "lines": [ + { + "bbox": [ + 206, + 173, + 406, + 233 + ], + "spans": [ + { + "bbox": [ + 206, + 173, + 406, + 233 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\mathrm { C h a i n } _ { \\mathrm { I n c } } ( i , j ) = \\displaystyle \\bigwedge _ { k = i } ^ { j - 1 } ( \\bar { x _ { k } } \\vee x _ { i + ( k + 1 ) } \\% ( j - i + 1 ) ) , \\mathrm { ~ a n d } } \\\\ & { \\mathrm { C h a i n } _ { \\mathrm { I n e c } } ( i , j ) = \\displaystyle \\bigwedge _ { k = i } ^ { j - 1 } ( x _ { k } \\vee \\bar { x } _ { i + ( k + 1 ) } \\% ( j - i + 1 ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "5a443b8e1d851aa053f60a9652603c292767f2223ecc0cf224db16d61bf1de93.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 173, + 406, + 188.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 206, + 188.0, + 406, + 203.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 206, + 203.0, + 406, + 218.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 206, + 218.0, + 406, + 233.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 238, + 473, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 475, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 407, + 252 + ], + "score": 1.0, + "content": "Additionally, a variable bridge is defined over an even number of variables", + "type": "text" + }, + { + "bbox": [ + 408, + 241, + 460, + 250 + ], + "score": 0.89, + "content": "x _ { 0 } , . . . , x _ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 237, + 475, + 252 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 237, + 475, + 252 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 256, + 403, + 285 + ], + "lines": [ + { + "bbox": [ + 207, + 256, + 403, + 285 + ], + "spans": [ + { + "bbox": [ + 207, + 256, + 403, + 285 + ], + "score": 0.93, + "content": "\\varphi _ { b r i d g e } = \\bigwedge _ { i = 0 } ^ { n - 1 } { \\big ( } ( x _ { i } \\vee x _ { 2 n - 1 - i } ) \\wedge ( { \\bar { x } } _ { i } \\vee { \\bar { x } } _ { 2 n - 1 - i } ) { \\big ) } .", + "type": "interline_equation", + "image_path": "3d2890cb3545abacee0aed89cd8d925528982926abce8d5f6f2af3d238b2937d.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 256, + 403, + 270.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 207, + 270.5, + 403, + 285.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 504, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 454, + 305 + ], + "score": 1.0, + "content": "A variable bridge makes the variables it connects forcibly have opposite values, e.g.,", + "type": "text" + }, + { + "bbox": [ + 455, + 293, + 489, + 302 + ], + "score": 0.91, + "content": "x _ { 0 } = \\bar { x _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 302, + 383, + 315 + ], + "spans": [ + { + "bbox": [ + 107, + 303, + 129, + 312 + ], + "score": 0.87, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 302, + 267, + 315 + ], + "score": 1.0, + "content": ". We denote a variable bridge over", + "type": "text" + }, + { + "bbox": [ + 267, + 304, + 319, + 314 + ], + "score": 0.88, + "content": "x _ { 0 } , . . . , x _ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 302, + 359, + 315 + ], + "score": 1.0, + "content": "as Bridge", + "type": "text" + }, + { + "bbox": [ + 359, + 303, + 379, + 315 + ], + "score": 0.7, + "content": "( 2 n )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 302, + 383, + 315 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 289, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 134, + 333 + ], + "score": 1.0, + "content": "To get", + "type": "text" + }, + { + "bbox": [ + 134, + 321, + 146, + 331 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 317, + 164, + 333 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 165, + 321, + 177, + 331 + ], + "score": 0.86, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 317, + 222, + 333 + ], + "score": 1.0, + "content": ", we define", + "type": "text" + }, + { + "bbox": [ + 222, + 321, + 234, + 331 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 317, + 506, + 333 + ], + "score": 1.0, + "content": "as a variable chain and bridge on all variables, yielding contrasting", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 264, + 342 + ], + "score": 1.0, + "content": "and unsatisfiable constraints. To define", + "type": "text" + }, + { + "bbox": [ + 264, + 332, + 276, + 342 + ], + "score": 0.86, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 330, + 458, + 342 + ], + "score": 1.0, + "content": ", we “cut” the chain in half, such that the first", + "type": "text" + }, + { + "bbox": [ + 459, + 332, + 466, + 340 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "variables", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 207, + 354 + ], + "score": 1.0, + "content": "can differ from the latter", + "type": "text" + }, + { + "bbox": [ + 207, + 344, + 214, + 352 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 340, + 505, + 354 + ], + "score": 1.0, + "content": ", satisfying the bridge. The second half of the “cut” chain is then flipped", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 352, + 365 + ], + "score": 1.0, + "content": "to a decrementing order, which preserves the satisfiability of", + "type": "text" + }, + { + "bbox": [ + 353, + 354, + 365, + 364 + ], + "score": 0.87, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ", but maintains the planarity of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 362, + 292, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 292, + 376 + ], + "score": 1.0, + "content": "resulting graph. More specifically, this yields:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 317, + 506, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 380, + 414, + 410 + ], + "lines": [ + { + "bbox": [ + 196, + 380, + 414, + 410 + ], + "spans": [ + { + "bbox": [ + 196, + 380, + 414, + 410 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\varphi _ { 1 } = \\mathbf { C h a i n } _ { \\mathrm { I n c } } ( 0 , 2 n ) \\wedge \\mathbf { B r i d g e } ( 2 n ) , \\mathrm { a n d } } \\\\ & { \\varphi _ { 2 } = \\mathbf { C h a i n } _ { \\mathrm { I n c } } ( 0 , n ) \\wedge \\mathbf { C h a i n } _ { \\mathrm { D e c } } ( n , 2 n ) \\wedge \\mathbf { B r i d g e } ( 2 n ) . } \\end{array}", + "type": "interline_equation", + "image_path": "a2e58547bb169db7ae266da9d0b79cb00beba7b954b1d7d862a337bd8fff6e3c.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 380, + 414, + 395.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 196, + 395.0, + 414, + 410.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 316, + 434 + ], + "score": 1.0, + "content": "Planar component. Following the generation of", + "type": "text" + }, + { + "bbox": [ + 316, + 423, + 328, + 433 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 421, + 348, + 434 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 423, + 361, + 433 + ], + "score": 0.86, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 421, + 505, + 434 + ], + "score": 1.0, + "content": ", a disjoint satisfiable planar graph", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 431, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 153, + 446 + ], + "score": 1.0, + "content": "component", + "type": "text" + }, + { + "bbox": [ + 154, + 433, + 179, + 444 + ], + "score": 0.9, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 431, + 218, + 446 + ], + "score": 1.0, + "content": "is added.", + "type": "text" + }, + { + "bbox": [ + 218, + 434, + 244, + 444 + ], + "score": 0.89, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 431, + 505, + 446 + ], + "score": 1.0, + "content": "shares no variables or disjunctions with the cores, so is primarily", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 370, + 457 + ], + "score": 1.0, + "content": "introduced to create noise and make learning more challenging.", + "type": "text" + }, + { + "bbox": [ + 370, + 444, + 396, + 456 + ], + "score": 0.88, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "is generated starting from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "random 2-connected (i.e., at least 2 edges must be removed to disconnect a component within the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "graph) bipartite planar graphs from the Plantri tool (Brinkmann et al., 2007), such that (i) the larger", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 475, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 504, + 489 + ], + "score": 1.0, + "content": "set of nodes in the graph is the variable set3, (ii) highly-connected disjunctions are split in a planarity-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "preserving fashion to maintain disjunction widths not exceeding 5, (iii) literal signs for variables are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 495, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 478, + 513 + ], + "score": 1.0, + "content": "uniformly randomly assigned, and (iv) redundant disjunctions, if any, are removed. If this", + "type": "text" + }, + { + "bbox": [ + 478, + 500, + 504, + 510 + ], + "score": 0.52, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 507, + 468, + 523 + ], + "score": 1.0, + "content": "is satisfiable, then it is accepted and used. Otherwise, the formula is discarded and a new", + "type": "text" + }, + { + "bbox": [ + 469, + 511, + 495, + 521 + ], + "score": 0.89, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 354, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 354, + 533 + ], + "score": 1.0, + "content": "analogously generated until a satisfiable formula is produced.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 421, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 504, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 199, + 551 + ], + "score": 1.0, + "content": "Since the core pair and", + "type": "text" + }, + { + "bbox": [ + 199, + 539, + 225, + 549 + ], + "score": 0.91, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 536, + 459, + 551 + ], + "score": 1.0, + "content": "are disjoint, it is easy to deduce that the graph encoding of", + "type": "text" + }, + { + "bbox": [ + 459, + 539, + 504, + 549 + ], + "score": 0.88, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 1 }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 123, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 549, + 169, + 560 + ], + "score": 0.83, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 547, + 404, + 561 + ], + "score": 1.0, + "content": "are both planar and 1-WL indistinguishable. Furthermore,", + "type": "text" + }, + { + "bbox": [ + 404, + 550, + 450, + 560 + ], + "score": 0.85, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "is satisfiable,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 123, + 572 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 561, + 170, + 571 + ], + "score": 0.88, + "content": "\\varphi _ { \\mathrm { p l a n a r } } \\wedge \\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 559, + 306, + 572 + ], + "score": 1.0, + "content": "is not. Hence, the introduction of", + "type": "text" + }, + { + "bbox": [ + 307, + 560, + 333, + 571 + ], + "score": 0.9, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "maintains all the desirable core properties,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 568, + 358, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 358, + 583 + ], + "score": 1.0, + "content": "all while making any generated EXP dataset more challenging.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 536, + 506, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 586, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 504, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 504, + 598 + ], + "score": 1.0, + "content": "The structural properties of the cores, combined with the combinatorial difficulty of SAT, make", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "EXP a challenging dataset. For example, even minor formula changes, such as flipping a literal,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "can lead to a change in the SAT outcome, which enables the creation of near-identical, yet seman-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "tically different instances. Moreover, SAT is NP-complete (Cook, 1971), and remains so on planar", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "instances (Hunt III et al., 1998). Hence, EXP is cast to be challenging, both from an expressiveness", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 641, + 233, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 233, + 654 + ], + "score": 1.0, + "content": "and computational perspective.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 587, + 505, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 205, + 679 + ], + "score": 1.0, + "content": "Remark 3. Intuitively,", + "type": "text" + }, + { + "bbox": [ + 206, + 668, + 218, + 678 + ], + "score": 0.86, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 666, + 235, + 679 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 236, + 668, + 248, + 678 + ], + "score": 0.85, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 666, + 505, + 679 + ], + "score": 1.0, + "content": ", generated as described, can be distinguished by 2-WL, as 2-WL", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "can detect the break in cycles resulting from the aforementioned “cut”. In other words, 2-WL can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "identify that the chain has been broken in between these two formulas, and thus will return distinct", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 363, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 184, + 712 + ], + "score": 1.0, + "content": "colourings. Hence,", + "type": "text" + }, + { + "bbox": [ + 185, + 700, + 196, + 711 + ], + "score": 0.85, + "content": "\\varphi _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 699, + 215, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 701, + 227, + 711 + ], + "score": 0.85, + "content": "\\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 699, + 363, + 712 + ], + "score": 1.0, + "content": "can be distinguished by 3-GCNs.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 666, + 505, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "Graph encoding. We use the following graph encoding, denoted by Enc: (i) Every variable is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "encoded by two nodes, representing its positive and negative literals, and connected by an edge, (ii)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Every disjunction is represented by a node, and an edge connects a literal node to a disjunction node", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "if the literal appears in the disjunction, and (iii) Variable and disjunction nodes are encoded with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "different types. We opt for this encoding, as it is commonly used in the literature (Selsam et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "2019), and is sufficient, for the sake of our empirical evaluation, to yield planar encodings for EXP", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 149, + 156, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 149, + 156, + 161 + ], + "score": 1.0, + "content": "graph pairs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 391, + 187 + ], + "score": 1.0, + "content": "Planar embeddings for core pair. We show planar embeddings for", + "type": "text" + }, + { + "bbox": [ + 392, + 174, + 430, + 186 + ], + "score": 0.91, + "content": "E n c ( \\varphi _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 174, + 450, + 187 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 174, + 489, + 186 + ], + "score": 0.9, + "content": "E n c ( \\varphi _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 174, + 506, + 187 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 129, + 196 + ], + "score": 0.87, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 185, + 396, + 199 + ], + "score": 1.0, + "content": "in Figure 4, and these embeddings can naturally be extended to any", + "type": "text" + }, + { + "bbox": [ + 396, + 187, + 403, + 195 + ], + "score": 0.51, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 185, + 408, + 199 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 408, + 186, + 447, + 198 + ], + "score": 0.9, + "content": "E n c ( \\varphi _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 185, + 464, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 186, + 504, + 198 + ], + "score": 0.91, + "content": "E n c ( \\varphi _ { 2 } )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 197, + 504, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 504, + 208 + ], + "score": 1.0, + "content": "can also be shown to be 1-WL indistinguishable. This can be observed intuitively, as node neighbor-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "hoods in both graphs are identical and very regular: all variable nodes are connected to exactly one", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "other variable node and two disjunction nodes, and all disjunction nodes are connected to exactly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 229, + 164, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 164, + 241 + ], + "score": 1.0, + "content": "two variables.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 255, + 252, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 254, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 254, + 269 + ], + "score": 1.0, + "content": "A.3.2 CONSTRUCTION OF CEXP", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 215, + 289 + ], + "score": 1.0, + "content": "Given a EXP dataset with", + "type": "text" + }, + { + "bbox": [ + 215, + 276, + 226, + 286 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 275, + 417, + 289 + ], + "score": 1.0, + "content": "pairs of graphs, we create CEXP by selecting", + "type": "text" + }, + { + "bbox": [ + 418, + 276, + 437, + 288 + ], + "score": 0.89, + "content": "N / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "graph pairs and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "modifying them to yield CORRUPT. The unmodified graph pairs are therefore exactly identical in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 298, + 412, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 388, + 311 + ], + "score": 1.0, + "content": "type to EXP instances, and we refer to these instances within CEXP as", + "type": "text" + }, + { + "bbox": [ + 389, + 298, + 408, + 309 + ], + "score": 0.58, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 299, + 412, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "For every graph pair, we discard the satisfiable graph and construct a new graph from a copy of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 327, + 229, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 229, + 338 + ], + "score": 1.0, + "content": "unsatisfiable graph as follows.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 130, + 349, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 129, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 129, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "1. Randomly introduce new literals to the existing disjunctions of the copy of the unsatisfiable", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 360, + 504, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 360, + 377, + 373 + ], + "score": 1.0, + "content": "graph, such that no redundancies are created (i.e., adding", + "type": "text" + }, + { + "bbox": [ + 377, + 362, + 384, + 370 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 360, + 476, + 373 + ], + "score": 1.0, + "content": "to a disjunction when", + "type": "text" + }, + { + "bbox": [ + 477, + 362, + 484, + 370 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 360, + 496, + 373 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 497, + 361, + 504, + 370 + ], + "score": 0.74, + "content": "\\bar { x }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "is already present), until 3 literals are added and the formula becomes satisfiable. Literal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 142, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "addition is done by creating new edges in the graph between disjunction and literal nodes.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 142, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "To do this, disjunctions with less than 5 literals are uniformly randomly selected, and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 141, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "literal to add is uniformly randomly sampled from the set of all non-redundant literals given", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 415, + 241, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 415, + 241, + 427 + ], + "score": 1.0, + "content": "the selected disjunction.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 130, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 130, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "2. Once a satisfiable formula is reached, iterate sequentially over all added edges, and elimi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "nate any edge whose removal does not restore unsatisfiability. This ensures that a minimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 453, + 446, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 446, + 466 + ], + "score": 1.0, + "content": "number of new edges, relative to the original unsatisfiable graph, are added.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Observe that these modifications have several interesting effects on the dataset. First, they preserve", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "the existing UNSAT core nodes and edges, while flipping the satisfiability of their overall formulas,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "which makes the learning task go beyond structure identification. Second, they introduce significant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "new variability to the dataset, in that the planar component and cores can share edges. Finally, they", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "make the graph pairs 1-WL distinguishable, which gives standard GNNs a chance to perform well", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 531, + 165, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 165, + 542 + ], + "score": 1.0, + "content": "on CORRUPT.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 109, + 557, + 321, + 568 + ], + "lines": [ + { + "bbox": [ + 107, + 556, + 323, + 569 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 323, + 569 + ], + "score": 1.0, + "content": "A.3.3 DATASET GENERATION FOR EXPERIMENTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 410, + 590 + ], + "score": 1.0, + "content": "To create the EXP dataset, we randomly generate 600 core pairs, where", + "type": "text" + }, + { + "bbox": [ + 410, + 579, + 418, + 587 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "(cf. Appendix A.3)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "is uniformly randomly set between 2 and 4 inclusive. Then, we generate the additional planar", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 246, + 612 + ], + "score": 1.0, + "content": "component using Plantri, such that", + "type": "text" + }, + { + "bbox": [ + 247, + 599, + 290, + 611 + ], + "score": 0.78, + "content": "5 0 0 \\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "formulas are generated from 12-node planar bipartite", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 610, + 432, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 432, + 623 + ], + "score": 1.0, + "content": "planar graphs, and the remaining 100 from planar bipartite graphs with 15 nodes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 672 + ], + "lines": [ + { + 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} + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 108, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 447, + 689 + ], + "score": 1.0, + "content": "Furthermore, the number of disjunctions also ranges from 10 (8 core disjunctions for", + "type": "text" + }, + { + "bbox": [ + 447, + 677, + 470, + 687 + ], + "score": 0.89, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "plus the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 280, + 702 + ], + "score": 1.0, + "content": "minimum 2 disjunctions for the case where", + "type": "text" + }, + { + "bbox": [ + 280, + 689, + 306, + 700 + ], + "score": 0.87, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 686, + 506, + 702 + ], + "score": 1.0, + "content": ", generated from 12-node graphs, has 10 variables", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 355, + 711 + ], + "score": 1.0, + "content": "and 2 disjunctions) to 30 disjunctions (16 core disjunctions for", + "type": "text" + }, + { + "bbox": [ + 355, + 699, + 378, + 709 + ], + "score": 0.89, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "plus at most 14 disjunctions for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 168, + 724 + ], + "score": 1.0, + "content": "the case where", + "type": "text" + }, + { + "bbox": [ + 169, + 711, + 194, + 722 + ], + "score": 0.9, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" 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26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "Graph encoding. We use the following graph encoding, denoted by Enc: (i) Every variable is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "encoded by two nodes, representing its positive and negative literals, and connected by an edge, (ii)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Every disjunction is represented by a node, and an edge connects a literal node to a disjunction node", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "if the literal appears in the disjunction, and (iii) Variable and disjunction nodes are encoded with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "different types. We opt for this encoding, as it is commonly used in the literature (Selsam et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "2019), and is sufficient, for the sake of our empirical evaluation, to yield planar encodings for EXP", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 149, + 156, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 149, + 156, + 161 + ], + "score": 1.0, + "content": "graph pairs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 82, + 506, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 391, + 187 + ], + "score": 1.0, + "content": "Planar embeddings for core pair. We show planar embeddings for", + "type": "text" + }, + { + "bbox": [ + 392, + 174, + 430, + 186 + ], + "score": 0.91, + "content": "E n c ( \\varphi _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 174, + 450, + 187 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 174, + 489, + 186 + ], + "score": 0.9, + "content": "E n c ( \\varphi _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 174, + 506, + 187 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 129, + 196 + ], + "score": 0.87, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 185, + 396, + 199 + ], + "score": 1.0, + "content": "in Figure 4, and these embeddings can naturally be extended to any", + "type": "text" + }, + { + "bbox": [ + 396, + 187, + 403, + 195 + ], + "score": 0.51, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 185, + 408, + 199 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 408, + 186, + 447, + 198 + ], + "score": 0.9, + "content": "E n c ( \\varphi _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 185, + 464, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 186, + 504, + 198 + ], + "score": 0.91, + "content": "E n c ( \\varphi _ { 2 } )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 197, + 504, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 504, + 208 + ], + "score": 1.0, + "content": "can also be shown to be 1-WL indistinguishable. This can be observed intuitively, as node neighbor-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "hoods in both graphs are identical and very regular: all variable nodes are connected to exactly one", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "other variable node and two disjunction nodes, and all disjunction nodes are connected to exactly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 229, + 164, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 164, + 241 + ], + "score": 1.0, + "content": "two variables.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 174, + 506, + 241 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 255, + 252, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 254, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 254, + 269 + ], + "score": 1.0, + "content": "A.3.2 CONSTRUCTION OF CEXP", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 215, + 289 + ], + "score": 1.0, + "content": "Given a EXP dataset with", + "type": "text" + }, + { + "bbox": [ + 215, + 276, + 226, + 286 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 275, + 417, + 289 + ], + "score": 1.0, + "content": "pairs of graphs, we create CEXP by selecting", + "type": "text" + }, + { + "bbox": [ + 418, + 276, + 437, + 288 + ], + "score": 0.89, + "content": "N / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "graph pairs and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "modifying them to yield CORRUPT. The unmodified graph pairs are therefore exactly identical in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 298, + 412, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 388, + 311 + ], + "score": 1.0, + "content": "type to EXP instances, and we refer to these instances within CEXP as", + "type": "text" + }, + { + "bbox": [ + 389, + 298, + 408, + 309 + ], + "score": 0.58, + "content": "\\overline { { \\mathrm { E x p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 299, + 412, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 275, + 505, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "For every graph pair, we discard the satisfiable graph and construct a new graph from a copy of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 327, + 229, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 229, + 338 + ], + "score": 1.0, + "content": "unsatisfiable graph as follows.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 315, + 505, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 349, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 129, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 129, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "1. Randomly introduce new literals to the existing disjunctions of the copy of the unsatisfiable", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 360, + 504, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 360, + 377, + 373 + ], + "score": 1.0, + "content": "graph, such that no redundancies are created (i.e., adding", + "type": "text" + }, + { + "bbox": [ + 377, + 362, + 384, + 370 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 360, + 476, + 373 + ], + "score": 1.0, + "content": "to a disjunction when", + "type": "text" + }, + { + "bbox": [ + 477, + 362, + 484, + 370 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 360, + 496, + 373 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 497, + 361, + 504, + 370 + ], + "score": 0.74, + "content": "\\bar { x }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "is already present), until 3 literals are added and the formula becomes satisfiable. Literal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 142, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "addition is done by creating new edges in the graph between disjunction and literal nodes.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 142, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "To do this, disjunctions with less than 5 literals are uniformly randomly selected, and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 141, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "literal to add is uniformly randomly sampled from the set of all non-redundant literals given", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 415, + 241, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 415, + 241, + 427 + ], + "score": 1.0, + "content": "the selected disjunction.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 130, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 130, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "2. Once a satisfiable formula is reached, iterate sequentially over all added edges, and elimi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "nate any edge whose removal does not restore unsatisfiability. This ensures that a minimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 453, + 446, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 446, + 466 + ], + "score": 1.0, + "content": "number of new edges, relative to the original unsatisfiable graph, are added.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 129, + 348, + 505, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Observe that these modifications have several interesting effects on the dataset. First, they preserve", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "the existing UNSAT core nodes and edges, while flipping the satisfiability of their overall formulas,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "which makes the learning task go beyond structure identification. Second, they introduce significant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "new variability to the dataset, in that the planar component and cores can share edges. Finally, they", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "make the graph pairs 1-WL distinguishable, which gives standard GNNs a chance to perform well", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 531, + 165, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 165, + 542 + ], + "score": 1.0, + "content": "on CORRUPT.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 475, + 505, + 542 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 557, + 321, + 568 + ], + "lines": [ + { + "bbox": [ + 107, + 556, + 323, + 569 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 323, + 569 + ], + "score": 1.0, + "content": "A.3.3 DATASET GENERATION FOR EXPERIMENTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 410, + 590 + ], + "score": 1.0, + "content": "To create the EXP dataset, we randomly generate 600 core pairs, where", + "type": "text" + }, + { + "bbox": [ + 410, + 579, + 418, + 587 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "(cf. Appendix A.3)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "is uniformly randomly set between 2 and 4 inclusive. Then, we generate the additional planar", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 246, + 612 + ], + "score": 1.0, + "content": "component using Plantri, such that", + "type": "text" + }, + { + "bbox": [ + 247, + 599, + 290, + 611 + ], + "score": 0.78, + "content": "5 0 0 \\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "formulas are generated from 12-node planar bipartite", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 610, + 432, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 432, + 623 + ], + "score": 1.0, + "content": "planar graphs, and the remaining 100 from planar bipartite graphs with 15 nodes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 576, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 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"content": "planar", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 387, + 661 + ], + "score": 1.0, + "content": "generation from 12-node graphs) and 22 variables (8 core variables for", + "type": "text" + }, + { + "bbox": [ + 387, + 649, + 410, + 659 + ], + "score": 0.89, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "plus a maximally-sized", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 395, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 231, + 673 + ], + "score": 1.0, + "content": "variable subset of 14 nodes for", + "type": "text" + }, + { + "bbox": [ + 231, + 662, + 257, + 672 + ], + "score": 0.83, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 660, + 395, + 673 + ], + "score": 1.0, + "content": "generation from 15-node graphs).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 626, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 447, + 689 + ], + "score": 1.0, + "content": "Furthermore, the number of disjunctions also ranges from 10 (8 core disjunctions for", + "type": "text" + }, + { + "bbox": [ + 447, + 677, + 470, + 687 + ], + "score": 0.89, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "plus the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 280, + 702 + ], + "score": 1.0, + "content": "minimum 2 disjunctions for the case where", + "type": "text" + }, + { + "bbox": [ + 280, + 689, + 306, + 700 + ], + "score": 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], + "score": 0.9, + "content": "\\varphi _ { \\mathrm { p l a n a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 709, + 506, + 724 + ], + "score": 1.0, + "content": ", generated from 15-node graphs, initially has 8 variables and 7 disjunctions,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 373, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 373, + 733 + ], + "score": 1.0, + "content": "which can at most lead to 14 final disjunctions following step (ii)).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 105, + 296, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 297, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 297, + 117 + ], + "score": 1.0, + "content": "In this subsection, we investigate the variability", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": 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From this figure,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 298, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 298, + 216 + ], + "score": 1.0, + "content": "we see that standard deviation spikes sharply at", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 298, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 298, + 227 + ], + "score": 1.0, + "content": "the start of training, and only begins dropping", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 225, + 297, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 297, + 237 + ], + "score": 1.0, + "content": "after 100 epochs. This suggests that the learn-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 236, + 297, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 173, + 248 + ], + "score": 1.0, + "content": "ing behavior of", + "type": "text" + }, + { + "bbox": [ + 174, + 236, + 218, + 247 + ], + "score": 0.51, + "content": "G C N { - } 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 236, + 297, + 248 + ], + "score": 1.0, + "content": "RNI is quite vari-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 297, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 297, + 260 + ], + "score": 1.0, + "content": "able, sometimes requiring few epochs to con-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 258, + 297, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 297, + 271 + ], + "score": 1.0, + "content": "verge, and in other cases requiring a very high", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 297, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 297, + 280 + ], + "score": 1.0, + "content": "number of epochs. Furthermore, standard de-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 297, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 297, + 292 + ], + "score": 1.0, + "content": "viation converges to almost zero following 200", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 297, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 297, + 303 + ], + "score": 1.0, + "content": "epochs, corresponding to the phase where all", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5 + }, + { + "type": "image", + "bbox": [ + 318, + 118, + 488, + 258 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 318, + 118, + 488, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 318, + 118, + 488, + 258 + ], + "spans": [ + { + "bbox": [ + 318, + 118, + 488, + 258 + ], + "score": 0.966, + "type": "image", + "image_path": "6721e91c6ef6ad166351a12e8a09d04875d6a5590bdce0dbf328d56a14ad16f7.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 318, + 118, + 488, + 132.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 318, + 132.0, + 488, + 146.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 318, + 146.0, + 488, + 160.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 318, + 160.0, + 488, + 174.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 318, + 174.0, + 488, + 188.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 318, + 188.0, + 488, + 202.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 318, + 202.0, + 488, + 216.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 318, + 216.0, + 488, + 230.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 318, + 230.0, + 488, + 244.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 318, + 244.0, + 488, + 258.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 303, + 268, + 505, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 304, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "Figure 5: Standard deviation of test accuracy over", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 304, + 279, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 304, + 279, + 430, + 291 + ], + "score": 1.0, + "content": "all 10 validation splits of GCN-", + "type": "text" + }, + { + "bbox": [ + 430, + 280, + 449, + 290 + ], + "score": 0.5, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 279, + 504, + 291 + ], + "score": 1.0, + "content": "RNI on EXP.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "validation folds have achieved near-perfect test performance. From these findings, we further con-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "firm that RNI introduces volatility to GCN training, this time manifesting in variable convergence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "times across validation folds, but that this volatility does not ultimately hinder convergence and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "performance, as all folds eventually reach satisfactory performance within a reasonable amount of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 345, + 296, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 296, + 358 + ], + "score": 1.0, + "content": "epochs, and subsequently stabilize at this level.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 253, + 386 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 254, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 254, + 387 + ], + "score": 1.0, + "content": "A.5 ADDITIONAL EXPERIMENTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "In addition to the experiments in the main body of the paper, we additionally evaluate RNI on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "sparser analog datasets to EXP and CEXP, namely SPARSEEXP and SPARSECEXP. These datasets", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 158, + 432 + ], + "score": 1.0, + "content": "only contain", + "type": "text" + }, + { + "bbox": [ + 158, + 419, + 178, + 430 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "of the number of instances of their original counterparts, and are used to study the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 429, + 302, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 302, + 442 + ], + "score": 1.0, + "content": "behavior and impact of RNI when data is sparse.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 108, + 457, + 274, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "score": 1.0, + "content": "A.5.1 EXPERIMENT 1 ON SPARSEEXP", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "In this experiment, we generate SPARSEEXP analogously to EXP, except that this dataset only con-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "sists of 150 graph pairs, i.e., 300 graphs in total. We then train 3-GCN for 200 epochs, and all other", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "systems for 1000 epochs on SPARSEEXP, as opposed to 100 and 500 respectively for EXP, to give", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "all evaluated models a better opportunity to compensate for the smaller dataset size. We show the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "learning curves for all models on SPARSEEXP, and reproduce the original figure for EXP, in Figure 6", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 533, + 197, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 197, + 546 + ], + "score": 1.0, + "content": "for easier comparison.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "First, we observe that all models converge slower on SPARSEEXP compared to EXP. This is not", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "surprising, as a lower data availability makes learning a well-performing function slower and more", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "challenging. More specifically, sparsity implies that (i) fewer weight updates are made per epoch,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "and (ii) these updates are of lower quality, as they are computed from a less representative and com-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "plete dataset. Nonetheless, the same relative convergence patterns between GCN-RNI models and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "3-GCN are also visible in this setting, further highighting the increased convergence time required", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 616, + 197, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 197, + 629 + ], + "score": 1.0, + "content": "by GCN-RNI models.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "We also observe that all GCN-RNI models, though also eventually converging, do so in a more", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "volatile fashion. Indeed, GCN-RNI models suffer from the sparseness of the dataset, as this makes", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "them more sensitive to RNI. As a result, these models require more training to effectively learn", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "robustness against RNI values, and learn this from a smaller sample set, increasing their variability", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "further. Moreover, the nature of SPARSEEXP makes learning more difficult, as it fully relies on RNI", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "for MPNNs to have a chance of achieving above-random performance, and thus encourages MPNNs", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "to fit specific RNI values. Hence, RNI introduces significant volatility and variability to training,", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 104, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "particularly with sparser data, and requires substantial training and epochs for GCN-RNI models to", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 721, + 323, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 323, + 733 + ], + "score": 1.0, + "content": "effectively develop a robustness to RNI instantiations.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 58 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 82, + 419, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 421, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 274, + 95 + ], + "score": 1.0, + "content": "A.4 STANDARD DEVIATION OF GCN-", + "type": "text" + }, + { + "bbox": [ + 274, + 82, + 294, + 93 + ], + "score": 0.63, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 82, + 421, + 95 + ], + "score": 1.0, + "content": "RNI ON EXP OVER TRAINING", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 105, + 296, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 297, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 297, + 117 + ], + "score": 1.0, + "content": "In this subsection, we investigate the variability", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 115, + 297, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 297, + 128 + ], + "score": 1.0, + "content": "of GCN-RNI learning across validation folds,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 126, + 297, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 297, + 138 + ], + "score": 1.0, + "content": "and do so with a representative model and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 137, + 297, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 297, + 149 + ], + "score": 1.0, + "content": "dataset, namely the semi-randomized GCN-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 147, + 297, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 144, + 159 + ], + "score": 0.53, + "content": "5 0 \\% \\mathrm { R N I }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 147, + 297, + 160 + ], + "score": 1.0, + "content": "model and the standard EXP dataset.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 159, + 297, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 297, + 172 + ], + "score": 1.0, + "content": "The standard deviation of the test accuracy", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 170, + 298, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 121, + 182 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 121, + 170, + 168, + 181 + ], + "score": 0.33, + "content": "\\mathrm { G C N } { - } 5 0 \\% \\mathrm { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 170, + 298, + 182 + ], + "score": 1.0, + "content": "RNI over EXP, across all 10", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 297, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 297, + 192 + ], + "score": 1.0, + "content": "cross-validation folds relative to the number of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 298, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 298, + 205 + ], + "score": 1.0, + "content": "epochs, is shown in Figure 5. From this figure,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 298, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 298, + 216 + ], + "score": 1.0, + "content": "we see that standard deviation spikes sharply at", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 298, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 298, + 227 + ], + "score": 1.0, + "content": "the start of training, and only begins dropping", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 225, + 297, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 297, + 237 + ], + "score": 1.0, + "content": "after 100 epochs. This suggests that the learn-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 236, + 297, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 173, + 248 + ], + "score": 1.0, + "content": "ing behavior of", + "type": "text" + }, + { + "bbox": [ + 174, + 236, + 218, + 247 + ], + "score": 0.51, + "content": "G C N { - } 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 236, + 297, + 248 + ], + "score": 1.0, + "content": "RNI is quite vari-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 297, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 297, + 260 + ], + "score": 1.0, + "content": "able, sometimes requiring few epochs to con-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 258, + 297, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 297, + 271 + ], + "score": 1.0, + "content": "verge, and in other cases requiring a very high", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 297, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 297, + 280 + ], + "score": 1.0, + "content": "number of epochs. Furthermore, standard de-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 297, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 297, + 292 + ], + "score": 1.0, + "content": "viation converges to almost zero following 200", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 297, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 297, + 303 + ], + "score": 1.0, + "content": "epochs, corresponding to the phase where all", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 104, + 298, + 303 + ] + }, + { + "type": "image", + "bbox": [ + 318, + 118, + 488, + 258 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 318, + 118, + 488, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 318, + 118, + 488, + 258 + ], + "spans": [ + { + "bbox": [ + 318, + 118, + 488, + 258 + ], + "score": 0.966, + "type": "image", + "image_path": "6721e91c6ef6ad166351a12e8a09d04875d6a5590bdce0dbf328d56a14ad16f7.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 318, + 118, + 488, + 132.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 318, + 132.0, + 488, + 146.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 318, + 146.0, + 488, + 160.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 318, + 160.0, + 488, + 174.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 318, + 174.0, + 488, + 188.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 318, + 188.0, + 488, + 202.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 318, + 202.0, + 488, + 216.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 318, + 216.0, + 488, + 230.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 318, + 230.0, + 488, + 244.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 318, + 244.0, + 488, + 258.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 303, + 268, + 505, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 304, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "Figure 5: Standard deviation of test accuracy over", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 304, + 279, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 304, + 279, + 430, + 291 + ], + "score": 1.0, + "content": "all 10 validation splits of GCN-", + "type": "text" + }, + { + "bbox": [ + 430, + 280, + 449, + 290 + ], + "score": 0.5, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 279, + 504, + 291 + ], + "score": 1.0, + "content": "RNI on EXP.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "validation folds have achieved near-perfect test performance. From these findings, we further con-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "firm that RNI introduces volatility to GCN training, this time manifesting in variable convergence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "times across validation folds, but that this volatility does not ultimately hinder convergence and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "performance, as all folds eventually reach satisfactory performance within a reasonable amount of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 345, + 296, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 296, + 358 + ], + "score": 1.0, + "content": "epochs, and subsequently stabilize at this level.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 302, + 506, + 358 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 253, + 386 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 254, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 254, + 387 + ], + "score": 1.0, + "content": "A.5 ADDITIONAL EXPERIMENTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "In addition to the experiments in the main body of the paper, we additionally evaluate RNI on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "sparser analog datasets to EXP and CEXP, namely SPARSEEXP and SPARSECEXP. These datasets", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 158, + 432 + ], + "score": 1.0, + "content": "only contain", + "type": "text" + }, + { + "bbox": [ + 158, + 419, + 178, + 430 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "of the number of instances of their original counterparts, and are used to study the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 429, + 302, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 302, + 442 + ], + "score": 1.0, + "content": "behavior and impact of RNI when data is sparse.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 397, + 505, + 442 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 457, + 274, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "score": 1.0, + "content": "A.5.1 EXPERIMENT 1 ON SPARSEEXP", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "In this experiment, we generate SPARSEEXP analogously to EXP, except that this dataset only con-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "sists of 150 graph pairs, i.e., 300 graphs in total. We then train 3-GCN for 200 epochs, and all other", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "systems for 1000 epochs on SPARSEEXP, as opposed to 100 and 500 respectively for EXP, to give", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "all evaluated models a better opportunity to compensate for the smaller dataset size. We show the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "learning curves for all models on SPARSEEXP, and reproduce the original figure for EXP, in Figure 6", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 533, + 197, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 197, + 546 + ], + "score": 1.0, + "content": "for easier comparison.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 479, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "First, we observe that all models converge slower on SPARSEEXP compared to EXP. This is not", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "surprising, as a lower data availability makes learning a well-performing function slower and more", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "challenging. More specifically, sparsity implies that (i) fewer weight updates are made per epoch,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "and (ii) these updates are of lower quality, as they are computed from a less representative and com-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "plete dataset. Nonetheless, the same relative convergence patterns between GCN-RNI models and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "3-GCN are also visible in this setting, further highighting the increased convergence time required", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 616, + 197, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 197, + 629 + ], + "score": 1.0, + "content": "by GCN-RNI models.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 550, + 506, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "We also observe that all GCN-RNI models, though also eventually converging, do so in a more", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "volatile fashion. Indeed, GCN-RNI models suffer from the sparseness of the dataset, as this makes", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "them more sensitive to RNI. As a result, these models require more training to effectively learn", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "robustness against RNI values, and learn this from a smaller sample set, increasing their variability", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "further. 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DatasetEXPCEXP
入p入p
GCN1×10-4N/A1×10-4N/A
GCN-12.5%RNI2×10-4N2×10-4N
GCN-50%RNI2×10-4N2×10-4N
GCN-87.5%RNI2×10-4N5×10-4N
GCN-RNI5×10-4N5×10-4N
3-GCN5×10-4N/A2×10-4N/A
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ModelTesting Accuracy (%)
GCN-RNI(U)92.7 ± 5.61
GCN-RNI(N)96.0 ± 2.11
GCN-RNI(XU)64.6 ± 19.9
GCN-RNI(XN)63.0 ± 20.9
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Finally, all experiments were run on a V100 GPU. Detailed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 619, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 269, + 634 + ], + "score": 1.0, + "content": "hyper-parameters, namely learning rate", + "type": "text" + }, + { + "bbox": [ + 270, + 621, + 277, + 630 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 619, + 366, + 634 + ], + "score": 1.0, + "content": "and RNI distribution", + "type": "text" + }, + { + "bbox": [ + 367, + 623, + 373, + 632 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 619, + 506, + 634 + ], + "score": 1.0, + "content": ", per model on every evaluation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 631, + 224, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 224, + 642 + ], + "score": 1.0, + "content": "dataset are shown in Table 2.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 107, + 655, + 427, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 430, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 430, + 668 + ], + "score": 1.0, + "content": "A.6.1 RESULTS FOR GCN-RNI WITH HYPERBOLIC TANGENT ACTIVATION", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 674, + 505, + 718 + ], + "lines": [ + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "score": 1.0, + "content": "In addition to experimenting with the RNI probability distribution, we also experimented with dif-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 684, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 699 + ], + "score": 1.0, + "content": "ferent activation functions for the GCN message passing iterations. Results are shown in Table 3.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "score": 1.0, + "content": "Performance with tanh is significantly more variable across distributions than ELU, which shows", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 707, + 402, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 402, + 720 + ], + "score": 1.0, + "content": "that RNI can be highly sensitive to practical choices of hyper-parameters.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 188, + 100, + 419, + 205 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 136, + 80, + 472, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 78, + 474, + 95 + ], + "spans": [ + { + "bbox": [ + 137, + 78, + 474, + 95 + ], + "score": 1.0, + "content": "Table 2: Hyper-parameter configurations for all GCN-RNIand 3-GCN experiments.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 188, + 100, + 419, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 188, + 100, + 419, + 205 + ], + "spans": [ + { + "bbox": [ + 188, + 100, + 419, + 205 + ], + "score": 0.975, + "html": "
DatasetEXPCEXP
入p入p
GCN1×10-4N/A1×10-4N/A
GCN-12.5%RNI2×10-4N2×10-4N
GCN-50%RNI2×10-4N2×10-4N
GCN-87.5%RNI2×10-4N5×10-4N
GCN-RNI5×10-4N5×10-4N
3-GCN5×10-4N/A2×10-4N/A
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ModelTesting Accuracy (%)
GCN-RNI(U)92.7 ± 5.61
GCN-RNI(N)96.0 ± 2.11
GCN-RNI(XU)64.6 ± 19.9
GCN-RNI(XN)63.0 ± 20.9
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However, we note that the “struggle”", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "phase described in the main paper, which only occurs during the first 100 epochs over CEXP, lasts", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "for around 500 epochs on SPARSEEXP. 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Furthermore, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 514, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 530 + ], + "score": 1.0, + "content": "final prediction for every graph is computed by aggregating all node embeddings following message", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "score": 1.0, + "content": "passing using the max function, and then passing the result through a multi-layer perceptron of 3", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 215, + 550 + ], + "score": 1.0, + "content": "layers with dimensionality", + "type": "text" + }, + { + "bbox": [ + 215, + 540, + 222, + 548 + ], + "score": 0.59, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 538, + 343, + 550 + ], + "score": 1.0, + "content": ", 32 and 2 respectively, where", + "type": "text" + }, + { + "bbox": [ + 343, + 539, + 350, + 547 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "is the embedding dimensionality used", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "in the given model. The activation function for the first two MLP layers is the ELU function (Clevert", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 559, + 486, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 486, + 572 + ], + "score": 1.0, + "content": "et al., 2016), and the softmax function is used to make a final prediction at the final MLP layer.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 482, + 506, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "All neural networks in this work are optimized using the Adam optimizer (Kingma & Ba, 2015).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 318, + 600 + ], + "score": 1.0, + "content": "All training is conducted with a fixed learning rate", + "type": "text" + }, + { + "bbox": [ + 319, + 588, + 326, + 597 + ], + "score": 0.74, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 587, + 505, + 600 + ], + "score": 1.0, + "content": ", for fairer comparison between all models.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "Initially, decaying learning rates were used, but these were discarded, as they yielded sub-optimal", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "convergence for all GCN-RNI models. Finally, all experiments were run on a V100 GPU. Detailed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 619, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 269, + 634 + ], + "score": 1.0, + "content": "hyper-parameters, namely learning rate", + "type": "text" + }, + { + "bbox": [ + 270, + 621, + 277, + 630 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 619, + 366, + 634 + ], + "score": 1.0, + "content": "and RNI distribution", + "type": "text" + }, + { + "bbox": [ + 367, + 623, + 373, + 632 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 619, + 506, + 634 + ], + "score": 1.0, + "content": ", per model on every evaluation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 631, + 224, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 224, + 642 + ], + "score": 1.0, + "content": "dataset are shown in Table 2.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 576, + 506, + 642 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 655, + 427, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 430, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 430, + 668 + ], + "score": 1.0, + "content": "A.6.1 RESULTS FOR GCN-RNI WITH HYPERBOLIC TANGENT ACTIVATION", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 674, + 505, + 718 + ], + "lines": [ + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "score": 1.0, + "content": "In addition to experimenting with the RNI probability distribution, we also experimented with dif-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 684, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 699 + ], + "score": 1.0, + "content": "ferent activation functions for the GCN message passing iterations. Results are shown in Table 3.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 709 + ], + "score": 1.0, + "content": "Performance with tanh is significantly more variable across distributions than ELU, which shows", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 707, + 402, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 402, + 720 + ], + "score": 1.0, + "content": "that RNI can be highly sensitive to practical choices of hyper-parameters.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 674, + 506, + 720 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_model.json b/parse/train/L7Irrt5sMQa/L7Irrt5sMQa_model.json new file mode 100644 index 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ModelTest Accuracy (%)
GCN-RNI(U)97.3 ± 2.55
GCN-RNI(N) GCN-RNI(XU)98.0 ± 1.85 97.0 ± 1.43
GCN-RNI(XN)96.6 ± 2.20
PPGN50.0
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ModelTesting Accuracy (%)
GCN-RNI(U)92.7 ± 5.61
GCN-RNI(N)96.0 ± 2.11
GCN-RNI(XU)64.6 ± 19.9
GCN-RNI(XN)63.0 ± 20.9
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DatasetEXPCEXP
入p入p
GCN1×10-4N/A1×10-4N/A
GCN-12.5%RNI2×10-4N2×10-4N
GCN-50%RNI2×10-4N2×10-4N
GCN-87.5%RNI2×10-4N5×10-4N
GCN-RNI5×10-4N5×10-4N
3-GCN5×10-4N/A2×10-4N/A
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Girgis UCLA amgirgis@g.ucla.edu + +Deepesh Data UCLA deepesh.data@gmail.com + +Suhas Diggavi UCLA suhasdiggavi@ucla.edu + +# Abstract + +We study privacy in a distributed learning framework, where clients collaboratively build a learning model iteratively through interactions with a server from whom we need privacy. Motivated by stochastic optimization and the federated learning (FL) paradigm, we focus on the case where a small fraction of data samples are randomly sub-sampled in each round to participate in the learning process, which also enables privacy amplification. To obtain even stronger local privacy guarantees, we study this in the shuffle privacy model, where each client randomizes its response using a local differentially private (LDP) mechanism and the server only receives a random permutation (shuffle) of the clients’ responses without their association to each client. The principal result of this paper is a privacyoptimization performance trade-off for discrete randomization mechanisms in this sub-sampled shuffle privacy model. This is enabled through a new theoretical technique to analyze the Rényi Differential Privacy (RDP) of the sub-sampled shuffle model. We numerically demonstrate that, for important regimes, with composition our bound yields significant improvement in privacy guarantee over the state-of-the-art approximate Differential Privacy (DP) guarantee (with strong composition) for sub-sampled shuffled models. We also demonstrate numerically significant improvement in privacy-learning performance operating point using real data sets. Despite these advances an open question is to bridge the gap between lower and upper privacy bounds in our RDP analysis. + +# 1 Introduction + +As learning moves towards the edge, there is a need to collaborate to build learning models1, such as in federated learning [36, 44, 33]. In this framework, the collaboration is typically mediated by a server. In particular, we want to collaboratively build a learning model by solving an empirical risk minimization (ERM) problem (see $( 2 )$ in Section $2 )$ . To obtain a model parametrized by $\theta$ using ERM, the commonly used mechanism is Stochastic Gradient Descent (SGD) [12]. However, one needs to solve this while enabling strong privacy guarantees on local data from the server, while also obtaining good learning performance, i.e., a suitable privacy-learning performance operating point. + +Differential privacy (DP) $\boxed { 1 8 }$ is the gold standard notion of data privacy that gives a rigorous framework through quantifying the information leakage about individual training data points from the observed interactions. Though DP was originally proposed in a framework where data resides centrally $\boxed { 1 8 }$ , for distributed learning the more appropriate notion is of local differential privacy (LDP) [35, 17]. Here, each client randomizes its interactions with the server from whom the data is to be kept private (e.g., see industrial implementations [23, 31, 16]). However, LDP mechanisms suffer from poor performance in comparison with the central DP mechanisms [17, 35, 32]. To overcome this, a new privacy framework using anonymization has been proposed in the so-called shuffled model $\underline { { { \sqrt { 2 2 } } } } | 2 5 | 6 | \dot { 2 } 6 | 5 | \dot { \overline { { { 1 5 } } } } | \overline { { { \mathrm { [ 7 ] } } } } | 8 |$ . In the shuffled model, each client sends her private message to a secure shuffler that randomly permutes all the received messages before forwarding them to the server. This model enables significantly better privacy-utility performance by amplifying DP through this shuffling. Therefore, in this paper we consider the shuffle privacy framework for distributed learning. + +In solving $( 2 )$ using (distributed) gradient descent, each exchange leaks information about the local data, but we need as many steps as possible to obtain a good model; setting up the tension between privacy and performance. The goal is to obtain as many such interactions as possible for a given privacy budget. This is quantified through analyzing the privacy of the composition of privacy mechanisms. Abadi et al. [1] developed a framework for tighter analysis of such compositions, and this was later reformulated in terms of Rényi Differential Privacy (RDP) $\textcircled { 1 3 7 }$ , and mapping this back to DP guarantee $\textcircled { 1 3 8 }$ . Therefore, studying RDP is important to obtaining strong composition privacy results, and is the focus of this paper. + +In distributed (and federated) learning, a fraction of the data samples are sampled; for example, with random client participation and stochastic gradient descent (SGD), which can be written as + +$$ +\theta _ { t + 1 } \theta _ { t } - \eta _ { t } \frac { 1 } { | \mathcal { T } | } \sum _ { i \in \mathcal { I } } \mathcal { R } ( \nabla f _ { i } ( \theta _ { t } ) ) , +$$ + +where $\mathcal { R }$ is the local randomization mechanism and $\mathcal { T }$ are the indices of the sampled data. This is a subsampled mechanism that enables another privacy amplification opportunity; which, in several cases, is shown to yield a privacy advantage proportional to the subsampling rate; see $\dot { \sqrt { 3 5 } } \boxed { 4 2 }$ . The central technical question addressed in this paper is how to analyze the RDP of an arbitrary discrete mechanism for the subsampled shuffle privacy model. This enables us to answer the overall question posed in this paper, which is an achievable privacy-learning performance trade-off point for solving in the shuffled privacy model for distributed learning (see Figure 1). Our contributions are: + +![](images/60fe5dd4ec96e99911e5639baf65cde4fbddb5ef4c7ef331a52a02859b1dfcf1.jpg) +Figure 1: An iteration from the CLDPSGD Algorithm, where 3 clients are randomly chosen at each iteration. Each client sends the private gradient $\mathcal { R } _ { p } \left( g _ { t } ( d _ { i } ) \right)$ to the shuffler that randomly permutes the gradients before passing them to the server. + +We analyze the RDP of subsampled mechanisms in the shuffle framework by developing a novel bound applicable to any discrete $\epsilon _ { 0 }$ -LDP mechanism as a function of the RDP order $\lambda$ , subsampling rate $\gamma$ , the LDP parameter $\epsilon _ { 0 }$ , and the number of clients $n$ ; see Theorem $\bigtriangledown$ The bound is explicit and amenable to numerics, including all constants.2 Furthermore, the bounds are valid for generic LDP mechanisms and all parameter regimes.3 We also provide a lower bound for the RDP in Theorem $\bigstar$ We prove our upper bound (Theorem 1) using the following novel analysis techniques: First, we reduce the problem of computing the RDP of sub-sampled shuffle mechanisms to the problem of computing ternary $| \chi | ^ { \alpha } \cdot \mathrm { D P } ^ { \bullet } | \overline { { { 4 3 } } } | ]$ of shuffle (non sub-sampled) mechanisms; see Lemma 2. Then we reduce the computation of the ternary $| \chi | ^ { \alpha }$ -DP of shuffle mechanisms for a generic triple of neighboring datasets to those that have a special structure (see Theorem $5 )$ – this reduction step is one of the core technical results of this paper. Then we bound the ternary $| \chi | ^ { \alpha }$ -DP of the shuffle mechanisms for triples of neighboring datasets having special structures by bounding the Pearson-Vajda divergence $\lvert \overline { { 4 3 } } \rvert$ using some concentration properties (see Theorem $\overline { { 6 ) } }$ . + +• Using the core technical result in Theorem $^ { 1 , }$ we analyze privacy-convergence trade-offs of the CLDP-SGD algorithm (see Algorithm 1) for Lipschitz convex functions in Theorem $3 .$ This partially resolves an open question posed in $| \vec { \boldsymbol { { \vert { 2 7 } } \| } }$ , to extend their privacy analysis to RDP and significantly strengthening their privacy guatantees. + +• Numerically, we save a factor $1 4 \times$ in privacy (✏) over the best known results for approximate DP for shuffling $\pm \overbrace { | 2 4 | }$ combined with strong composition $\pm$ for $T = 1 0 ^ { 5 } , \gamma = 0 . 0 0 \bar { 1 } , n = 1 0 ^ { 6 }$ , and a factor of $2 . 5 \times$ better than the best known RDP for shuffling bound $\lVert 2 9 \rVert$ combined with the sub-sampling result in $\textcircled { 1 4 3 } \textcircled { 1 }$ . Translating these to privacy-performance operating point in distributed optimization, over the MNIST data set with $\ell _ { \infty }$ clipping we numerically show gains: For the same privacy budget of $\epsilon = 1 . 4$ , we get a test performance of $8 0 \%$ whereas using strong composition the test performance of $\mathbb { \lVert 2 4 \rVert }$ is $7 0 \%$ ; furthermore, we achieves $9 0 \%$ accuracy with the total privacy budget $\epsilon = 2 . 9 1$ , whereas, $\mathring { \| 2 4 \| }$ (with strong composition) achieves the same accuracy with a total privacy budget of $\epsilon = 4 . 8 2$ . See Section 4 and the supplementary material for more results. + +Related work: We give a more complete literature review in Appendix A, and focus here on the works that are closest to the results presented in this paper. + +Private optimization in the shuffled model: Recently, $\pmb { \mathbb { Z } } 1 \mathbf { l }$ and $ { \mathbb { P } } ^ { \geq \sum { \left. \left. 2 8 \right. \right. } }$ have proposed differentially private SGD algorithms for federated learning, where at each iteration, each client applies an LDP mechanism on the gradients with the existence of a secure shuffler between the clients and the central server. However, the privacy analyses in these works developed approximate DP using advanced composition theorems for DP (e.g., $\pm \pm \pm \pm ) )$ ), which are known to be loose for composition $\textcircled { 1 }$ . To the best of our knowledge, analyzing the private optimization framework using RDP and subsampling in the shuffled model is new to this paper. + +Subsampled RDP: The works [38, 43, 45] have studied the RDP of subsampled mechanisms without shuffling. They demonstrated that this provides a tighter bound on the total privacy loss than the bound that can be obtained using the standard strong composition theorems. The RDP analysis of subsampled mechanisms in the shuffled privacy framework has not been studied before,4 and is new to this paper. The RDP of the shuffled model was very recently studied in $\left[ \left[ 2 9 \right] \right]$ , but without incorporating subsampling, which poses new technical challenges, as directly bounding the RDP of subsampled shuffle mechanisms is non-trivial. We overcome this by reducing our problem of computing RDP to bounding the ternary $| \chi | ^ { \alpha }$ -DP, and bounding the latter is a core technical contribution of our paper. + +Paper organization: We give preliminaries and problem formulation in Section 2, main results (upper and lower bounds, and privacy-convergence tradeoff) in Section $\textcircled { 3 }$ numerical results in Section 4, proof of the upper bound in Section $5 _ { : }$ and proof of the ternary DP of the shuffle model in Section 6. Omitted details/proofs from this paper are given in the supplementary material. + +# 2 Preliminaries and Problem Formulation + +We use several privacy definitions throughout this paper. Among these, the local and central differential privacy definitions are standard and we defer them to Appendix $\begin{array} { l } { \mathbf { B } . } \\ { . } \end{array}$ The other privacy definitions (Rényi DP and ternary $| \chi | ^ { \alpha }$ -DP) are relatively less standard and we define them below. + +We say that two datasets $\mathcal { D } = \{ d _ { 1 } , \ldots , d _ { n } \} \in \mathcal { X } ^ { n }$ and $\mathcal { D } ^ { \prime } = \{ d _ { 1 } ^ { \prime } , \ldots , d _ { n } ^ { \prime } \} \in \mathcal { X } ^ { n }$ are neighboring (and denoted by $\mathcal { D } \sim \mathcal { D } ^ { \prime }$ ) if they differ in one data point, i.e., there exists an $i \in [ n ]$ such that $d _ { i } \neq d _ { i } ^ { \prime }$ and for every $j \in [ n ] , j \neq i$ , we have $d _ { j } = d _ { j } ^ { \prime }$ . + +Definition 1 $( \lambda , \epsilon )$ -RDP (Rényi Differential Privacy) $\pmb { \mathbb { B } } \pmb { \mathbb { Z } } )$ ). A randomized mechanism $\mathcal { M } : \mathcal { X } ^ { n } \mathcal { Y }$ is said to have $\epsilon$ -Rényi differential privacy of order $\lambda \in ( 1 , \infty )$ (in short, $( \lambda , \epsilon ( \lambda ) )$ -RDP), if for any neighboring datasets $\mathcal { D }$ , $\mathcal { D } ^ { \prime } \in \mathcal { X } ^ { n }$ , the Rényi divergence of order $\lambda$ between $\mathcal { M } ( \mathcal { D } )$ and $\mathcal { M } ( \mathcal { D ^ { \prime } } )$ is upper-bounded by $\epsilon ( \lambda )$ , i.e., + +$$ +D _ { \lambda } ( \mathcal { M } ( \mathcal { D } ) | | \mathcal { M } ( \mathcal { D } ^ { \prime } ) ) = \frac { 1 } { \lambda - 1 } \log \left( \mathbb { E } _ { \theta \sim \mathcal { M } ( \mathcal { D } ^ { \prime } ) } \left[ \left( \frac { \mathcal { M } ( \mathcal { D } ) ( \theta ) } { \mathcal { M } ( \mathcal { D } ^ { \prime } ) ( \theta ) } \right) ^ { \lambda } \right] \right) \leq \epsilon ( \lambda ) , +$$ + +where $\mathcal { M } ( \mathcal { D } ) ( \theta )$ denotes the probability that $\mathcal { M }$ on input $\mathcal { D }$ generates the output $\theta$ + +Definition 2 ( $\zeta$ -Ternary $| \chi | ^ { \alpha }$ -differential privacy $\mathbb { \left[ 4 3 \right] }$ ). A randomized mechanism $\mathcal { M } : \mathcal { X } ^ { n } \mathcal { Y }$ is said to have $\zeta$ -ternary- $| \chi | ^ { \alpha }$ -DP, if for any triple of mutually adjacent datasets $\mathcal { D } , \mathcal { D } ^ { \prime } , \mathcal { D } ^ { \prime \prime } \in \mathcal { X } ^ { n }$ (i.e., they mutually differ in the same location), the ternary- $\cdot | \chi | ^ { \alpha }$ divergence of $\mathcal { M } ( \mathcal { D } ) , \mathcal { M } ( \mathcal { D ^ { \prime } } ) , \mathcal { M } ( \mathcal { D ^ { \prime } } )$ is upper-bounded by $( \zeta ( \alpha ) ) ^ { \alpha }$ for all $\alpha \geq 1$ (where $\zeta$ is a function from $\mathbb { R } ^ { + }$ to $\mathbb { R } ^ { + }$ ), i.e., + +$$ +D _ { | \chi | ^ { \alpha } } \left( \mathcal { M } ( \mathcal { D } ) , \mathcal { M } ( \mathcal { D } ^ { \prime } ) | | \mathcal { M } ( \mathcal { D } ^ { \prime \prime } ) \right) : = \mathbb { E } _ { \mathcal { M } ( \mathcal { D } ^ { \prime \prime } ) } \left[ \left| \frac { \mathcal { M } ( \mathcal { D } ) - \mathcal { M } ( \mathcal { D } ^ { \prime } ) } { \mathcal { M } ( \mathcal { D } ^ { \prime \prime } ) } \right| ^ { \alpha } \right] \leq \left( \zeta ( \alpha ) \right) ^ { \alpha } . +$$ + +The ternary $| \chi | ^ { \alpha }$ -DP was proposed in $\underline { { \| 4 3 \| } }$ to characterize the RDP of the sub-sampled mechanism without shuffling. In this work, we analyze the ternary $| \chi | ^ { \alpha }$ -DP of the shuffled mechanism to bound the RDP of the sub-sampled shuffle model. + +We can use the following result for converting the RDP guarantees of a mechanism to its central DP guarantees. To the best of our knowledge, this result gives the best conversion. + +Lemma 1 (From RDP to DP [13, 4]). Suppose for any $\lambda > 1$ , a mechanism $\mathcal { M }$ is $( \lambda , \epsilon \left( \lambda \right) )$ -RDP. Then, the mechanism $\mathcal { M }$ is $( \epsilon , \delta )$ -DP, where $\delta > 0$ is arbitrary and $\epsilon$ is given by + +$$ +\epsilon = \operatorname* { m i n } _ { \lambda } \left( \epsilon \left( \lambda \right) + \frac { \log \left( 1 / \delta \right) + \left( \lambda - 1 \right) \log \left( 1 - 1 / \lambda \right) - \log \left( \lambda \right) } { \lambda - 1 } \right) . +$$ + +Problem formulation: We consider a distributed private learning setup comprising a set of $n$ clients, where the ith client has a data point $d _ { i }$ drawn from a universe $\mathcal { X }$ for $i \in [ n ]$ ; see also Figure 1. Let $\mathcal { D } = ( d _ { 1 } , \ldots , d _ { n } )$ denote the entire training dataset. The clients are connected to an untrusted server in order to solve the following empirical risk minimization (ERM) problem + +$$ +\operatorname* { m i n } _ { \theta \in { \mathcal { C } } } { \Big ( } F ( \theta , { \mathcal { D } } ) : = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } f ( \theta , d _ { i } ) { \Big ) } , +$$ + +where $\mathcal { C } \subset \mathbb { R } ^ { d }$ is a closed convex set, and $f : \mathcal { C } \times \mathcal { D } \mathbb { R }$ is the loss function. Our goal is to construct a global learning model $\theta$ via stochastic gradient descent (SGD) while preserving privacy of individual data points in the training dataset $\mathcal { D }$ by providing strong DP guarantees. + +We revisit the CLDP-SGD algorithm presented in $\pmb { \Vert 2 7 }$ and described in Algorithm $\bar { \mathbf { \xi } } _ { 1 }$ to solve the ERM $\boxed { 2 }$ . In each step of CLDP-SGD, we choose uniformly at random a set $\mathcal { U } _ { t }$ of $\boldsymbol { k } \quad \le \quad n$ clients out of $n$ clients. Each client $i \in \ U _ { t }$ computes and clips the $\ell _ { p }$ -norm of the gradient $\nabla _ { { \boldsymbol { \theta } } _ { t } } f \left( { \boldsymbol { \theta } } _ { t } , d _ { i } \right)$ to apply the LDP mechanism $\mathcal { R } _ { p }$ where $\mathcal { R } _ { p } \ : \ B _ { p _ { . } } ^ { d } \ \to \ \{ 0 , 1 \} ^ { \bar { b } }$ is an $\epsilon _ { \mathrm { 0 } }$ -LDP mechanism when inputs come from an $\ell _ { p }$ -norm ball. In $\textstyle \left\| 2 7 \right\|$ , the authors proposed different $\epsilon _ { \mathrm { 0 } }$ -LDP mechanisms for general $\ell _ { p }$ -norm balls. After that, the shuffler randomly permutes the received $k$ gradients $\{ \mathcal { R } _ { p } \left( \tilde { \mathbf { g } } _ { t } \left( d _ { i } \right) \right) \} _ { i \in \mathcal { U } _ { t } }$ and sends them to the server. Finally, + +Algorithm 1 Acldp: CLDP-SGD + +Input: Datasets $\mathcal { D } = ( d _ { 1 } , \ldots , d _ { n } ) .$ , LDP privacy parameter $\epsilon _ { \mathrm { 0 } }$ gradient norm bound $C$ , and learning rate schedule $\left\{ \eta _ { t } \right\}$ . + +1: Initialize: $\theta _ { 0 } \in { \mathcal { C } }$ +2: for $t \in [ T ]$ do +3: Client sampling: A random set $\mathcal { U } _ { t }$ of $k$ clients is chosen. +4: for clients $i \in \mathcal { U } _ { t }$ do +5: Compute gradient: $\mathbf { g } _ { t } \left( d _ { i } \right) \gets \nabla _ { \theta _ { t } } f \left( \theta _ { t } , d _ { i } \right)$ +6: Clip gradient: g˜t (di) gt (di) / max n 1, kgt(di)kpC o +7: Client $i$ sends $\mathcal { R } _ { p } \left( \tilde { \bf g } _ { t } \left( d _ { i } \right) \right)$ to the shuffler. +8: end for +9: Shuffling: The shuffler sends random permutation of +$\{ \mathcal { R } _ { p } \left( \tilde { \bf g } _ { t } \left( d _ { i } \right) \right) : i \in \mathcal { U } _ { t } \}$ to the server. +10: Aggregate: $\begin{array} { r } { \overline { { \bf g } } _ { t } \frac { 1 } { k } \sum _ { i \in \mathcal { U } _ { t } } \mathcal { R } _ { p } ( \tilde { \bf g } _ { t } ( d _ { i } ) ) } \end{array}$ +11: Descent Step: $\theta _ { t + 1 } \prod _ { \mathcal { C } } ( \theta _ { t } - \eta _ { t } \overline { { \mathbf { g } } } _ { t } )$ , where $\Pi _ { c }$ is the +projection operator onto the set $\mathcal { C }$ . + +# 12: end for + +Output: The model $\theta _ { T }$ and the privacy parameters $\epsilon , \delta$ + +the server takes the average of the received gradients and updates the parameter vector. Our main contribution in this work is to present a stronger privacy analysis of the CLDP-SGD algorithm by characterizing the RDP of the sub-sampled shuffle model. + +# 3 Main Results + +In this section, we present our main results. First, we characterize the RDP of the subsampled shuffle mechanism by presenting an upper bound in Theorem 1 and a lower bound in Theorem $\dot { 2 }$ We then present the privacy-convergence trade-offs of the CLDP-SGD Algorithm in Theorem 3. + +Consider an arbitrary $\epsilon _ { \mathrm { 0 } }$ -LDP mechanism $\mathcal { R }$ , whose range is a discrete set $[ B ] = \{ 1 , \dots , B \}$ for some $B \in \mathbb { N } : = \{ 1 , 2 , 3 , . . . \}$ . Here, $[ B ]$ could be the whole of $\mathbb { N }$ . Let $\mathcal { M } ( \mathcal { D } )$ be a subsampled shuffle mechanism defined as follows: First subsample $k \leq n$ clients of the $n$ clients (without replacement), where $\textstyle \gamma = { \frac { k } { n } }$ denotes the sampling parameter. Each client $i$ out of the $k$ selected clients applies $\mathcal { R }$ on $d _ { i }$ and sends $\mathcal { R } ( d _ { i } )$ to the shuffler,5 who randomly permutes the received $k$ inputs and outputs the result. To formalize this, let $\mathcal { H } _ { k } : \overline { { \mathcal { V } } } ^ { k } \to \mathcal { V } ^ { k }$ denote the shuffling operation that takes $k$ inputs and outputs their uniformly random permutation. Let $\operatorname { s a m p } _ { k } ^ { n } : \mathcal { X } ^ { n } \to ^ { \mathbf { \bar { \alpha } } } \mathcal { X } ^ { k }$ denote the sampling operation for choosing a random subset of $k$ elements from a set of $n$ elements. We define the subsampled-shuffle mechanism as + +$$ +\mathcal { M } \left( \mathcal { D } \right) : = \mathcal { H } _ { k } \circ \operatorname { s a m p } _ { k } ^ { n } \left( \mathcal { R } \left( d _ { 1 } \right) , \dotsc , \mathcal { R } \left( d _ { n } \right) \right) . +$$ + +Observe that each iteration of Algorithm 1 can be represented as an output of the subsampled shuffle mechanism $\mathcal { M }$ . Thus, to analyze the privacy of Algorithm $\bigstar$ it is sufficient to analyze the privacy of a sequence of identical $T$ subsampled shuffle mechanisms, and then apply composition theorems. + +Histogram notation. It will be useful to define the following notation. Since the output of $\mathcal { H } _ { k }$ is a random permutation of the $k$ outputs of $\mathcal { R }$ (subsampling is not important here), the server cannot associate the $k$ messages to the clients; and the only information it can use from the messages is the histogram, i.e., the number of messages that give any particular output in $[ B ]$ . We define a set $\mathcal { A } _ { B } ^ { k }$ as + +$$ +\mathcal { A } _ { B } ^ { k } = \bigg \{ h = ( h _ { 1 } , \ldots , h _ { B } ) : \sum _ { j = 1 } ^ { B } h _ { j } = k \bigg \} , +$$ + +to denote the set of all possible histograms of the output of the shuffler with $k$ inputs. Therefore, we can assume, without loss of generality (w.l.o.g.), that the output of $\mathcal { M }$ is a distribution over $\mathcal { A } _ { B } ^ { k }$ . + +Our main results for the RDP of the subsampled shuffled mechanism (defined in $( 3 )$ ) are given below. Our first result provides an upper bound (stated in Theorem 1 and proved in Section $. 5 )$ and the second result provides a lower bound (stated in Theorem 2 and proved in Appendix D) + +Theorem 1 (Upper Bound). For any $n \in \mathbb { N } , k \leq n , \epsilon _ { 0 } \geq 0 ,$ and any integer $\lambda \geq 2$ , the RDP of the subsampled shuffle mechanism $\mathcal { M }$ (defined in $\textcircled { 3 }$ ) is upper-bounded by + +$\epsilon ( \lambda ) \leq \frac { 1 } { \lambda - 1 } \log \left( 1 + 4 \binom { \lambda } { 2 } \gamma ^ { 2 } \frac { \left( e ^ { \epsilon _ { 0 } } - 1 \right) ^ { 2 } } { \bar { k } e ^ { \epsilon _ { 0 } } } + \sum _ { j = 3 } ^ { \lambda } \binom { \lambda } { j } \gamma ^ { j } j \Gamma \left( j / 2 \right) \left( \frac { 2 \left( e ^ { 2 \epsilon _ { 0 } } - 1 \right) ^ { 2 } } { \bar { k } e ^ { 2 \epsilon _ { 0 } } } \right) ^ { j / 2 } + \Upsilon \right) ,$ where $\begin{array} { r } { \overline { { k } } = \lfloor \frac { k - 1 } { 2 e ^ { \epsilon _ { 0 } } } \rfloor + 1 , \gamma = \frac { k } { n } } \end{array}$ , and $\begin{array} { r } { \Gamma \left( z \right) = \int _ { 0 } ^ { \infty } x ^ { z - 1 } e ^ { - x } d x } \end{array}$ is the Gamma function. The term $\Upsilon$ is given by $\begin{array} { r } { \Upsilon = \left( \left( 1 + \gamma \frac { e ^ { 2 \epsilon _ { 0 } } - 1 } { e ^ { \epsilon _ { 0 } } } \right) ^ { \lambda } - 1 - \lambda \gamma \frac { e ^ { 2 \epsilon _ { 0 } } - 1 } { e ^ { \epsilon _ { 0 } } } \right) e ^ { - \frac { k - 1 } { 8 e ^ { \epsilon _ { 0 } } } } . } \end{array}$ . + +Theorem 2 (Lower Bound). For any $n \in \mathbb { N } , k \leq n , \epsilon _ { 0 } \geq 0$ , and any integer $\lambda \geq 2$ , the RDP of the subsampled shuffle mechanism $\mathcal { M }$ (defined in $\textcircled { 3 }$ ) is lower-bounded by + +$$ +\mathrm { \Sigma } ( \lambda ) \geq \frac { 1 } { \lambda - 1 } \log \left( 1 + { \binom { \lambda } { 2 } } \gamma ^ { 2 } \frac { \left( e ^ { \epsilon _ { 0 } } - 1 \right) ^ { 2 } } { k e ^ { \epsilon _ { 0 } } } + \sum _ { j = 3 } ^ { \lambda } { \binom { \lambda } { j } } \gamma ^ { j } \left( \frac { \left( e ^ { 2 \epsilon _ { 0 } } - 1 \right) } { k e ^ { \epsilon _ { 0 } } } \right) ^ { j } \mathbb { E } \left( m - \frac { k } { e ^ { \epsilon _ { 0 } } + 1 } \right) ^ { j } \right) , +$$ + +where expectation is taken w.r.t. the binomial r.v. $m \sim B i n \left( k , p \right)$ with parameter $\begin{array} { r } { p = \frac { 1 } { e ^ { \epsilon _ { 0 } } + 1 } } \end{array}$ . + +Our CLDP-SGD algorithm and its privacy-convergence trade-offs (stated in Theorem 3 below) are given for a general local randomizer $\mathcal { R } _ { p }$ (whose inputs comes from an $\ell _ { p }$ -ball for any $p \in [ 1 , \infty ] )$ that satisfies the following conditions: (i) The randomized mechanism $\mathcal { R } _ { p }$ is an $\scriptstyle \epsilon _ { 0 } - \mathrm { \mathrm { L D P } }$ mechanism. (ii) The randomized mechanism $\mathcal { R } _ { p }$ is unbiased, i.e., $\vec { \mathfrak { L } } [ \mathcal { R } _ { p } ( \mathbf { x } ) | \mathbf { x } | = \mathbf { x }$ for all $\mathbf { x } \in B _ { p } ( a )$ , where $a$ is the radius of the ball $B _ { p }$ . (iii) The output of the randomized mechanism $\mathcal { R } _ { p }$ can be represented using $B \in$ $\mathbb { N } ^ { + }$ bits. (iv) The randomized $\mathcal { R } _ { p }$ has a bounded variance: $\begin{array} { r } { \operatorname* { s u p } _ { \mathbf { x } \in \mathcal { B } _ { p } ( a ) } \mathbb { E } \| \mathcal { R } _ { p } \left( \mathbf { x } \right) - \mathbf { x } \| _ { 2 } ^ { 2 } \leq G _ { p } ^ { 2 } ( a ) } \end{array}$ , where $G _ { p } ^ { 2 }$ is a function from $\mathbb { R } ^ { + }$ to $\mathbb { R } ^ { + }$ . + +Girgis et al. $\textstyle \left\| 2 7 \right\|$ proposed unbiased $\epsilon _ { 0 }$ -LDP mechanisms $\mathcal { R } _ { p }$ for several values of norms $p \in [ 1 , \infty ]$ that require $b = \mathcal { O } \left( \log \left( d \right) \right)$ bits of communication and satisfy the above conditions. In this paper, achieving communication efficiency is not our goal (though we also achieve that since the $\epsilon _ { 0 } { \mathrm { - L D P } }$ mechanism $\mathcal { R } _ { p }$ that we use takes values in a discrete set), as our main focus is on analyzing the RDP of the subsampled shuffle mechanism. If we use the $\epsilon _ { 0 }$ -LDP mechanism $\mathcal { R } _ { p }$ from $\dot { \underline { 1 2 7 } }$ , we would also get similar gains in communication as were obtained in $\lVert 2 7 \rVert$ . + +The privacy-convergence trade-off of our algorithm $\mathcal { A } _ { \mathrm { c l d p } }$ is given below. + +![](images/7e175b7ca5f6830d336d8ae1c400c95ba5217b0196bb789ebee85750d10f85c6.jpg) +(a) Approx. DP as a function of $T$ (b) Approx. DP as a function of $T$ (c) Approx. DP as a function of $n$ for $\epsilon _ { 0 } = 2 $ , $\gamma = 0 . 0 0 1$ , $n = 1 0 ^ { 6 }$ for $\epsilon _ { 0 } = 1$ , $\gamma = 0 . 0 0 1$ , $n = 1 0 ^ { 7 }$ for $\epsilon _ { 0 } = 2 $ , $\gamma n = 1 0 ^ { 3 }$ , $T = 1 0 ^ { 5 }$ +Figure 2: Comparison of several bounds on the Approximate $( \epsilon , \delta )$ -DP for composition of a sequence of subsampled shuffle mechanisms for $\delta = 1 0 ^ { - 8 }$ : (i) Approximate DP obtained from our upper bound on the RDP in Theorem $\perp$ (blue); (ii) Approximate DP obtained from our lower bound on the RDP in Theorem $\bigstar$ (red); (iii) Approximate DP obtained from the upper bound on the RDP given in $\left[ \left[ 2 9 \right] \right]$ with RDP amplification by subsampling from $\pmb { \boxed { 4 3 } }$ (black); and (iv) Applying the strong composition theorem $\textcircled { 1 3 4 } |$ after getting the Approximate DP of the shuffled model given in $\pm$ with subsampling $\mathbf { \widehat { | 4 2 | } }$ (magenta). + +Theorem 3 (Privacy-Convergence tradeoffs). Let the set $\mathcal { C }$ be convex with diameter $D$ and the function $f ( \theta ; . ) : \mathcal { C } \times \mathcal { D } \mathbb { R }$ be convex and $L$ -Lipschitz continuous with respect to the $\ell _ { g }$ -norm, which is the dual of the $\ell _ { p }$ -norm. Let $\theta ^ { * } = \arg \operatorname* { m i n } _ { \theta \in { \mathcal { C } } } F \left( \theta \right)$ denote the minimizer of the problem $\textcircled { 2 }$ For $\textstyle \gamma = { \frac { k } { n } }$ , if we run Algorithm $\mathcal { A } _ { \mathrm { c l d p } }$ over $T$ iterations, then we have + +$$ +\epsilon = \operatorname* { m i n } _ { \lambda } \left( T \epsilon \left( \lambda \right) + \frac { \log \left( 1 / \delta \right) + \left( \lambda \stackrel { \star } { - } 1 \right) \log \left( 1 - 1 / \lambda \right) - \log \left( \lambda \right) } { \lambda - 1 } \right) , +$$ + +where $\epsilon \left( \lambda \right)$ is the RDP of the subsampled shuffle mechanism given in Theorem 1. + +2. Convergence: If we run $\mathcal { A } _ { \mathrm { c l d p } }$ with $\begin{array} { r } { \eta _ { t } = \frac { D } { G \sqrt { t } } } \end{array}$ , where $\begin{array} { r } { G ^ { 2 } = \operatorname* { m a x } \lbrace d ^ { 1 - \frac { 2 } { p } } , 1 \rbrace L ^ { 2 } + \frac { G _ { p } ^ { 2 } ( L ) } { \gamma n } } \end{array}$ , we get $\mathbb { E } \left[ F \left( \theta _ { T } \right) \right] - F \left( \theta ^ { * } \right) \leq \mathcal { O } \left( \frac { D G \log ( T ) } { \sqrt { T } } \right) .$ + +The proof outline of Theorem $3$ is as follows: Note that $\mathcal { A } _ { \mathrm { c l d p } }$ is an iterative algorithm, where in each iteration we use the subsampled shuffle mechanism as defined in $\textcircled{3}$ , for which we have computed the RDP guarantees in Theorem $\bigtriangledown$ Now, for the privacy analysis of $\mathcal { A } _ { \mathrm { c l d p } }$ , we use the adaptive composition theorem from $\textcircled { 1 3 7 }$ Proposition 1] and then use the RDP to DP conversion given in Lemma $^ { 1 . }$ For the convergence analysis, we use a standard non-private SGD convergence result and compute the required parameters for that. See Appendix F for a complete proof of Theorem 3. + +Remark 1. Note that our convergence bound is affected by the variance of the $\epsilon _ { \mathrm { 0 } }$ -LDP mechanism $\mathcal { R } _ { p }$ . For example, when $f$ is $L$ -Lipschitz continuous w.r.t. the $\ell _ { 2 }$ -norm, we can use the LDP mechanism $\mathcal { R } _ { 2 }$ proposed in $\boxed { 1 1 }$ that has variance $\begin{array} { r } { G _ { 2 } ^ { 2 } ( L ) = 1 4 L ^ { 2 } d \big ( \frac { e ^ { \epsilon _ { 0 } } + 1 } { e ^ { \epsilon _ { 0 } } - 1 } \big ) ^ { 2 } } \end{array}$ ; and when $f$ is $L$ -Lipschitz continuous w.r.t. the $\ell _ { 1 }$ -norm or $\ell _ { \infty }$ -norm, we can use the LDP mechanisms $\mathcal { R } _ { \infty }$ or $\mathcal { R } _ { 1 }$ , respectively, proposed in $\lVert 2 7 \rVert$ 1 that have variances $\begin{array} { r } { G _ { \infty } ^ { 2 } ( L ) = L ^ { 2 } d ^ { 2 } \big ( \frac { e ^ { \epsilon _ { 0 } } + 1 } { e ^ { \epsilon _ { 0 } } - 1 } \big ) ^ { 2 } } \end{array}$ 2 and G21(L) = L2d e✏0 +1e✏0 1 2 , respectively. By plugging these variances $G _ { p } ^ { 2 } ( L )$ (for $p = 1 , 2 , \infty )$ into Theorem $3 .$ we get the convergence rate of the $L$ -Lipschitz continuous loss function w.r.t. the $\ell _ { p }$ -norm (for $p \equiv \infty , 2 , 1 $ ). + +Remark 2. The privacy parameter in $( 5 )$ is not in a closed form expression and could be obtained by solving an optimization problem. However, we numerically compute it for several interesting regimes of parameters in our numerical experiments; see Section 4 for more details. + +# 4 Numerical Results + +In this section, we present numerical experiments to show the performance of our bounds on RDP of the subsampled shuffle mechanism and its usage for getting approximate DP of Algorithm 1 for training machine learning models. + +Composition of a sequence of subsampled shuffle models: In Figure 2, we plot several bounds on the approximate $( \epsilon , \delta )$ -DP for a composition of $T$ mechanisms $( \mathcal { M } _ { 1 } , \ldots , \mathcal { M } _ { T } )$ , where $\mathcal { M } _ { t }$ is a subsampled shuffle mechanism for $t \in [ T ]$ . In all our experiments reported in Figure $2 .$ we fix $\delta = 1 0 ^ { - 8 }$ . We observe that our new bound on the RDP of the subsampled shuffle mechanism achieves a significant saving in total privacy $\epsilon$ compared to the state-of-the-art. For example, we save a factor of $1 4 \times$ compared to the bound on DP $\pm \overbrace { \lVert 2 4 \rVert }$ with strong composition theorem $\textcircled { 1 3 4 } \textcircled { 1 }$ and $2 . 5 \times$ compared to the bound on the RDP given in $\lVert 2 9 \rVert$ with subsampled RDP $\textcircled { 1 4 3 } |$ in computing the overall privacy parameter $\epsilon$ for number of iterations $\bar { T } = 1 0 ^ { 5 }$ , subsampling parameter $\gamma = 0 . 0 0 1$ , LDP parameter $\epsilon _ { 0 } = 2 $ , and number of clients $n = 1 0 ^ { 6 }$ . We observe in Figure $\boxed { 2 \mathrm { b } }$ that the bound given in $\pmb { \mathbb { Z 4 } }$ with the strong composition theorem $\pm$ behaves better than the bound on the RDP $\pmb { \bigtriangledown }$ with subsampled RDP bound $\boxed { \boxed { 4 3 } }$ when the number of subsampled clients per iteration is equal to $k = \gamma n = \mathrm { \bar { 1 0 ^ { 4 } } }$ ; however, our bound beats both of them.6 In Figure $\boxed { 2 \mathrm { c } }$ we fix the number of subsampled clients per iteration to be $k = \gamma n = 1 0 ^ { 3 }$ , and hence, the subsampling parameter $\gamma$ varies with $n$ . + +Distributed private learning: We numerically evaluate the proposed privacy-learning performance on training machine learning models. We consider the standard MNIST handwritten digit dataset that has 60, 000 training images and 10, 000 test images. We train a simple neural network that was also used in $\pm \pm \pm \textcircled { 3 9 }$ and described in Table $\bigstar$ This model has $d = 1 3$ , 170 parameters and achieves an accuracy of $9 9 \%$ for non-private, uncompressed vanilla SGD. We assume that we have $n = 6 0 , 0 0 0$ clients, where each client has one sample. At each step of the Algorithm $\bigstar \bigstar$ we choose uniformly at random 10, 000 clients, where each client clips the $\ell _ { \infty }$ -norm of the gradient with clipping parameter $C = 1 / 1 0 0$ and applies the $\mathcal { R } _ { \infty }$ $\epsilon _ { \mathrm { 0 } }$ -LDP mechanism proposed in $\mathbb { \ Z } \mathbb { \ Z }$ with $\epsilon _ { 0 } = 1 . 5$ . We run Algorithm $^ 1$ with $\bar { \delta } = 1 0 ^ { - 5 }$ for 200 epochs, with learning rate $\eta = 0 . 3$ for the first 70 epochs, and + +then decrease it to 0.18 in the remaining epochs. + +Table 1: Model architecture for MNIST + +
LayerParameters
Convolution Max-Pooling Convolution Max-Pooling Fully connected Softmax16 filters of 8 × 8,Stride 2 2×2 32 filters of 4 × 4, Stride 2 2×2 32 units 10 units
+ +![](images/e2f0e8974b2af63fd0a36f8a233db485da410d667a0ad9a2b433a98053b80b94.jpg) +Figure 3: Privacy-Utility trade-offs on the MNIST dataset with $\ell _ { \infty }$ -norm clipping. + +Figure 3 plots the mean and the standard deviation of privacy-accuracy trade-offs averaged over 10 runs. For our privacy analysis, the total privacy budget is computed by optimizing over RDP order $\lambda$ using our upper bound given in Theorem $^ { 1 . }$ For privacy analysis of $\dot { \lVert 2 4 \rVert }$ , we first compute the privacy amplification by shuffling numerically given in $\lVert 2 4 \rVert$ ; then we compute its privacy obtained when amplified via subsampling $\lVert \overline { { 4 2 } } \rVert$ ; and finally we use the strong composition theorem $\overleftarrow { \mathbb { B } 4 }$ to obtain the central privacy parameter $\epsilon$ . + +We observe that we achieve an accuracy of $8 0 \% ( \pm 1 . 8 )$ with a total privacy budget of $\epsilon = 1 . 4$ using our new privacy analysis, whereas, $\dot { \lVert 2 4 \rVert }$ achieves an accuracy of only $7 0 . 7 \% ( \pm 2 . 1 )$ with the same privacy budget of $\epsilon = 1 . 4$ using the standard composition theorems. Furthermore, we can see that we achieves accuracy $9 0 \% ( \pm 0 . { \bar { 5 } } )$ with total privacy budget $\epsilon = 2 . 9 1$ using our new privacy analysis, whereas, $\pmb { \left[ 2 4 \right] }$ (together with the standard strong composition theorem) achieves the same accuracy with a total privacy budget of $\epsilon = 4 . 8 2$ . + +# 5 Proof of Theorem 1: Upper Bound + +For any dataset $\mathcal { D } _ { k } = ( d _ { 1 } , \ldots , d _ { k } ) \in \mathcal { X } ^ { k }$ containing of $k$ data points, we define a shuffle mechanism $\mathcal { M } _ { s h } ( \mathcal { D } _ { k } )$ as follows: + +$$ +\mathcal { M } _ { s h } ( \mathcal { D } _ { k } ) = \mathcal { H } _ { k } \left( \mathcal { R } \left( d _ { 1 } \right) , \ldots , \mathcal { R } \left( d _ { k } \right) \right) , +$$ + +where $\mathcal { H } _ { k }$ takes $k$ inputs and outputs a uniformly random permutation of them. Recall from $\textcircled{3}$ , for any dataset $\mathcal { D } _ { n } = ( \bar { d } _ { 1 } , \ldots , d _ { n } ) \in \mathcal { X } ^ { n }$ containing $n$ data points, the subsampled-shuffle mechanism is defined as $\mathcal { M } \left( \mathcal { D } \right) : = \mathcal { H } _ { k } \circ \mathrm { s a m p } _ { k } ^ { n }$ $( \mathcal { R } \left( d _ { 1 } \right) , \ldots , \mathcal { R } \left( d _ { n } \right) )$ . + +The proof of Theorem 1 consists of two steps. First, we bound the ternary- $\cdot | \chi | ^ { \alpha }$ -DP of the shuffle mechanism $\mathcal { M } _ { s h }$ (see Theorem $\bigstar$ , which is the main technical contribution in this proof. Then, using this, we bound the RDP of the subsampled shuffle mechanism $\mathcal { M }$ . + +Theorem 4 $\zeta$ -ternary- $| \chi | ^ { \alpha }$ -DP of the shuffle mechanism $\mathcal { M } _ { s h }$ ). For any integer $k \geq 2$ , $\epsilon _ { 0 } > 0$ , and all $\alpha \geq 2$ , the $\zeta$ -ternary- $\cdot | \chi | ^ { \alpha }$ -DP of the shuffle mechanism $\mathcal { M } _ { s h }$ is bounded by: + +$$ +\zeta \left( \alpha \right) ^ { \alpha } \leq \left\{ \begin{array} { l l } { 4 \frac { \left( e ^ { \epsilon _ { 0 } } - 1 \right) ^ { 2 } } { \bar { k } e ^ { \epsilon _ { 0 } } } + ( e ^ { \epsilon _ { 0 } } - e ^ { - \epsilon _ { 0 } } ) ^ { \alpha } e ^ { - \frac { k - 1 } { 8 e ^ { \epsilon _ { 0 } } } } } & { i f \alpha = 2 , } \\ { \alpha \Gamma \left( \alpha / 2 \right) \left( \frac { 2 \left( e ^ { 2 \epsilon _ { 0 } } - 1 \right) ^ { 2 } } { \bar { k } e ^ { 2 \epsilon _ { 0 } } } \right) ^ { \alpha / 2 } + ( e ^ { \epsilon _ { 0 } } - e ^ { - \epsilon _ { 0 } } ) ^ { \alpha } e ^ { - \frac { k - 1 } { 8 e ^ { \epsilon _ { 0 } } } } } & { o t h e r w i s e , } \end{array} \right. +$$ + +where $\begin{array} { r } { \overline { { k } } = \lfloor \frac { k - 1 } { 2 e ^ { \epsilon _ { 0 } } } \rfloor + 1 } \end{array}$ and $\begin{array} { r } { \Gamma \left( z \right) = \int _ { 0 } ^ { \infty } x ^ { z - 1 } e ^ { - x } d x } \end{array}$ is the Gamma function. + +Theorem 4 is one of the core technical results of this paper, and we prove it in Section 6. + +It was shown in $\boxed { \boxed { 4 3 } }$ Proposition 16] that if a mechanism obeys $\zeta$ -ternary- $\cdot | \chi | ^ { \alpha }$ -DP, then its subsampled version (with subsampling parameter $\gamma$ ) will obey $\gamma \zeta .$ -ternary- $| \chi | ^ { \alpha }$ -DP. Using that result, the authors then bounded the RDP of the subsampled mechanism in $\underline { { \bar { 1 4 3 } } } \underline { { \bar { 1 } } }$ Eq. (9)]. Adapting that result to our setting, we have the following lemma. + +Lemma 2 (From $\zeta$ -ternary- $| \chi | ^ { \alpha }$ -DP to subsampled RDP). Suppose the shuffle mechanism $\mathcal { M } _ { s h }$ obeys $\zeta$ -ternary- $| \chi | ^ { \alpha }$ -DP. For any $\lambda \geq 2 , k \leq n$ , RDP of the subsampled shuffle mechanism $\mathcal { M }$ (with subsampling parameter $\gamma = k / n ,$ ) is bounded by: $\begin{array} { r } { \epsilon ( \lambda ) \leq \frac { 1 } { \lambda - 1 } \log \left( 1 + \sum _ { \alpha = 2 } ^ { \lambda } \binom { \lambda } { \alpha } \gamma ^ { \alpha } \zeta ( \alpha ) ^ { \alpha } \right) } \end{array}$ . + +Lemma 2 can be seen as a corollary to $\boxed { \ 4 3 }$ Proposition 16 and Eq. (9)]. However, for completeness, we prove it in Appendix $\boxed { \mathrm { E . 1 } }$ Substituting the bound on $\zeta ( \alpha )$ from Theorem 4 into Lemma 2 together with some algebraic manipulation gives proves Theorem 1; see Appendix $\mathrm { E } . 2$ for details. + +# 6 Proof of Theorem $\mathbf { 4 } ;$ Ternary $| \chi | ^ { \alpha }$ -DP of the Shuffle Model + +The proof has two main steps. In the first step, we reduce the problem of deriving ternary divergence for arbitrary neighboring datasets to the problem of deriving the ternary divergence for specific neighboring datasets, $\mathcal { D } \stackrel { - } { \sim } \mathcal { D } ^ { \prime } \sim \mathcal { D } ^ { \prime \prime }$ , where all elements in $\mathcal { D }$ are the same and $\mathcal { D } ^ { \prime } , \mathcal { D } ^ { \prime \prime }$ differ from $\mathcal { D }$ in one entry. In the second step, we derive the ternary divergence for the special neighboring datasets. + +The specific neighboring datasets to which we reduce our general problem has the following form: + +$$ +\begin{array} { r } { \mathcal { D } _ { \mathrm { s a m e } } ^ { m } = \{ ( \mathcal { D } _ { m } , \mathcal { D } _ { m } ^ { \prime } , \mathcal { D } _ { m } ^ { \prime \prime } ) : \mathcal { D } _ { m } = ( d , \ldots , d , d ) \in \mathcal { X } ^ { m } , \mathcal { D } _ { m } ^ { \prime } = ( d , \ldots , d , d ^ { \prime } ) \in \mathcal { X } ^ { m } , \mathrm { ~ a ~ n ~ d ~ } \mathcal { X } = 1 , \ldots , d , 1 \} , } \\ { \mathcal { D } _ { m } ^ { \prime \prime } = ( d , \ldots , d , d ^ { \prime \prime } ) \in \mathcal { X } ^ { m } , \mathrm { ~ w h e r e ~ } d , d ^ { \prime } , d ^ { \prime \prime } \in \mathcal { X } \} } \end{array} +$$ + +Consider arbitrary neighboring datasets $\mathcal { D } = \left( d _ { 1 } , \ldots , d _ { k - 1 } , d _ { k } \right)$ , $\mathcal { D } ^ { \prime } = ( d _ { 1 } , \ldots , d _ { k - 1 } , d _ { k } ^ { \prime } )$ , and $\mathcal { D } ^ { \prime \prime } = ( d _ { 1 } , \dots , d _ { k - 1 } ^ { - } , d _ { k } ^ { \overline { { \prime } } \prime } )$ , each having $k$ elements. For any $m \in \{ 0 , \ldots , k - 1 \}$ , we define new neighboring dataeach having = (d00k , . . . , d00k , dk), D0(k)m+1 = (d00k , . . . , d00k , d0k), and D00(k)m+1 $\mathcal { D } _ { m + 1 } ^ { \prime \prime ( k ) } = ( d _ { k } ^ { \prime \prime } , \dots , d _ { k } ^ { \prime \prime } )$ $m + 1$ $( \mathcal { D } _ { m + 1 } ^ { \prime \prime ( k ) } , \mathcal { D } _ { m + 1 } ^ { \prime ( k ) } , \mathcal { D } _ { m + 1 } ^ { ( k ) } ) \in \mathcal { D } _ { \mathrm { s a m e } } ^ { m }$ + +The first step of the proof is given in the following theorem. + +Theorem 5 (Reduction to the Special Case). Let $\begin{array} { r } { q = \frac { 1 } { e ^ { \epsilon _ { 0 } } } } \end{array}$ . We have: + +$$ +\begin{array} { r l } & { \mathbb { E } _ { h \sim \mathcal { M } _ { s h } ( \mathcal { D } ^ { \prime \prime } ) } \left[ \left| \frac { \mathcal { M } _ { s h } ( \mathcal { D } ) ( h ) - \mathcal { M } _ { s h } ( \mathcal { D } ^ { \prime } ) ( h ) } { \mathcal { M } _ { s h } ( \mathcal { D } ^ { \prime \prime } ) ( h ) } \right| ^ { \alpha } \right] } \\ & { \qquad \leq \mathbb { E } _ { m \sim \mathrm { B i n } ( k - 1 , q ) } \left[ \mathbb { E } _ { h \sim \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { \prime \prime ( k ) } ) } \left[ \left| \frac { \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { ( k ) } ) ( h ) - \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { \prime ( k ) } ) ( h ) } { \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { \prime \prime ( k ) } ) ( h ) } \right| ^ { \alpha } \right] \right] . } \end{array} +$$ + +We know (by Chernoff bound) that the binomial r.v. is concentrated around its mean, which implies that the terms in the RHS of $\textcircled { 9 }$ that correspond to $m \ < \ ( 1 - \tau ) q ( k - 1 )$ (we will take $\tau ~ = ~ 1 / 2 )$ will contribute in a negligible amount. Then we show that $E _ { m } : =$ $\mathbb { E } _ { \boldsymbol { h } \sim \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { \prime \prime ( k ) } ) } \left[ \left| \frac { \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { ( k ) } ) ( \boldsymbol { h } ) - \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { \prime ( k ) } ) ( \boldsymbol { h } ) } { \mathcal { M } _ { s h } ( \mathcal { D } _ { m + 1 } ^ { \prime \prime ( k ) } ) ( \underline { { h } } ) } \right| ^ { \alpha } \right]$ is a non-increasing function of $m$ . These observation together imply that the RHS in $( 9 )$ is approximately equal to $E _ { ( 1 - \tau ) q ( k - 1 ) }$ . + +Since $E _ { m }$ is precisely what is required to bound the ternary DP for the specific neighboring datasets, we have reduced the problem of computing the ternary DP for arbitrary neighboring datasets to the problem of computing ternary DP for specific neighboring datasets. The second step of the proof bounds $E _ { ( 1 - \tau ) q ( n - 1 ) }$ , which follows from the result below that holds for any $m \in \mathbb { N }$ . + +$[ | \chi | ^ { \alpha }$ $m \in \mathbb { N }$ $\alpha \geq 2$ $( \mathcal { D } _ { m } ^ { \prime \prime } , \mathcal { D } _ { m } ^ { \prime } , \mathcal { D } _ { m } ) \in \mathcal { D } _ { s a m e } ^ { m }$ + +$$ +\mathbb { E } _ { h \sim M _ { s h } ( \mathcal { D } _ { m } ) } \left[ \left| \frac { \mathcal { M } _ { s h } ( \mathcal { D } _ { m } ^ { \prime } ) ( h ) - \mathcal { M } _ { s h } ( \mathcal { D } _ { m } ^ { \prime \prime } ) ( h ) } { \mathcal { M } _ { s h } ( \mathcal { D } _ { m } ) ( h ) } \right| ^ { \alpha } \right] \leq \left\{ \begin{array} { l l } { 4 \frac { ( e ^ { \epsilon _ { 0 } } - 1 ) ^ { 2 } } { m \epsilon ^ { \epsilon _ { 0 } } } } & { i f \alpha = 2 , } \\ { \alpha \Gamma ( \alpha / 2 ) \left( \frac { 2 ( e ^ { z _ { 0 } } - 1 ) ^ { 2 } } { m e ^ { 2 z _ { 0 } } } \right) ^ { \alpha / 2 } } & { o t h e r w i s e . } \end{array} \right. +$$ + +Missing details of how Theorem $^ 4$ follows from Theorems 5, 6 can be found in Appendix C.4. + +Proof sketch of Theorem $\boxed { 5 }$ Let $\pmb { p } _ { i } , i \in [ k ] , \pmb { p } _ { k } ^ { \prime } , \pmb { p } _ { k } ^ { \prime \prime }$ denote the distributions of $\mathcal { R }$ when the input data point is $d _ { i } , d _ { k } ^ { \prime } , d _ { k } ^ { \prime \prime }$ , respectively. The main idea of the proof is the observation that each $\pmb { p } _ { i }$ can be written as a mixture distribution $\begin{array} { r } { \pmb { p } _ { i } = \frac { 1 } { e ^ { \epsilon _ { 0 } } } \pmb { p } _ { k } ^ { \prime \prime } + \left( 1 - \frac { 1 } { e ^ { \epsilon _ { 0 } } } \right) \tilde { \pmb { p } } _ { i } } \end{array}$ , where $\tilde { \pmb { p } } _ { i }$ is defined in terms of ${ \pmb p } _ { i } , { \pmb p } _ { k } ^ { \prime \prime }$ . So, instead of client $i \in [ k - 1 ]$ mapping its data point $d _ { i }$ according to $\mathbf { \nabla } _ { \pmb { p } _ { i } }$ , we can view it as the client $i$ maps $d _ { i }$ according to $\pmb { p } _ { k } ^ { \prime \prime }$ with probability (w.p.) $1 / e ^ { \epsilon _ { 0 } }$ and according to $\tilde { \pmb { p } } _ { i }$ w.p. $\left( 1 - 1 / e ^ { \epsilon _ { 0 } } \right)$ . As a result, the number of clients that sample from the distribution $\pmb { p } _ { k } ^ { \prime \prime }$ follows a binomial distribution $\mathrm { B i n } ( k - 1 , 1 / e ^ { \epsilon _ { 0 } } )$ . This allows us to write the distribution of $\mathcal { M } _ { s h }$ when clients map their data points according to $\pmb { p } _ { 1 } , \ldots , \pmb { p } _ { k } , \pmb { p } _ { k } ^ { \prime } , \pmb { p } _ { k } ^ { \prime \prime }$ as a convex combination of the distribution of $\mathcal { M }$ when clients map their data points according to $\tilde { { p } } _ { 1 } , \ldots , { p } _ { k - 1 } , { p } _ { k } , { p } _ { k } ^ { \prime } , { p } _ { k } ^ { \prime \prime }$ ; see Lemma $\boxed { 4 }$ Then using a joint convexity argument (see Lemma $\textcircled { 3 }$ , we write the ternary divergence between the original triple of distributions of $\mathcal { M } _ { s h }$ in terms of the same convex combination of the ternary divergence between the resulting triples of distributions of $\mathcal { M } _ { s h }$ as in Lemma $\bigstar$ Using a monotonicity argument (see Lemma 5), we can remove the effect of clients that do not sample from the distribution $p _ { k } ^ { \prime \prime }$ without decreasing the ternary divergence. By this chain of arguments, we have reduced the problem to the one involving the computation of ternary divergence only for the special form of neighboring datasets (as in Theorem 6), which proves Theorem $\bigtriangledown$ See Appendix C.1 for a complete proof. + +Proof sketch of Theorem $6 .$ Consider $( \mathcal { D } _ { m } ^ { \prime \prime } , \mathcal { D } _ { m } ^ { \prime } , \mathcal { D } _ { m } ) \in \mathcal { D } _ { \mathrm { s a m e } } ^ { m }$ as in the statement of Theorem First we observe that for any $\alpha \geq 1$ and any three distributions $p , q , r$ over the same domain, we can write $\begin{array} { r } { \mathbb { E } _ { r } \left[ \left| \frac { p - q } { r } \right| ^ { \alpha } \right] \leq 2 ^ { \alpha - 1 } \left( \mathbb { E } _ { r } \left[ \left| \frac { p } { r } - 1 \right| ^ { \alpha } \right] + \mathbb { E } _ { r } \left[ \left| \frac { q } { r } - 1 \right| ^ { \alpha } \right] \right) } \end{array}$ . This is a straight-forward application of the standard inequality $| x + y | ^ { \alpha } \leq 2 ^ { \alpha - 1 } ( | x | ^ { \alpha } + | y | ^ { \alpha } )$ which holds for all $x , y \in \mathbb { R }$ and $\alpha \geq 1$ . Now, by taking $p = \mathcal { M } _ { s h } ( \mathcal { D } _ { m } ^ { \prime } )$ , $q = \mathcal { M } _ { s h } ( \mathcal { D } _ { m } ^ { \prime \prime } )$ , and $r = \mathcal { M } _ { s h } ( \mathcal { D } _ { m } )$ , we reduce the problem of computing the ternary $| \chi | ^ { \alpha } .$ -divergence (which we need to bound) to the problem of computing the Pearson-Vajda divergence $| | \overline { { 4 3 } } | |$ , which we can write in terms of the $\alpha$ -th absolute moment of the r.v. $X : \mathcal { A } _ { B } ^ { m } \mathbb { R }$ , defined as $\begin{array} { r } { X ( \pmb { h } ) : = \big ( \frac { \mathcal { M } _ { s h } ( \mathcal { D } ^ { \prime } ) ( \pmb { h } ) } { \mathcal { M } _ { s h } ( \mathcal { D } _ { m } ) ( \pmb { h } ) } - 1 \big ) } \end{array}$ for all $\pmb { h } \in \mathcal { A } _ { B } ^ { m }$ (where $\mathcal { D } ^ { \prime } \in \{ \mathcal { D } _ { m } ^ { \prime } , \mathcal { D } _ { m } ^ { \prime \prime } \} )$ and distributed according to $X ( \pmb { h } ) \sim \mathcal { M } _ { s h } ( \mathcal { D } _ { m } ) ( \pmb { h } )$ . In $\mathbb { E 9 }$ , the authors have bounded the absolute moments of the r.v. $X ( h )$ by showing that $X ( h )$ is sub-Gaussian r.v. and using standard concentration results. See Appendix C.3 for a complete proof. + +# 7 Discussion + +In this paper, we analyzed the Rényi differential privacy of the subsampled shuffle model by bounding the ternary $| \chi | ^ { \alpha }$ -DP of the shuffle model. We numerically demonstrated the importance of our proposed bound, where we obtain a significant improvement over using the state-of-the-art in practical regimes. Furthermore, we used our privacy analysis to study the privacy-accuracy trade-offs on the MNIST dataset, where we obtained $9 0 \%$ accuracy with total privacy budget of $\epsilon = 2 . 9 1$ , which is an improvement over an analysis yielding 4.82, using standard strong composition theorem. + +Closing the gap (shown numerically) between our lower bound in Theorem $\bigstar$ and the achievable upper bound in Theorem $\triangledown$ is an important unresolved question. Another direction to explore would be to analyze the RDP of the subsampled shuffle model for different sub-sampling techniques such as Poisson subsampling $\underline { { \mathbb { G 5 } } } ] |$ , random check-in $\textcircled { 9 }$ , or client self-sampling $| \overline { { 3 0 } } \|$ . + +Societal Impact. Collaborative learning comes with significant societal risks of privacy violations, which is the main topic addressed in this paper. However, such learning is only as good as the data used for training, and if the data is not unbiased, this could lead to significant issues related to fairness and could also lead to societally undesirable outcomes. Such an issue is exacerbated when privacy is guaranteed on the data used for training, making a-priori fairness checks on data infeasible. This can be ameliorated by properly testing models finally obtained against fairness criteria and rejecting models that fail the test. This paper did not consider the issue of robustness to security, and this could also be an important societal issue in collaborative learning, where a small subset of users could insert malicious inputs to disrupt the learning process or worse bias the learned model covertly. This could also lead to negative outcomes. This issue of robustness to malicious participants has been studied in several papers, and incorporating this into the framework of the paper is an important future research topic. + +# Acknowledgment + +This work was supported in part by NSF grants #2007714 and #1955632 and a Google Faculty research award and an Amazon Research Award. + +# References + +[1] M. Abadi, A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC conference on computer and communications security, pages 308–318, 2016. +[2] N. Agarwal, A. T. Suresh, F. X. X. Yu, S. Kumar, and B. McMahan. cpsgd: Communicationefficient and differentially-private distributed sgd. In Advances in Neural Information Processing Systems, pages 7564–7575, 2018. +[3] S. Asoodeh, J. Liao, F. P. Calmon, O. Kosut, and L. Sankar. Three variants of differential privacy: Lossless conversion and applications. 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PMLR, 2019. \ No newline at end of file diff --git a/parse/train/SPrVNsXnGd/SPrVNsXnGd_content_list.json b/parse/train/SPrVNsXnGd/SPrVNsXnGd_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6b59ddad5f841ec639aaf6282e70a236795973b2 --- /dev/null +++ b/parse/train/SPrVNsXnGd/SPrVNsXnGd_content_list.json @@ -0,0 +1,1380 @@ +[ + { + "type": "text", + "text": "Rényi Differential Privacy of the Subsampled Shuffle Model in Distributed Learning ", + "text_level": 1, + "bbox": [ + 179, + 122, + 818, + 174 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Antonious M. Girgis UCLA amgirgis@g.ucla.edu ", + "bbox": [ + 196, + 227, + 359, + 270 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deepesh Data UCLA deepesh.data@gmail.com ", + "bbox": [ + 390, + 227, + 581, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Suhas Diggavi UCLA suhasdiggavi@ucla.edu ", + "bbox": [ + 611, + 227, + 794, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 304, + 535, + 320 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We study privacy in a distributed learning framework, where clients collaboratively build a learning model iteratively through interactions with a server from whom we need privacy. Motivated by stochastic optimization and the federated learning (FL) paradigm, we focus on the case where a small fraction of data samples are randomly sub-sampled in each round to participate in the learning process, which also enables privacy amplification. To obtain even stronger local privacy guarantees, we study this in the shuffle privacy model, where each client randomizes its response using a local differentially private (LDP) mechanism and the server only receives a random permutation (shuffle) of the clients’ responses without their association to each client. The principal result of this paper is a privacyoptimization performance trade-off for discrete randomization mechanisms in this sub-sampled shuffle privacy model. This is enabled through a new theoretical technique to analyze the Rényi Differential Privacy (RDP) of the sub-sampled shuffle model. We numerically demonstrate that, for important regimes, with composition our bound yields significant improvement in privacy guarantee over the state-of-the-art approximate Differential Privacy (DP) guarantee (with strong composition) for sub-sampled shuffled models. We also demonstrate numerically significant improvement in privacy-learning performance operating point using real data sets. Despite these advances an open question is to bridge the gap between lower and upper privacy bounds in our RDP analysis. ", + "bbox": [ + 232, + 337, + 766, + 613 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 641, + 310, + 657 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As learning moves towards the edge, there is a need to collaborate to build learning models1, such as in federated learning [36, 44, 33]. In this framework, the collaboration is typically mediated by a server. In particular, we want to collaboratively build a learning model by solving an empirical risk minimization (ERM) problem (see $( 2 )$ in Section $2 )$ . To obtain a model parametrized by $\\theta$ using ERM, the commonly used mechanism is Stochastic Gradient Descent (SGD) [12]. However, one needs to solve this while enabling strong privacy guarantees on local data from the server, while also obtaining good learning performance, i.e., a suitable privacy-learning performance operating point. ", + "bbox": [ + 174, + 672, + 825, + 771 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Differential privacy (DP) $\\boxed { 1 8 }$ is the gold standard notion of data privacy that gives a rigorous framework through quantifying the information leakage about individual training data points from the observed interactions. Though DP was originally proposed in a framework where data resides centrally $\\boxed { 1 8 }$ , for distributed learning the more appropriate notion is of local differential privacy (LDP) [35, 17]. Here, each client randomizes its interactions with the server from whom the data is to be kept private (e.g., see industrial implementations [23, 31, 16]). However, LDP mechanisms suffer from poor performance in comparison with the central DP mechanisms [17, 35, 32]. To overcome this, a new privacy framework using anonymization has been proposed in the so-called shuffled model $\\underline { { { \\sqrt { 2 2 } } } } | 2 5 | 6 | \\dot { 2 } 6 | 5 | \\dot { \\overline { { { 1 5 } } } } | \\overline { { { \\mathrm { [ 7 ] } } } } | 8 |$ . In the shuffled model, each client sends her private message to a secure shuffler that randomly permutes all the received messages before forwarding them to the server. This model enables significantly better privacy-utility performance by amplifying DP through this shuffling. Therefore, in this paper we consider the shuffle privacy framework for distributed learning. ", + "bbox": [ + 174, + 775, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 175 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In solving $( 2 )$ using (distributed) gradient descent, each exchange leaks information about the local data, but we need as many steps as possible to obtain a good model; setting up the tension between privacy and performance. The goal is to obtain as many such interactions as possible for a given privacy budget. This is quantified through analyzing the privacy of the composition of privacy mechanisms. Abadi et al. [1] developed a framework for tighter analysis of such compositions, and this was later reformulated in terms of Rényi Differential Privacy (RDP) $\\textcircled { 1 3 7 }$ , and mapping this back to DP guarantee $\\textcircled { 1 3 8 }$ . Therefore, studying RDP is important to obtaining strong composition privacy results, and is the focus of this paper. ", + "bbox": [ + 173, + 180, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In distributed (and federated) learning, a fraction of the data samples are sampled; for example, with random client participation and stochastic gradient descent (SGD), which can be written as ", + "bbox": [ + 173, + 297, + 537, + 353 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/0a2d0378521d86e20fa4cb16150f84dd4f2bc66fc85b5b9c8f717f82818e994d.jpg", + "text": "$$\n\\theta _ { t + 1 } \\theta _ { t } - \\eta _ { t } \\frac { 1 } { | \\mathcal { T } | } \\sum _ { i \\in \\mathcal { I } } \\mathcal { R } ( \\nabla f _ { i } ( \\theta _ { t } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 233, + 357, + 475, + 395 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mathcal { R }$ is the local randomization mechanism and $\\mathcal { T }$ are the indices of the sampled data. This is a subsampled mechanism that enables another privacy amplification opportunity; which, in several cases, is shown to yield a privacy advantage proportional to the subsampling rate; see $\\dot { \\sqrt { 3 5 } } \\boxed { 4 2 }$ . The central technical question addressed in this paper is how to analyze the RDP of an arbitrary discrete mechanism for the subsampled shuffle privacy model. This enables us to answer the overall question posed in this paper, which is an achievable privacy-learning performance trade-off point for solving in the shuffled privacy model for distributed learning (see Figure 1). Our contributions are: ", + "bbox": [ + 174, + 398, + 535, + 534 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/60fe5dd4ec96e99911e5639baf65cde4fbddb5ef4c7ef331a52a02859b1dfcf1.jpg", + "image_caption": [ + "Figure 1: An iteration from the CLDPSGD Algorithm, where 3 clients are randomly chosen at each iteration. Each client sends the private gradient $\\mathcal { R } _ { p } \\left( g _ { t } ( d _ { i } ) \\right)$ to the shuffler that randomly permutes the gradients before passing them to the server. " + ], + "image_footnote": [], + "bbox": [ + 550, + 294, + 823, + 417 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 179, + 525, + 825, + 553 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We analyze the RDP of subsampled mechanisms in the shuffle framework by developing a novel bound applicable to any discrete $\\epsilon _ { 0 }$ -LDP mechanism as a function of the RDP order $\\lambda$ , subsampling rate $\\gamma$ , the LDP parameter $\\epsilon _ { 0 }$ , and the number of clients $n$ ; see Theorem $\\bigtriangledown$ The bound is explicit and amenable to numerics, including all constants.2 Furthermore, the bounds are valid for generic LDP mechanisms and all parameter regimes.3 We also provide a lower bound for the RDP in Theorem $\\bigstar$ We prove our upper bound (Theorem 1) using the following novel analysis techniques: First, we reduce the problem of computing the RDP of sub-sampled shuffle mechanisms to the problem of computing ternary $| \\chi | ^ { \\alpha } \\cdot \\mathrm { D P } ^ { \\bullet } | \\overline { { { 4 3 } } } | ]$ of shuffle (non sub-sampled) mechanisms; see Lemma 2. Then we reduce the computation of the ternary $| \\chi | ^ { \\alpha }$ -DP of shuffle mechanisms for a generic triple of neighboring datasets to those that have a special structure (see Theorem $5 )$ – this reduction step is one of the core technical results of this paper. Then we bound the ternary $| \\chi | ^ { \\alpha }$ -DP of the shuffle mechanisms for triples of neighboring datasets having special structures by bounding the Pearson-Vajda divergence $\\lvert \\overline { { 4 3 } } \\rvert$ using some concentration properties (see Theorem $\\overline { { 6 ) } }$ . ", + "bbox": [ + 173, + 558, + 825, + 738 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Using the core technical result in Theorem $^ { 1 , }$ we analyze privacy-convergence trade-offs of the CLDP-SGD algorithm (see Algorithm 1) for Lipschitz convex functions in Theorem $3 .$ This partially resolves an open question posed in $| \\vec { \\boldsymbol { { \\vert { 2 7 } } \\| } }$ , to extend their privacy analysis to RDP and significantly strengthening their privacy guatantees. ", + "bbox": [ + 174, + 743, + 825, + 800 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Numerically, we save a factor $1 4 \\times$ in privacy (✏) over the best known results for approximate DP for shuffling $\\pm \\overbrace { | 2 4 | }$ combined with strong composition $\\pm$ for $T = 1 0 ^ { 5 } , \\gamma = 0 . 0 0 \\bar { 1 } , n = 1 0 ^ { 6 }$ , and a factor of $2 . 5 \\times$ better than the best known RDP for shuffling bound $\\lVert 2 9 \\rVert$ combined with the sub-sampling result in $\\textcircled { 1 4 3 } \\textcircled { 1 }$ . Translating these to privacy-performance operating point in distributed optimization, over the MNIST data set with $\\ell _ { \\infty }$ clipping we numerically show gains: For the same privacy budget of $\\epsilon = 1 . 4$ , we get a test performance of $8 0 \\%$ whereas using strong composition the test performance of $\\mathbb { \\lVert 2 4 \\rVert }$ is $7 0 \\%$ ; furthermore, we achieves $9 0 \\%$ accuracy with the total privacy budget $\\epsilon = 2 . 9 1$ , whereas, $\\mathring { \\| 2 4 \\| }$ (with strong composition) achieves the same accuracy with a total privacy budget of $\\epsilon = 4 . 8 2$ . See Section 4 and the supplementary material for more results. ", + "bbox": [ + 174, + 806, + 825, + 862 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 161 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Related work: We give a more complete literature review in Appendix A, and focus here on the works that are closest to the results presented in this paper. ", + "bbox": [ + 173, + 166, + 823, + 195 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Private optimization in the shuffled model: Recently, $\\pmb { \\mathbb { Z } } 1 \\mathbf { l }$ and $ { \\mathbb { P } } ^ { \\geq \\sum { \\left. \\left. 2 8 \\right. \\right. } }$ have proposed differentially private SGD algorithms for federated learning, where at each iteration, each client applies an LDP mechanism on the gradients with the existence of a secure shuffler between the clients and the central server. However, the privacy analyses in these works developed approximate DP using advanced composition theorems for DP (e.g., $\\pm \\pm \\pm \\pm ) )$ ), which are known to be loose for composition $\\textcircled { 1 }$ . To the best of our knowledge, analyzing the private optimization framework using RDP and subsampling in the shuffled model is new to this paper. ", + "bbox": [ + 173, + 200, + 825, + 299 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Subsampled RDP: The works [38, 43, 45] have studied the RDP of subsampled mechanisms without shuffling. They demonstrated that this provides a tighter bound on the total privacy loss than the bound that can be obtained using the standard strong composition theorems. The RDP analysis of subsampled mechanisms in the shuffled privacy framework has not been studied before,4 and is new to this paper. The RDP of the shuffled model was very recently studied in $\\left[ \\left[ 2 9 \\right] \\right]$ , but without incorporating subsampling, which poses new technical challenges, as directly bounding the RDP of subsampled shuffle mechanisms is non-trivial. We overcome this by reducing our problem of computing RDP to bounding the ternary $| \\chi | ^ { \\alpha }$ -DP, and bounding the latter is a core technical contribution of our paper. ", + "bbox": [ + 173, + 304, + 825, + 416 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Paper organization: We give preliminaries and problem formulation in Section 2, main results (upper and lower bounds, and privacy-convergence tradeoff) in Section $\\textcircled { 3 }$ numerical results in Section 4, proof of the upper bound in Section $5 _ { : }$ and proof of the ternary DP of the shuffle model in Section 6. Omitted details/proofs from this paper are given in the supplementary material. ", + "bbox": [ + 173, + 421, + 826, + 478 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Preliminaries and Problem Formulation ", + "text_level": 1, + "bbox": [ + 174, + 501, + 540, + 518 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We use several privacy definitions throughout this paper. Among these, the local and central differential privacy definitions are standard and we defer them to Appendix $\\begin{array} { l } { \\mathbf { B } . } \\\\ { . } \\end{array}$ The other privacy definitions (Rényi DP and ternary $| \\chi | ^ { \\alpha }$ -DP) are relatively less standard and we define them below. ", + "bbox": [ + 173, + 535, + 826, + 577 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We say that two datasets $\\mathcal { D } = \\{ d _ { 1 } , \\ldots , d _ { n } \\} \\in \\mathcal { X } ^ { n }$ and $\\mathcal { D } ^ { \\prime } = \\{ d _ { 1 } ^ { \\prime } , \\ldots , d _ { n } ^ { \\prime } \\} \\in \\mathcal { X } ^ { n }$ are neighboring (and denoted by $\\mathcal { D } \\sim \\mathcal { D } ^ { \\prime }$ ) if they differ in one data point, i.e., there exists an $i \\in [ n ]$ such that $d _ { i } \\neq d _ { i } ^ { \\prime }$ and for every $j \\in [ n ] , j \\neq i$ , we have $d _ { j } = d _ { j } ^ { \\prime }$ . ", + "bbox": [ + 173, + 582, + 825, + 627 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 1 $( \\lambda , \\epsilon )$ -RDP (Rényi Differential Privacy) $\\pmb { \\mathbb { B } } \\pmb { \\mathbb { Z } } )$ ). A randomized mechanism $\\mathcal { M } : \\mathcal { X } ^ { n } \\mathcal { Y }$ is said to have $\\epsilon$ -Rényi differential privacy of order $\\lambda \\in ( 1 , \\infty )$ (in short, $( \\lambda , \\epsilon ( \\lambda ) )$ -RDP), if for any neighboring datasets $\\mathcal { D }$ , $\\mathcal { D } ^ { \\prime } \\in \\mathcal { X } ^ { n }$ , the Rényi divergence of order $\\lambda$ between $\\mathcal { M } ( \\mathcal { D } )$ and $\\mathcal { M } ( \\mathcal { D ^ { \\prime } } )$ is upper-bounded by $\\epsilon ( \\lambda )$ , i.e., ", + "bbox": [ + 174, + 632, + 825, + 690 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e9a2794fb7be0bbe83f3c20b4f17329f4eb96121ead9cbc1fa3c89aec04ffd11.jpg", + "text": "$$\nD _ { \\lambda } ( \\mathcal { M } ( \\mathcal { D } ) | | \\mathcal { M } ( \\mathcal { D } ^ { \\prime } ) ) = \\frac { 1 } { \\lambda - 1 } \\log \\left( \\mathbb { E } _ { \\theta \\sim \\mathcal { M } ( \\mathcal { D } ^ { \\prime } ) } \\left[ \\left( \\frac { \\mathcal { M } ( \\mathcal { D } ) ( \\theta ) } { \\mathcal { M } ( \\mathcal { D } ^ { \\prime } ) ( \\theta ) } \\right) ^ { \\lambda } \\right] \\right) \\leq \\epsilon ( \\lambda ) ,\n$$", + "text_format": "latex", + "bbox": [ + 241, + 699, + 754, + 742 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { M } ( \\mathcal { D } ) ( \\theta )$ denotes the probability that $\\mathcal { M }$ on input $\\mathcal { D }$ generates the output $\\theta$ ", + "bbox": [ + 173, + 751, + 717, + 767 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2 ( $\\zeta$ -Ternary $| \\chi | ^ { \\alpha }$ -differential privacy $\\mathbb { \\left[ 4 3 \\right] }$ ). A randomized mechanism $\\mathcal { M } : \\mathcal { X } ^ { n } \\mathcal { Y }$ is said to have $\\zeta$ -ternary- $| \\chi | ^ { \\alpha }$ -DP, if for any triple of mutually adjacent datasets $\\mathcal { D } , \\mathcal { D } ^ { \\prime } , \\mathcal { D } ^ { \\prime \\prime } \\in \\mathcal { X } ^ { n }$ (i.e., they mutually differ in the same location), the ternary- $\\cdot | \\chi | ^ { \\alpha }$ divergence of $\\mathcal { M } ( \\mathcal { D } ) , \\mathcal { M } ( \\mathcal { D ^ { \\prime } } ) , \\mathcal { M } ( \\mathcal { D ^ { \\prime } } )$ is upper-bounded by $( \\zeta ( \\alpha ) ) ^ { \\alpha }$ for all $\\alpha \\geq 1$ (where $\\zeta$ is a function from $\\mathbb { R } ^ { + }$ to $\\mathbb { R } ^ { + }$ ), i.e., ", + "bbox": [ + 173, + 772, + 826, + 829 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c94ab7413eb4a2a62b3895076d4050869c15b08e506c241227a1b4f77dc76651.jpg", + "text": "$$\nD _ { | \\chi | ^ { \\alpha } } \\left( \\mathcal { M } ( \\mathcal { D } ) , \\mathcal { M } ( \\mathcal { D } ^ { \\prime } ) | | \\mathcal { M } ( \\mathcal { D } ^ { \\prime \\prime } ) \\right) : = \\mathbb { E } _ { \\mathcal { M } ( \\mathcal { D } ^ { \\prime \\prime } ) } \\left[ \\left| \\frac { \\mathcal { M } ( \\mathcal { D } ) - \\mathcal { M } ( \\mathcal { D } ^ { \\prime } ) } { \\mathcal { M } ( \\mathcal { D } ^ { \\prime \\prime } ) } \\right| ^ { \\alpha } \\right] \\leq \\left( \\zeta ( \\alpha ) \\right) ^ { \\alpha } .\n$$", + "text_format": "latex", + "bbox": [ + 225, + 839, + 771, + 875 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The ternary $| \\chi | ^ { \\alpha }$ -DP was proposed in $\\underline { { \\| 4 3 \\| } }$ to characterize the RDP of the sub-sampled mechanism without shuffling. In this work, we analyze the ternary $| \\chi | ^ { \\alpha }$ -DP of the shuffled mechanism to bound the RDP of the sub-sampled shuffle model. ", + "bbox": [ + 174, + 90, + 823, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can use the following result for converting the RDP guarantees of a mechanism to its central DP guarantees. To the best of our knowledge, this result gives the best conversion. ", + "bbox": [ + 171, + 138, + 821, + 167 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 1 (From RDP to DP [13, 4]). Suppose for any $\\lambda > 1$ , a mechanism $\\mathcal { M }$ is $( \\lambda , \\epsilon \\left( \\lambda \\right) )$ -RDP. Then, the mechanism $\\mathcal { M }$ is $( \\epsilon , \\delta )$ -DP, where $\\delta > 0$ is arbitrary and $\\epsilon$ is given by ", + "bbox": [ + 171, + 171, + 823, + 202 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c8eccc3fa4e2c61b245a80dc59b8c6df46f9b3327671ff8731d7dfa22344182a.jpg", + "text": "$$\n\\epsilon = \\operatorname* { m i n } _ { \\lambda } \\left( \\epsilon \\left( \\lambda \\right) + \\frac { \\log \\left( 1 / \\delta \\right) + \\left( \\lambda - 1 \\right) \\log \\left( 1 - 1 / \\lambda \\right) - \\log \\left( \\lambda \\right) } { \\lambda - 1 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 208, + 717, + 244 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Problem formulation: We consider a distributed private learning setup comprising a set of $n$ clients, where the ith client has a data point $d _ { i }$ drawn from a universe $\\mathcal { X }$ for $i \\in [ n ]$ ; see also Figure 1. Let $\\mathcal { D } = ( d _ { 1 } , \\ldots , d _ { n } )$ denote the entire training dataset. The clients are connected to an untrusted server in order to solve the following empirical risk minimization (ERM) problem ", + "bbox": [ + 173, + 257, + 826, + 314 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/cf564570a3bfd9f0a4b8ac93d0e2e23265adca090fdce6c429f8fd4f7c62ad6b.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in { \\mathcal { C } } } { \\Big ( } F ( \\theta , { \\mathcal { D } } ) : = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } f ( \\theta , d _ { i } ) { \\Big ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 320, + 614, + 363 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathcal { C } \\subset \\mathbb { R } ^ { d }$ is a closed convex set, and $f : \\mathcal { C } \\times \\mathcal { D } \\mathbb { R }$ is the loss function. Our goal is to construct a global learning model $\\theta$ via stochastic gradient descent (SGD) while preserving privacy of individual data points in the training dataset $\\mathcal { D }$ by providing strong DP guarantees. ", + "bbox": [ + 174, + 371, + 825, + 415 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We revisit the CLDP-SGD algorithm presented in $\\pmb { \\Vert 2 7 }$ and described in Algorithm $\\bar { \\mathbf { \\xi } } _ { 1 }$ to solve the ERM $\\boxed { 2 }$ . In each step of CLDP-SGD, we choose uniformly at random a set $\\mathcal { U } _ { t }$ of $\\boldsymbol { k } \\quad \\le \\quad n$ clients out of $n$ clients. Each client $i \\in \\ U _ { t }$ computes and clips the $\\ell _ { p }$ -norm of the gradient $\\nabla _ { { \\boldsymbol { \\theta } } _ { t } } f \\left( { \\boldsymbol { \\theta } } _ { t } , d _ { i } \\right)$ to apply the LDP mechanism $\\mathcal { R } _ { p }$ where $\\mathcal { R } _ { p } \\ : \\ B _ { p _ { . } } ^ { d } \\ \\to \\ \\{ 0 , 1 \\} ^ { \\bar { b } }$ is an $\\epsilon _ { \\mathrm { 0 } }$ -LDP mechanism when inputs come from an $\\ell _ { p }$ -norm ball. In $\\textstyle \\left\\| 2 7 \\right\\|$ , the authors proposed different $\\epsilon _ { \\mathrm { 0 } }$ -LDP mechanisms for general $\\ell _ { p }$ -norm balls. After that, the shuffler randomly permutes the received $k$ gradients $\\{ \\mathcal { R } _ { p } \\left( \\tilde { \\mathbf { g } } _ { t } \\left( d _ { i } \\right) \\right) \\} _ { i \\in \\mathcal { U } _ { t } }$ and sends them to the server. Finally, ", + "bbox": [ + 174, + 420, + 387, + 712 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 Acldp: CLDP-SGD ", + "bbox": [ + 400, + 425, + 614, + 441 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Input: Datasets $\\mathcal { D } = ( d _ { 1 } , \\ldots , d _ { n } ) .$ , LDP privacy parameter $\\epsilon _ { \\mathrm { 0 } }$ gradient norm bound $C$ , and learning rate schedule $\\left\\{ \\eta _ { t } \\right\\}$ . ", + "bbox": [ + 400, + 444, + 823, + 473 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1: Initialize: $\\theta _ { 0 } \\in { \\mathcal { C } }$ \n2: for $t \\in [ T ]$ do \n3: Client sampling: A random set $\\mathcal { U } _ { t }$ of $k$ clients is chosen. \n4: for clients $i \\in \\mathcal { U } _ { t }$ do \n5: Compute gradient: $\\mathbf { g } _ { t } \\left( d _ { i } \\right) \\gets \\nabla _ { \\theta _ { t } } f \\left( \\theta _ { t } , d _ { i } \\right)$ \n6: Clip gradient: g˜t (di) gt (di) / max n 1, kgt(di)kpC o \n7: Client $i$ sends $\\mathcal { R } _ { p } \\left( \\tilde { \\bf g } _ { t } \\left( d _ { i } \\right) \\right)$ to the shuffler. \n8: end for \n9: Shuffling: The shuffler sends random permutation of \n$\\{ \\mathcal { R } _ { p } \\left( \\tilde { \\bf g } _ { t } \\left( d _ { i } \\right) \\right) : i \\in \\mathcal { U } _ { t } \\}$ to the server. \n10: Aggregate: $\\begin{array} { r } { \\overline { { \\bf g } } _ { t } \\frac { 1 } { k } \\sum _ { i \\in \\mathcal { U } _ { t } } \\mathcal { R } _ { p } ( \\tilde { \\bf g } _ { t } ( d _ { i } ) ) } \\end{array}$ \n11: Descent Step: $\\theta _ { t + 1 } \\prod _ { \\mathcal { C } } ( \\theta _ { t } - \\eta _ { t } \\overline { { \\mathbf { g } } } _ { t } )$ , where $\\Pi _ { c }$ is the \nprojection operator onto the set $\\mathcal { C }$ . ", + "bbox": [ + 406, + 478, + 823, + 671 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "12: end for ", + "text_level": 1, + "bbox": [ + 401, + 672, + 480, + 684 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Output: The model $\\theta _ { T }$ and the privacy parameters $\\epsilon , \\delta$ ", + "bbox": [ + 400, + 688, + 766, + 702 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "the server takes the average of the received gradients and updates the parameter vector. Our main contribution in this work is to present a stronger privacy analysis of the CLDP-SGD algorithm by characterizing the RDP of the sub-sampled shuffle model. ", + "bbox": [ + 176, + 712, + 825, + 753 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Main Results ", + "text_level": 1, + "bbox": [ + 174, + 773, + 315, + 791 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we present our main results. First, we characterize the RDP of the subsampled shuffle mechanism by presenting an upper bound in Theorem 1 and a lower bound in Theorem $\\dot { 2 }$ We then present the privacy-convergence trade-offs of the CLDP-SGD Algorithm in Theorem 3. ", + "bbox": [ + 174, + 806, + 825, + 849 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Consider an arbitrary $\\epsilon _ { \\mathrm { 0 } }$ -LDP mechanism $\\mathcal { R }$ , whose range is a discrete set $[ B ] = \\{ 1 , \\dots , B \\}$ for some $B \\in \\mathbb { N } : = \\{ 1 , 2 , 3 , . . . \\}$ . Here, $[ B ]$ could be the whole of $\\mathbb { N }$ . Let $\\mathcal { M } ( \\mathcal { D } )$ be a subsampled shuffle mechanism defined as follows: First subsample $k \\leq n$ clients of the $n$ clients (without replacement), where $\\textstyle \\gamma = { \\frac { k } { n } }$ denotes the sampling parameter. Each client $i$ out of the $k$ selected clients applies $\\mathcal { R }$ on $d _ { i }$ and sends $\\mathcal { R } ( d _ { i } )$ to the shuffler,5 who randomly permutes the received $k$ inputs and outputs the result. To formalize this, let $\\mathcal { H } _ { k } : \\overline { { \\mathcal { V } } } ^ { k } \\to \\mathcal { V } ^ { k }$ denote the shuffling operation that takes $k$ inputs and outputs their uniformly random permutation. Let $\\operatorname { s a m p } _ { k } ^ { n } : \\mathcal { X } ^ { n } \\to ^ { \\mathbf { \\bar { \\alpha } } } \\mathcal { X } ^ { k }$ denote the sampling operation for choosing a random subset of $k$ elements from a set of $n$ elements. We define the subsampled-shuffle mechanism as ", + "bbox": [ + 174, + 853, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 88, + 825, + 161 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0223bb31f1664a464c00b8082a8b37d0849c1e33f21bbc166f164cecae74c6d9.jpg", + "text": "$$\n\\mathcal { M } \\left( \\mathcal { D } \\right) : = \\mathcal { H } _ { k } \\circ \\operatorname { s a m p } _ { k } ^ { n } \\left( \\mathcal { R } \\left( d _ { 1 } \\right) , \\dotsc , \\mathcal { R } \\left( d _ { n } \\right) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 165, + 656, + 181 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Observe that each iteration of Algorithm 1 can be represented as an output of the subsampled shuffle mechanism $\\mathcal { M }$ . Thus, to analyze the privacy of Algorithm $\\bigstar$ it is sufficient to analyze the privacy of a sequence of identical $T$ subsampled shuffle mechanisms, and then apply composition theorems. ", + "bbox": [ + 174, + 184, + 826, + 227 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Histogram notation. It will be useful to define the following notation. Since the output of $\\mathcal { H } _ { k }$ is a random permutation of the $k$ outputs of $\\mathcal { R }$ (subsampling is not important here), the server cannot associate the $k$ messages to the clients; and the only information it can use from the messages is the histogram, i.e., the number of messages that give any particular output in $[ B ]$ . We define a set $\\mathcal { A } _ { B } ^ { k }$ as ", + "bbox": [ + 173, + 232, + 825, + 289 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/61484f74ad9e21f7d49d42a77cda88e4a4c37cf2b60810897a74b6186cf90955.jpg", + "text": "$$\n\\mathcal { A } _ { B } ^ { k } = \\bigg \\{ h = ( h _ { 1 } , \\ldots , h _ { B } ) : \\sum _ { j = 1 } ^ { B } h _ { j } = k \\bigg \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 356, + 291, + 640, + 337 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "to denote the set of all possible histograms of the output of the shuffler with $k$ inputs. Therefore, we can assume, without loss of generality (w.l.o.g.), that the output of $\\mathcal { M }$ is a distribution over $\\mathcal { A } _ { B } ^ { k }$ . ", + "bbox": [ + 171, + 338, + 823, + 368 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our main results for the RDP of the subsampled shuffled mechanism (defined in $( 3 )$ ) are given below. Our first result provides an upper bound (stated in Theorem 1 and proved in Section $. 5 )$ and the second result provides a lower bound (stated in Theorem 2 and proved in Appendix D) ", + "bbox": [ + 173, + 372, + 826, + 416 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1 (Upper Bound). For any $n \\in \\mathbb { N } , k \\leq n , \\epsilon _ { 0 } \\geq 0 ,$ and any integer $\\lambda \\geq 2$ , the RDP of the subsampled shuffle mechanism $\\mathcal { M }$ (defined in $\\textcircled { 3 }$ ) is upper-bounded by ", + "bbox": [ + 173, + 417, + 823, + 446 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$\\epsilon ( \\lambda ) \\leq \\frac { 1 } { \\lambda - 1 } \\log \\left( 1 + 4 \\binom { \\lambda } { 2 } \\gamma ^ { 2 } \\frac { \\left( e ^ { \\epsilon _ { 0 } } - 1 \\right) ^ { 2 } } { \\bar { k } e ^ { \\epsilon _ { 0 } } } + \\sum _ { j = 3 } ^ { \\lambda } \\binom { \\lambda } { j } \\gamma ^ { j } j \\Gamma \\left( j / 2 \\right) \\left( \\frac { 2 \\left( e ^ { 2 \\epsilon _ { 0 } } - 1 \\right) ^ { 2 } } { \\bar { k } e ^ { 2 \\epsilon _ { 0 } } } \\right) ^ { j / 2 } + \\Upsilon \\right) ,$ where $\\begin{array} { r } { \\overline { { k } } = \\lfloor \\frac { k - 1 } { 2 e ^ { \\epsilon _ { 0 } } } \\rfloor + 1 , \\gamma = \\frac { k } { n } } \\end{array}$ , and $\\begin{array} { r } { \\Gamma \\left( z \\right) = \\int _ { 0 } ^ { \\infty } x ^ { z - 1 } e ^ { - x } d x } \\end{array}$ is the Gamma function. The term $\\Upsilon$ is given by $\\begin{array} { r } { \\Upsilon = \\left( \\left( 1 + \\gamma \\frac { e ^ { 2 \\epsilon _ { 0 } } - 1 } { e ^ { \\epsilon _ { 0 } } } \\right) ^ { \\lambda } - 1 - \\lambda \\gamma \\frac { e ^ { 2 \\epsilon _ { 0 } } - 1 } { e ^ { \\epsilon _ { 0 } } } \\right) e ^ { - \\frac { k - 1 } { 8 e ^ { \\epsilon _ { 0 } } } } . } \\end{array}$ . ", + "bbox": [ + 173, + 449, + 825, + 549 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 2 (Lower Bound). For any $n \\in \\mathbb { N } , k \\leq n , \\epsilon _ { 0 } \\geq 0$ , and any integer $\\lambda \\geq 2$ , the RDP of the subsampled shuffle mechanism $\\mathcal { M }$ (defined in $\\textcircled { 3 }$ ) is lower-bounded by ", + "bbox": [ + 173, + 553, + 823, + 582 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d852f2de6ba0e5eb41ce19b89415bfded0ba2f915472fdff465eec7043154e8b.jpg", + "text": "$$\n\\mathrm { \\Sigma } ( \\lambda ) \\geq \\frac { 1 } { \\lambda - 1 } \\log \\left( 1 + { \\binom { \\lambda } { 2 } } \\gamma ^ { 2 } \\frac { \\left( e ^ { \\epsilon _ { 0 } } - 1 \\right) ^ { 2 } } { k e ^ { \\epsilon _ { 0 } } } + \\sum _ { j = 3 } ^ { \\lambda } { \\binom { \\lambda } { j } } \\gamma ^ { j } \\left( \\frac { \\left( e ^ { 2 \\epsilon _ { 0 } } - 1 \\right) } { k e ^ { \\epsilon _ { 0 } } } \\right) ^ { j } \\mathbb { E } \\left( m - \\frac { k } { e ^ { \\epsilon _ { 0 } } + 1 } \\right) ^ { j } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 583, + 823, + 631 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where expectation is taken w.r.t. the binomial r.v. $m \\sim B i n \\left( k , p \\right)$ with parameter $\\begin{array} { r } { p = \\frac { 1 } { e ^ { \\epsilon _ { 0 } } + 1 } } \\end{array}$ . ", + "bbox": [ + 173, + 636, + 776, + 651 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our CLDP-SGD algorithm and its privacy-convergence trade-offs (stated in Theorem 3 below) are given for a general local randomizer $\\mathcal { R } _ { p }$ (whose inputs comes from an $\\ell _ { p }$ -ball for any $p \\in [ 1 , \\infty ] )$ that satisfies the following conditions: (i) The randomized mechanism $\\mathcal { R } _ { p }$ is an $\\scriptstyle \\epsilon _ { 0 } - \\mathrm { \\mathrm { L D P } }$ mechanism. (ii) The randomized mechanism $\\mathcal { R } _ { p }$ is unbiased, i.e., $\\vec { \\mathfrak { L } } [ \\mathcal { R } _ { p } ( \\mathbf { x } ) | \\mathbf { x } | = \\mathbf { x }$ for all $\\mathbf { x } \\in B _ { p } ( a )$ , where $a$ is the radius of the ball $B _ { p }$ . (iii) The output of the randomized mechanism $\\mathcal { R } _ { p }$ can be represented using $B \\in$ $\\mathbb { N } ^ { + }$ bits. (iv) The randomized $\\mathcal { R } _ { p }$ has a bounded variance: $\\begin{array} { r } { \\operatorname* { s u p } _ { \\mathbf { x } \\in \\mathcal { B } _ { p } ( a ) } \\mathbb { E } \\| \\mathcal { R } _ { p } \\left( \\mathbf { x } \\right) - \\mathbf { x } \\| _ { 2 } ^ { 2 } \\leq G _ { p } ^ { 2 } ( a ) } \\end{array}$ , where $G _ { p } ^ { 2 }$ is a function from $\\mathbb { R } ^ { + }$ to $\\mathbb { R } ^ { + }$ . ", + "bbox": [ + 173, + 661, + 826, + 763 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Girgis et al. $\\textstyle \\left\\| 2 7 \\right\\|$ proposed unbiased $\\epsilon _ { 0 }$ -LDP mechanisms $\\mathcal { R } _ { p }$ for several values of norms $p \\in [ 1 , \\infty ]$ that require $b = \\mathcal { O } \\left( \\log \\left( d \\right) \\right)$ bits of communication and satisfy the above conditions. In this paper, achieving communication efficiency is not our goal (though we also achieve that since the $\\epsilon _ { 0 } { \\mathrm { - L D P } }$ mechanism $\\mathcal { R } _ { p }$ that we use takes values in a discrete set), as our main focus is on analyzing the RDP of the subsampled shuffle mechanism. If we use the $\\epsilon _ { 0 }$ -LDP mechanism $\\mathcal { R } _ { p }$ from $\\dot { \\underline { 1 2 7 } }$ , we would also get similar gains in communication as were obtained in $\\lVert 2 7 \\rVert$ . ", + "bbox": [ + 173, + 770, + 826, + 854 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The privacy-convergence trade-off of our algorithm $\\mathcal { A } _ { \\mathrm { c l d p } }$ is given below. ", + "bbox": [ + 174, + 861, + 650, + 876 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/7e175b7ca5f6830d336d8ae1c400c95ba5217b0196bb789ebee85750d10f85c6.jpg", + "image_caption": [ + "(a) Approx. DP as a function of $T$ (b) Approx. DP as a function of $T$ (c) Approx. DP as a function of $n$ for $\\epsilon _ { 0 } = 2 $ , $\\gamma = 0 . 0 0 1$ , $n = 1 0 ^ { 6 }$ for $\\epsilon _ { 0 } = 1$ , $\\gamma = 0 . 0 0 1$ , $n = 1 0 ^ { 7 }$ for $\\epsilon _ { 0 } = 2 $ , $\\gamma n = 1 0 ^ { 3 }$ , $T = 1 0 ^ { 5 }$ ", + "Figure 2: Comparison of several bounds on the Approximate $( \\epsilon , \\delta )$ -DP for composition of a sequence of subsampled shuffle mechanisms for $\\delta = 1 0 ^ { - 8 }$ : (i) Approximate DP obtained from our upper bound on the RDP in Theorem $\\perp$ (blue); (ii) Approximate DP obtained from our lower bound on the RDP in Theorem $\\bigstar$ (red); (iii) Approximate DP obtained from the upper bound on the RDP given in $\\left[ \\left[ 2 9 \\right] \\right]$ with RDP amplification by subsampling from $\\pmb { \\boxed { 4 3 } }$ (black); and (iv) Applying the strong composition theorem $\\textcircled { 1 3 4 } |$ after getting the Approximate DP of the shuffled model given in $\\pm$ with subsampling $\\mathbf { \\widehat { | 4 2 | } }$ (magenta). " + ], + "image_footnote": [], + "bbox": [ + 181, + 94, + 816, + 208 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3 (Privacy-Convergence tradeoffs). Let the set $\\mathcal { C }$ be convex with diameter $D$ and the function $f ( \\theta ; . ) : \\mathcal { C } \\times \\mathcal { D } \\mathbb { R }$ be convex and $L$ -Lipschitz continuous with respect to the $\\ell _ { g }$ -norm, which is the dual of the $\\ell _ { p }$ -norm. Let $\\theta ^ { * } = \\arg \\operatorname* { m i n } _ { \\theta \\in { \\mathcal { C } } } F \\left( \\theta \\right)$ denote the minimizer of the problem $\\textcircled { 2 }$ For $\\textstyle \\gamma = { \\frac { k } { n } }$ , if we run Algorithm $\\mathcal { A } _ { \\mathrm { c l d p } }$ over $T$ iterations, then we have ", + "bbox": [ + 171, + 342, + 825, + 402 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/aca1d683fb6d9f05ecce0242fa7faf3ba048e12073db693f367503baf80680f9.jpg", + "text": "$$\n\\epsilon = \\operatorname* { m i n } _ { \\lambda } \\left( T \\epsilon \\left( \\lambda \\right) + \\frac { \\log \\left( 1 / \\delta \\right) + \\left( \\lambda \\stackrel { \\star } { - } 1 \\right) \\log \\left( 1 - 1 / \\lambda \\right) - \\log \\left( \\lambda \\right) } { \\lambda - 1 } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 284, + 417, + 732, + 452 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\epsilon \\left( \\lambda \\right)$ is the RDP of the subsampled shuffle mechanism given in Theorem 1. ", + "bbox": [ + 192, + 455, + 723, + 470 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2. Convergence: If we run $\\mathcal { A } _ { \\mathrm { c l d p } }$ with $\\begin{array} { r } { \\eta _ { t } = \\frac { D } { G \\sqrt { t } } } \\end{array}$ , where $\\begin{array} { r } { G ^ { 2 } = \\operatorname* { m a x } \\lbrace d ^ { 1 - \\frac { 2 } { p } } , 1 \\rbrace L ^ { 2 } + \\frac { G _ { p } ^ { 2 } ( L ) } { \\gamma n } } \\end{array}$ , we get $\\mathbb { E } \\left[ F \\left( \\theta _ { T } \\right) \\right] - F \\left( \\theta ^ { * } \\right) \\leq \\mathcal { O } \\left( \\frac { D G \\log ( T ) } { \\sqrt { T } } \\right) .$ ", + "bbox": [ + 173, + 479, + 815, + 535 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The proof outline of Theorem $3$ is as follows: Note that $\\mathcal { A } _ { \\mathrm { c l d p } }$ is an iterative algorithm, where in each iteration we use the subsampled shuffle mechanism as defined in $\\textcircled{3}$ , for which we have computed the RDP guarantees in Theorem $\\bigtriangledown$ Now, for the privacy analysis of $\\mathcal { A } _ { \\mathrm { c l d p } }$ , we use the adaptive composition theorem from $\\textcircled { 1 3 7 }$ Proposition 1] and then use the RDP to DP conversion given in Lemma $^ { 1 . }$ For the convergence analysis, we use a standard non-private SGD convergence result and compute the required parameters for that. See Appendix F for a complete proof of Theorem 3. ", + "bbox": [ + 173, + 544, + 825, + 628 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Remark 1. Note that our convergence bound is affected by the variance of the $\\epsilon _ { \\mathrm { 0 } }$ -LDP mechanism $\\mathcal { R } _ { p }$ . For example, when $f$ is $L$ -Lipschitz continuous w.r.t. the $\\ell _ { 2 }$ -norm, we can use the LDP mechanism $\\mathcal { R } _ { 2 }$ proposed in $\\boxed { 1 1 }$ that has variance $\\begin{array} { r } { G _ { 2 } ^ { 2 } ( L ) = 1 4 L ^ { 2 } d \\big ( \\frac { e ^ { \\epsilon _ { 0 } } + 1 } { e ^ { \\epsilon _ { 0 } } - 1 } \\big ) ^ { 2 } } \\end{array}$ ; and when $f$ is $L$ -Lipschitz continuous w.r.t. the $\\ell _ { 1 }$ -norm or $\\ell _ { \\infty }$ -norm, we can use the LDP mechanisms $\\mathcal { R } _ { \\infty }$ or $\\mathcal { R } _ { 1 }$ , respectively, proposed in $\\lVert 2 7 \\rVert$ 1 that have variances $\\begin{array} { r } { G _ { \\infty } ^ { 2 } ( L ) = L ^ { 2 } d ^ { 2 } \\big ( \\frac { e ^ { \\epsilon _ { 0 } } + 1 } { e ^ { \\epsilon _ { 0 } } - 1 } \\big ) ^ { 2 } } \\end{array}$ 2 and G21(L) = L2d \u0000 e✏0 +1e✏0 \u00001 \u00002 , respectively. By plugging these variances $G _ { p } ^ { 2 } ( L )$ (for $p = 1 , 2 , \\infty )$ into Theorem $3 .$ we get the convergence rate of the $L$ -Lipschitz continuous loss function w.r.t. the $\\ell _ { p }$ -norm (for $p \\equiv \\infty , 2 , 1 $ ). ", + "bbox": [ + 173, + 631, + 826, + 741 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Remark 2. The privacy parameter in $( 5 )$ is not in a closed form expression and could be obtained by solving an optimization problem. However, we numerically compute it for several interesting regimes of parameters in our numerical experiments; see Section 4 for more details. ", + "bbox": [ + 174, + 742, + 825, + 786 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Numerical Results ", + "text_level": 1, + "bbox": [ + 174, + 804, + 359, + 820 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we present numerical experiments to show the performance of our bounds on RDP of the subsampled shuffle mechanism and its usage for getting approximate DP of Algorithm 1 for training machine learning models. ", + "bbox": [ + 174, + 834, + 825, + 877 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Composition of a sequence of subsampled shuffle models: In Figure 2, we plot several bounds on the approximate $( \\epsilon , \\delta )$ -DP for a composition of $T$ mechanisms $( \\mathcal { M } _ { 1 } , \\ldots , \\mathcal { M } _ { T } )$ , where $\\mathcal { M } _ { t }$ is a subsampled shuffle mechanism for $t \\in [ T ]$ . In all our experiments reported in Figure $2 .$ we fix $\\delta = 1 0 ^ { - 8 }$ . We observe that our new bound on the RDP of the subsampled shuffle mechanism achieves a significant saving in total privacy $\\epsilon$ compared to the state-of-the-art. For example, we save a factor of $1 4 \\times$ compared to the bound on DP $\\pm \\overbrace { \\lVert 2 4 \\rVert }$ with strong composition theorem $\\textcircled { 1 3 4 } \\textcircled { 1 }$ and $2 . 5 \\times$ compared to the bound on the RDP given in $\\lVert 2 9 \\rVert$ with subsampled RDP $\\textcircled { 1 4 3 } |$ in computing the overall privacy parameter $\\epsilon$ for number of iterations $\\bar { T } = 1 0 ^ { 5 }$ , subsampling parameter $\\gamma = 0 . 0 0 1$ , LDP parameter $\\epsilon _ { 0 } = 2 $ , and number of clients $n = 1 0 ^ { 6 }$ . We observe in Figure $\\boxed { 2 \\mathrm { b } }$ that the bound given in $\\pmb { \\mathbb { Z 4 } }$ with the strong composition theorem $\\pm$ behaves better than the bound on the RDP $\\pmb { \\bigtriangledown }$ with subsampled RDP bound $\\boxed { \\boxed { 4 3 } }$ when the number of subsampled clients per iteration is equal to $k = \\gamma n = \\mathrm { \\bar { 1 0 ^ { 4 } } }$ ; however, our bound beats both of them.6 In Figure $\\boxed { 2 \\mathrm { c } }$ we fix the number of subsampled clients per iteration to be $k = \\gamma n = 1 0 ^ { 3 }$ , and hence, the subsampling parameter $\\gamma$ varies with $n$ . ", + "bbox": [ + 174, + 882, + 821, + 912 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 89, + 825, + 244 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Distributed private learning: We numerically evaluate the proposed privacy-learning performance on training machine learning models. We consider the standard MNIST handwritten digit dataset that has 60, 000 training images and 10, 000 test images. We train a simple neural network that was also used in $\\pm \\pm \\pm \\textcircled { 3 9 }$ and described in Table $\\bigstar$ This model has $d = 1 3$ , 170 parameters and achieves an accuracy of $9 9 \\%$ for non-private, uncompressed vanilla SGD. We assume that we have $n = 6 0 , 0 0 0$ clients, where each client has one sample. At each step of the Algorithm $\\bigstar \\bigstar$ we choose uniformly at random 10, 000 clients, where each client clips the $\\ell _ { \\infty }$ -norm of the gradient with clipping parameter $C = 1 / 1 0 0$ and applies the $\\mathcal { R } _ { \\infty }$ $\\epsilon _ { \\mathrm { 0 } }$ -LDP mechanism proposed in $\\mathbb { \\ Z } \\mathbb { \\ Z }$ with $\\epsilon _ { 0 } = 1 . 5$ . We run Algorithm $^ 1$ with $\\bar { \\delta } = 1 0 ^ { - 5 }$ for 200 epochs, with learning rate $\\eta = 0 . 3$ for the first 70 epochs, and ", + "bbox": [ + 173, + 250, + 825, + 376 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "then decrease it to 0.18 in the remaining epochs. ", + "bbox": [ + 178, + 375, + 490, + 388 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/4c9f9a736abe26e5c2ef830c409740c93a5ede536760f5b6876bdb5a01813138.jpg", + "table_caption": [ + "Table 1: Model architecture for MNIST " + ], + "table_footnote": [], + "table_body": "
LayerParameters
Convolution Max-Pooling Convolution Max-Pooling Fully connected Softmax16 filters of 8 × 8,Stride 2 2×2 32 filters of 4 × 4, Stride 2 2×2 32 units 10 units
", + "bbox": [ + 174, + 406, + 495, + 508 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e2f0e8974b2af63fd0a36f8a233db485da410d667a0ad9a2b433a98053b80b94.jpg", + "image_caption": [ + "Figure 3: Privacy-Utility trade-offs on the MNIST dataset with $\\ell _ { \\infty }$ -norm clipping. " + ], + "image_footnote": [], + "bbox": [ + 524, + 381, + 830, + 539 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 3 plots the mean and the standard deviation of privacy-accuracy trade-offs averaged over 10 runs. For our privacy analysis, the total privacy budget is computed by optimizing over RDP order $\\lambda$ using our upper bound given in Theorem $^ { 1 . }$ For privacy analysis of $\\dot { \\lVert 2 4 \\rVert }$ , we first compute the privacy amplification by shuffling numerically given in $\\lVert 2 4 \\rVert$ ; then we compute its privacy obtained when amplified via subsampling $\\lVert \\overline { { 4 2 } } \\rVert$ ; and finally we use the strong composition theorem $\\overleftarrow { \\mathbb { B } 4 }$ to obtain the central privacy parameter $\\epsilon$ . ", + "bbox": [ + 173, + 583, + 825, + 666 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We observe that we achieve an accuracy of $8 0 \\% ( \\pm 1 . 8 )$ with a total privacy budget of $\\epsilon = 1 . 4$ using our new privacy analysis, whereas, $\\dot { \\lVert 2 4 \\rVert }$ achieves an accuracy of only $7 0 . 7 \\% ( \\pm 2 . 1 )$ with the same privacy budget of $\\epsilon = 1 . 4$ using the standard composition theorems. Furthermore, we can see that we achieves accuracy $9 0 \\% ( \\pm 0 . { \\bar { 5 } } )$ with total privacy budget $\\epsilon = 2 . 9 1$ using our new privacy analysis, whereas, $\\pmb { \\left[ 2 4 \\right] }$ (together with the standard strong composition theorem) achieves the same accuracy with a total privacy budget of $\\epsilon = 4 . 8 2$ . ", + "bbox": [ + 173, + 672, + 825, + 757 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 Proof of Theorem 1: Upper Bound ", + "text_level": 1, + "bbox": [ + 173, + 780, + 495, + 800 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For any dataset $\\mathcal { D } _ { k } = ( d _ { 1 } , \\ldots , d _ { k } ) \\in \\mathcal { X } ^ { k }$ containing of $k$ data points, we define a shuffle mechanism $\\mathcal { M } _ { s h } ( \\mathcal { D } _ { k } )$ as follows: ", + "bbox": [ + 171, + 814, + 825, + 844 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/f272419b015b9d39f5513104c3df49cd2a7314a4f591ba4d23e4f30ac7f31650.jpg", + "text": "$$\n\\mathcal { M } _ { s h } ( \\mathcal { D } _ { k } ) = \\mathcal { H } _ { k } \\left( \\mathcal { R } \\left( d _ { 1 } \\right) , \\ldots , \\mathcal { R } \\left( d _ { k } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 849, + 632, + 866 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\mathcal { H } _ { k }$ takes $k$ inputs and outputs a uniformly random permutation of them. Recall from $\\textcircled{3}$ , for any dataset $\\mathcal { D } _ { n } = ( \\bar { d } _ { 1 } , \\ldots , d _ { n } ) \\in \\mathcal { X } ^ { n }$ containing $n$ data points, the subsampled-shuffle mechanism is defined as $\\mathcal { M } \\left( \\mathcal { D } \\right) : = \\mathcal { H } _ { k } \\circ \\mathrm { s a m p } _ { k } ^ { n }$ $( \\mathcal { R } \\left( d _ { 1 } \\right) , \\ldots , \\mathcal { R } \\left( d _ { n } \\right) )$ . ", + "bbox": [ + 173, + 90, + 825, + 135 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The proof of Theorem 1 consists of two steps. First, we bound the ternary- $\\cdot | \\chi | ^ { \\alpha }$ -DP of the shuffle mechanism $\\mathcal { M } _ { s h }$ (see Theorem $\\bigstar$ , which is the main technical contribution in this proof. Then, using this, we bound the RDP of the subsampled shuffle mechanism $\\mathcal { M }$ . ", + "bbox": [ + 174, + 138, + 823, + 181 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 4 $\\zeta$ -ternary- $| \\chi | ^ { \\alpha }$ -DP of the shuffle mechanism $\\mathcal { M } _ { s h }$ ). For any integer $k \\geq 2$ , $\\epsilon _ { 0 } > 0$ , and all $\\alpha \\geq 2$ , the $\\zeta$ -ternary- $\\cdot | \\chi | ^ { \\alpha }$ -DP of the shuffle mechanism $\\mathcal { M } _ { s h }$ is bounded by: ", + "bbox": [ + 171, + 185, + 826, + 214 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/fa5b4a536ec8c61be0ee2788fbf1e2878397b1ec48982646abb3549d81cba0d6.jpg", + "text": "$$\n\\zeta \\left( \\alpha \\right) ^ { \\alpha } \\leq \\left\\{ \\begin{array} { l l } { 4 \\frac { \\left( e ^ { \\epsilon _ { 0 } } - 1 \\right) ^ { 2 } } { \\bar { k } e ^ { \\epsilon _ { 0 } } } + ( e ^ { \\epsilon _ { 0 } } - e ^ { - \\epsilon _ { 0 } } ) ^ { \\alpha } e ^ { - \\frac { k - 1 } { 8 e ^ { \\epsilon _ { 0 } } } } } & { i f \\alpha = 2 , } \\\\ { \\alpha \\Gamma \\left( \\alpha / 2 \\right) \\left( \\frac { 2 \\left( e ^ { 2 \\epsilon _ { 0 } } - 1 \\right) ^ { 2 } } { \\bar { k } e ^ { 2 \\epsilon _ { 0 } } } \\right) ^ { \\alpha / 2 } + ( e ^ { \\epsilon _ { 0 } } - e ^ { - \\epsilon _ { 0 } } ) ^ { \\alpha } e ^ { - \\frac { k - 1 } { 8 e ^ { \\epsilon _ { 0 } } } } } & { o t h e r w i s e , } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 235, + 219, + 751, + 279 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\overline { { k } } = \\lfloor \\frac { k - 1 } { 2 e ^ { \\epsilon _ { 0 } } } \\rfloor + 1 } \\end{array}$ and $\\begin{array} { r } { \\Gamma \\left( z \\right) = \\int _ { 0 } ^ { \\infty } x ^ { z - 1 } e ^ { - x } d x } \\end{array}$ is the Gamma function. ", + "bbox": [ + 174, + 284, + 671, + 303 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 4 is one of the core technical results of this paper, and we prove it in Section 6. ", + "bbox": [ + 174, + 310, + 753, + 327 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "It was shown in $\\boxed { \\boxed { 4 3 } }$ Proposition 16] that if a mechanism obeys $\\zeta$ -ternary- $\\cdot | \\chi | ^ { \\alpha }$ -DP, then its subsampled version (with subsampling parameter $\\gamma$ ) will obey $\\gamma \\zeta .$ -ternary- $| \\chi | ^ { \\alpha }$ -DP. Using that result, the authors then bounded the RDP of the subsampled mechanism in $\\underline { { \\bar { 1 4 3 } } } \\underline { { \\bar { 1 } } }$ Eq. (9)]. Adapting that result to our setting, we have the following lemma. ", + "bbox": [ + 173, + 330, + 825, + 387 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Lemma 2 (From $\\zeta$ -ternary- $| \\chi | ^ { \\alpha }$ -DP to subsampled RDP). Suppose the shuffle mechanism $\\mathcal { M } _ { s h }$ obeys $\\zeta$ -ternary- $| \\chi | ^ { \\alpha }$ -DP. For any $\\lambda \\geq 2 , k \\leq n$ , RDP of the subsampled shuffle mechanism $\\mathcal { M }$ (with subsampling parameter $\\gamma = k / n ,$ ) is bounded by: $\\begin{array} { r } { \\epsilon ( \\lambda ) \\leq \\frac { 1 } { \\lambda - 1 } \\log \\left( 1 + \\sum _ { \\alpha = 2 } ^ { \\lambda } \\binom { \\lambda } { \\alpha } \\gamma ^ { \\alpha } \\zeta ( \\alpha ) ^ { \\alpha } \\right) } \\end{array}$ . ", + "bbox": [ + 173, + 391, + 825, + 439 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Lemma 2 can be seen as a corollary to $\\boxed { \\ 4 3 }$ Proposition 16 and Eq. (9)]. However, for completeness, we prove it in Appendix $\\boxed { \\mathrm { E . 1 } }$ Substituting the bound on $\\zeta ( \\alpha )$ from Theorem 4 into Lemma 2 together with some algebraic manipulation gives proves Theorem 1; see Appendix $\\mathrm { E } . 2$ for details. ", + "bbox": [ + 173, + 446, + 826, + 493 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 Proof of Theorem $\\mathbf { 4 } ;$ Ternary $| \\chi | ^ { \\alpha }$ -DP of the Shuffle Model ", + "text_level": 1, + "bbox": [ + 173, + 508, + 692, + 530 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The proof has two main steps. In the first step, we reduce the problem of deriving ternary divergence for arbitrary neighboring datasets to the problem of deriving the ternary divergence for specific neighboring datasets, $\\mathcal { D } \\stackrel { - } { \\sim } \\mathcal { D } ^ { \\prime } \\sim \\mathcal { D } ^ { \\prime \\prime }$ , where all elements in $\\mathcal { D }$ are the same and $\\mathcal { D } ^ { \\prime } , \\mathcal { D } ^ { \\prime \\prime }$ differ from $\\mathcal { D }$ in one entry. In the second step, we derive the ternary divergence for the special neighboring datasets. ", + "bbox": [ + 173, + 541, + 826, + 598 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The specific neighboring datasets to which we reduce our general problem has the following form: ", + "bbox": [ + 176, + 603, + 820, + 619 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/64302209a85420d5699588a5bf97c3c2506407f83fcdb28ccaf3a2355c83b02b.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m } = \\{ ( \\mathcal { D } _ { m } , \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ^ { \\prime \\prime } ) : \\mathcal { D } _ { m } = ( d , \\ldots , d , d ) \\in \\mathcal { X } ^ { m } , \\mathcal { D } _ { m } ^ { \\prime } = ( d , \\ldots , d , d ^ { \\prime } ) \\in \\mathcal { X } ^ { m } , \\mathrm { ~ a ~ n ~ d ~ } \\mathcal { X } = 1 , \\ldots , d , 1 \\} , } \\\\ { \\mathcal { D } _ { m } ^ { \\prime \\prime } = ( d , \\ldots , d , d ^ { \\prime \\prime } ) \\in \\mathcal { X } ^ { m } , \\mathrm { ~ w h e r e ~ } d , d ^ { \\prime } , d ^ { \\prime \\prime } \\in \\mathcal { X } \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 623, + 766, + 661 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Consider arbitrary neighboring datasets $\\mathcal { D } = \\left( d _ { 1 } , \\ldots , d _ { k - 1 } , d _ { k } \\right)$ , $\\mathcal { D } ^ { \\prime } = ( d _ { 1 } , \\ldots , d _ { k - 1 } , d _ { k } ^ { \\prime } )$ , and $\\mathcal { D } ^ { \\prime \\prime } = ( d _ { 1 } , \\dots , d _ { k - 1 } ^ { - } , d _ { k } ^ { \\overline { { \\prime } } \\prime } )$ , each having $k$ elements. For any $m \\in \\{ 0 , \\ldots , k - 1 \\}$ \u0000, we define new neighboring dataeach having = (d00k , . . . , d00k , dk), D0(k)m+1 = (d00k , . . . , d00k , d0k), and D00(k)m+1 $\\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } = ( d _ { k } ^ { \\prime \\prime } , \\dots , d _ { k } ^ { \\prime \\prime } )$ $m + 1$ $( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } , \\mathcal { D } _ { m + 1 } ^ { \\prime ( k ) } , \\mathcal { D } _ { m + 1 } ^ { ( k ) } ) \\in \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m }$ ", + "bbox": [ + 173, + 666, + 828, + 733 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The first step of the proof is given in the following theorem. ", + "bbox": [ + 173, + 737, + 568, + 752 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 5 (Reduction to the Special Case). Let $\\begin{array} { r } { q = \\frac { 1 } { e ^ { \\epsilon _ { 0 } } } } \\end{array}$ . We have: ", + "bbox": [ + 173, + 755, + 619, + 771 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/6165b501b24104cf860f31af8810369280ef2911bf671a46948162972cfadbaf.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { h \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime \\prime } ) } \\left[ \\left| \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } ) ( h ) - \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime } ) ( h ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime \\prime } ) ( h ) } \\right| ^ { \\alpha } \\right] } \\\\ & { \\qquad \\leq \\mathbb { E } _ { m \\sim \\mathrm { B i n } ( k - 1 , q ) } \\left[ \\mathbb { E } _ { h \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) } \\left[ \\left| \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { ( k ) } ) ( h ) - \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime ( k ) } ) ( h ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) ( h ) } \\right| ^ { \\alpha } \\right] \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 776, + 784, + 859 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We know (by Chernoff bound) that the binomial r.v. is concentrated around its mean, which implies that the terms in the RHS of $\\textcircled { 9 }$ that correspond to $m \\ < \\ ( 1 - \\tau ) q ( k - 1 )$ (we will take $\\tau ~ = ~ 1 / 2 )$ will contribute in a negligible amount. Then we show that $E _ { m } : =$ $\\mathbb { E } _ { \\boldsymbol { h } \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) } \\left[ \\left| \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { ( k ) } ) ( \\boldsymbol { h } ) - \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime ( k ) } ) ( \\boldsymbol { h } ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) ( \\underline { { h } } ) } \\right| ^ { \\alpha } \\right]$ is a non-increasing function of $m$ . These observation together imply that the RHS in $( 9 )$ is approximately equal to $E _ { ( 1 - \\tau ) q ( k - 1 ) }$ . ", + "bbox": [ + 176, + 869, + 823, + 912 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 88, + 826, + 136 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Since $E _ { m }$ is precisely what is required to bound the ternary DP for the specific neighboring datasets, we have reduced the problem of computing the ternary DP for arbitrary neighboring datasets to the problem of computing ternary DP for specific neighboring datasets. The second step of the proof bounds $E _ { ( 1 - \\tau ) q ( n - 1 ) }$ , which follows from the result below that holds for any $m \\in \\mathbb { N }$ . ", + "bbox": [ + 173, + 140, + 826, + 196 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "$[ | \\chi | ^ { \\alpha }$ $m \\in \\mathbb { N }$ $\\alpha \\geq 2$ $( \\mathcal { D } _ { m } ^ { \\prime \\prime } , \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ) \\in \\mathcal { D } _ { s a m e } ^ { m }$ ", + "bbox": [ + 173, + 200, + 825, + 215 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/c4a98378a5434f1a2114b75379bb5580b5016360de3ae1e1fd2f16e009ba53be.jpg", + "text": "$$\n\\mathbb { E } _ { h \\sim M _ { s h } ( \\mathcal { D } _ { m } ) } \\left[ \\left| \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime } ) ( h ) - \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime \\prime } ) ( h ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ) ( h ) } \\right| ^ { \\alpha } \\right] \\leq \\left\\{ \\begin{array} { l l } { 4 \\frac { ( e ^ { \\epsilon _ { 0 } } - 1 ) ^ { 2 } } { m \\epsilon ^ { \\epsilon _ { 0 } } } } & { i f \\alpha = 2 , } \\\\ { \\alpha \\Gamma ( \\alpha / 2 ) \\left( \\frac { 2 ( e ^ { z _ { 0 } } - 1 ) ^ { 2 } } { m e ^ { 2 z _ { 0 } } } \\right) ^ { \\alpha / 2 } } & { o t h e r w i s e . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 181, + 218, + 825, + 268 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Missing details of how Theorem $^ 4$ follows from Theorems 5, 6 can be found in Appendix C.4. ", + "bbox": [ + 173, + 276, + 792, + 295 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Proof sketch of Theorem $\\boxed { 5 }$ Let $\\pmb { p } _ { i } , i \\in [ k ] , \\pmb { p } _ { k } ^ { \\prime } , \\pmb { p } _ { k } ^ { \\prime \\prime }$ denote the distributions of $\\mathcal { R }$ when the input data point is $d _ { i } , d _ { k } ^ { \\prime } , d _ { k } ^ { \\prime \\prime }$ , respectively. The main idea of the proof is the observation that each $\\pmb { p } _ { i }$ can be written as a mixture distribution $\\begin{array} { r } { \\pmb { p } _ { i } = \\frac { 1 } { e ^ { \\epsilon _ { 0 } } } \\pmb { p } _ { k } ^ { \\prime \\prime } + \\left( 1 - \\frac { 1 } { e ^ { \\epsilon _ { 0 } } } \\right) \\tilde { \\pmb { p } } _ { i } } \\end{array}$ , where $\\tilde { \\pmb { p } } _ { i }$ is defined in terms of ${ \\pmb p } _ { i } , { \\pmb p } _ { k } ^ { \\prime \\prime }$ . So, instead of client $i \\in [ k - 1 ]$ mapping its data point $d _ { i }$ according to $\\mathbf { \\nabla } _ { \\pmb { p } _ { i } }$ , we can view it as the client $i$ maps $d _ { i }$ according to $\\pmb { p } _ { k } ^ { \\prime \\prime }$ with probability (w.p.) $1 / e ^ { \\epsilon _ { 0 } }$ and according to $\\tilde { \\pmb { p } } _ { i }$ w.p. $\\left( 1 - 1 / e ^ { \\epsilon _ { 0 } } \\right)$ . As a result, the number of clients that sample from the distribution $\\pmb { p } _ { k } ^ { \\prime \\prime }$ follows a binomial distribution $\\mathrm { B i n } ( k - 1 , 1 / e ^ { \\epsilon _ { 0 } } )$ . This allows us to write the distribution of $\\mathcal { M } _ { s h }$ when clients map their data points according to $\\pmb { p } _ { 1 } , \\ldots , \\pmb { p } _ { k } , \\pmb { p } _ { k } ^ { \\prime } , \\pmb { p } _ { k } ^ { \\prime \\prime }$ as a convex combination of the distribution of $\\mathcal { M }$ when clients map their data points according to $\\tilde { { p } } _ { 1 } , \\ldots , { p } _ { k - 1 } , { p } _ { k } , { p } _ { k } ^ { \\prime } , { p } _ { k } ^ { \\prime \\prime }$ ; see Lemma $\\boxed { 4 }$ Then using a joint convexity argument (see Lemma $\\textcircled { 3 }$ , we write the ternary divergence between the original triple of distributions of $\\mathcal { M } _ { s h }$ in terms of the same convex combination of the ternary divergence between the resulting triples of distributions of $\\mathcal { M } _ { s h }$ as in Lemma $\\bigstar$ Using a monotonicity argument (see Lemma 5), we can remove the effect of clients that do not sample from the distribution $p _ { k } ^ { \\prime \\prime }$ without decreasing the ternary divergence. By this chain of arguments, we have reduced the problem to the one involving the computation of ternary divergence only for the special form of neighboring datasets (as in Theorem 6), which proves Theorem $\\bigtriangledown$ See Appendix C.1 for a complete proof. ", + "bbox": [ + 173, + 306, + 826, + 534 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Proof sketch of Theorem $6 .$ Consider $( \\mathcal { D } _ { m } ^ { \\prime \\prime } , \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ) \\in \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m }$ as in the statement of Theorem First we observe that for any $\\alpha \\geq 1$ and any three distributions $p , q , r$ over the same domain, we can write $\\begin{array} { r } { \\mathbb { E } _ { r } \\left[ \\left| \\frac { p - q } { r } \\right| ^ { \\alpha } \\right] \\leq 2 ^ { \\alpha - 1 } \\left( \\mathbb { E } _ { r } \\left[ \\left| \\frac { p } { r } - 1 \\right| ^ { \\alpha } \\right] + \\mathbb { E } _ { r } \\left[ \\left| \\frac { q } { r } - 1 \\right| ^ { \\alpha } \\right] \\right) } \\end{array}$ . This is a straight-forward application of the standard inequality $| x + y | ^ { \\alpha } \\leq 2 ^ { \\alpha - 1 } ( | x | ^ { \\alpha } + | y | ^ { \\alpha } )$ which holds for all $x , y \\in \\mathbb { R }$ and $\\alpha \\geq 1$ . Now, by taking $p = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime } )$ , $q = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime \\prime } )$ , and $r = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } )$ , we reduce the problem of computing the ternary $| \\chi | ^ { \\alpha } .$ -divergence (which we need to bound) to the problem of computing the Pearson-Vajda divergence $| | \\overline { { 4 3 } } | |$ , which we can write in terms of the $\\alpha$ -th absolute moment of the r.v. $X : \\mathcal { A } _ { B } ^ { m } \\mathbb { R }$ , defined as $\\begin{array} { r } { X ( \\pmb { h } ) : = \\big ( \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime } ) ( \\pmb { h } ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ) ( \\pmb { h } ) } - 1 \\big ) } \\end{array}$ for all $\\pmb { h } \\in \\mathcal { A } _ { B } ^ { m }$ (where $\\mathcal { D } ^ { \\prime } \\in \\{ \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ^ { \\prime \\prime } \\} )$ and distributed according to $X ( \\pmb { h } ) \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ) ( \\pmb { h } )$ . In $\\mathbb { E 9 }$ , the authors have bounded the absolute moments of the r.v. $X ( h )$ by showing that $X ( h )$ is sub-Gaussian r.v. and using standard concentration results. See Appendix C.3 for a complete proof. ", + "bbox": [ + 173, + 545, + 826, + 718 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Discussion ", + "text_level": 1, + "bbox": [ + 174, + 734, + 294, + 752 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we analyzed the Rényi differential privacy of the subsampled shuffle model by bounding the ternary $| \\chi | ^ { \\alpha }$ -DP of the shuffle model. We numerically demonstrated the importance of our proposed bound, where we obtain a significant improvement over using the state-of-the-art in practical regimes. Furthermore, we used our privacy analysis to study the privacy-accuracy trade-offs on the MNIST dataset, where we obtained $9 0 \\%$ accuracy with total privacy budget of $\\epsilon = 2 . 9 1$ , which is an improvement over an analysis yielding 4.82, using standard strong composition theorem. ", + "bbox": [ + 173, + 765, + 825, + 849 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Closing the gap (shown numerically) between our lower bound in Theorem $\\bigstar$ and the achievable upper bound in Theorem $\\triangledown$ is an important unresolved question. Another direction to explore would be to analyze the RDP of the subsampled shuffle model for different sub-sampling techniques such as Poisson subsampling $\\underline { { \\mathbb { G 5 } } } ] |$ , random check-in $\\textcircled { 9 }$ , or client self-sampling $| \\overline { { 3 0 } } \\|$ . ", + "bbox": [ + 174, + 854, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Societal Impact. Collaborative learning comes with significant societal risks of privacy violations, which is the main topic addressed in this paper. However, such learning is only as good as the data used for training, and if the data is not unbiased, this could lead to significant issues related to fairness and could also lead to societally undesirable outcomes. Such an issue is exacerbated when privacy is guaranteed on the data used for training, making a-priori fairness checks on data infeasible. This can be ameliorated by properly testing models finally obtained against fairness criteria and rejecting models that fail the test. This paper did not consider the issue of robustness to security, and this could also be an important societal issue in collaborative learning, where a small subset of users could insert malicious inputs to disrupt the learning process or worse bias the learned model covertly. This could also lead to negative outcomes. This issue of robustness to malicious participants has been studied in several papers, and incorporating this into the framework of the paper is an important future research topic. ", + "bbox": [ + 173, + 90, + 825, + 257 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgment ", + "text_level": 1, + "bbox": [ + 176, + 276, + 320, + 292 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported in part by NSF grants #2007714 and #1955632 and a Google Faculty research award and an Amazon Research Award. ", + "bbox": [ + 173, + 306, + 823, + 335 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 354, + 266, + 371 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] M. Abadi, A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang. Deep learning with differential privacy. 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Federated machine learning: Concept and applications. ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):1–19, 2019. \n[45] Y. Zhu and Y.-X. Wang. Poission subsampled rényi differential privacy. In International Conference on Machine Learning, pages 7634–7642. PMLR, 2019. 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Motivated by stochastic optimization and the federated learning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "score": 1.0, + "content": "(FL) paradigm, we focus on the case where a small fraction of data samples", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 312, + 470, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 470, + 324 + ], + "score": 1.0, + "content": "are randomly sub-sampled in each round to participate in the learning process,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 321, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 335 + ], + "score": 1.0, + "content": "which also enables privacy amplification. To obtain even stronger local privacy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "score": 1.0, + "content": "guarantees, we study this in the shuffle privacy model, where each client randomizes", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 345, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 470, + 356 + ], + "score": 1.0, + "content": "its response using a local differentially private (LDP) mechanism and the server", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 142, + 355, + 469, + 367 + ], + "score": 1.0, + "content": "only receives a random permutation (shuffle) of the clients’ responses without", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 365, + 470, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 470, + 378 + ], + "score": 1.0, + "content": "their association to each client. The principal result of this paper is a privacy-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 377, + 470, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 470, + 388 + ], + "score": 1.0, + "content": "optimization performance trade-off for discrete randomization mechanisms in this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 388, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 469, + 399 + ], + "score": 1.0, + "content": "sub-sampled shuffle privacy model. This is enabled through a new theoretical", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 398, + 469, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 469, + 410 + ], + "score": 1.0, + "content": "technique to analyze the Rényi Differential Privacy (RDP) of the sub-sampled", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 408, + 469, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 469, + 422 + ], + "score": 1.0, + "content": "shuffle model. We numerically demonstrate that, for important regimes, with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "score": 1.0, + "content": "composition our bound yields significant improvement in privacy guarantee over", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 430, + 470, + 444 + ], + "spans": [ + { + "bbox": [ + 141, + 430, + 470, + 444 + ], + "score": 1.0, + "content": "the state-of-the-art approximate Differential Privacy (DP) guarantee (with strong", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 442, + 469, + 454 + ], + "spans": [ + { + "bbox": [ + 142, + 442, + 469, + 454 + ], + "score": 1.0, + "content": "composition) for sub-sampled shuffled models. We also demonstrate numerically", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 453, + 469, + 465 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 469, + 465 + ], + "score": 1.0, + "content": "significant improvement in privacy-learning performance operating point using real", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 464, + 469, + 477 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 469, + 477 + ], + "score": 1.0, + "content": "data sets. Despite these advances an open question is to bridge the gap between", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 475, + 358, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 358, + 488 + ], + "score": 1.0, + "content": "lower and upper privacy bounds in our RDP analysis.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 21.5, + "bbox_fs": [ + 141, + 267, + 470, + 488 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 508, + 190, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 192, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 192, + 523 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "As learning moves towards the edge, there is a need to collaborate to build learning models1, such", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "as in federated learning [36, 44, 33]. In this framework, the collaboration is typically mediated by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "a server. In particular, we want to collaboratively build a learning model by solving an empirical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 263, + 579 + ], + "score": 1.0, + "content": "risk minimization (ERM) problem (see", + "type": "text" + }, + { + "bbox": [ + 263, + 565, + 276, + 579 + ], + "score": 0.78, + "content": "( 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 565, + 318, + 579 + ], + "score": 1.0, + "content": "in Section", + "type": "text" + }, + { + "bbox": [ + 318, + 565, + 329, + 579 + ], + "score": 0.53, + "content": "2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 565, + 473, + 579 + ], + "score": 1.0, + "content": ". To obtain a model parametrized by", + "type": "text" + }, + { + "bbox": [ + 474, + 567, + 480, + 576 + ], + "score": 0.74, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "ERM, the commonly used mechanism is Stochastic Gradient Descent (SGD) [12]. However, one", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "needs to solve this while enabling strong privacy guarantees on local data from the server, while also", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 599, + 503, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 503, + 612 + ], + "score": 1.0, + "content": "obtaining good learning performance, i.e., a suitable privacy-learning performance operating point.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 532, + 505, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 614, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 214, + 628 + ], + "score": 1.0, + "content": "Differential privacy (DP)", + "type": "text" + }, + { + "bbox": [ + 214, + 614, + 232, + 627 + ], + "score": 0.55, + "content": "\\boxed { 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "is the gold standard notion of data privacy that gives a rigorous", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "framework through quantifying the information leakage about individual training data points from", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "the observed interactions. Though DP was originally proposed in a framework where data resides", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 647, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 145, + 661 + ], + "score": 1.0, + "content": "centrally", + "type": "text" + }, + { + "bbox": [ + 145, + 647, + 163, + 659 + ], + "score": 0.37, + "content": "\\boxed { 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 647, + 505, + 661 + ], + "score": 1.0, + "content": ", for distributed learning the more appropriate notion is of local differential privacy", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "(LDP) [35, 17]. Here, each client randomizes its interactions with the server from whom the data is to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 312, + 682 + ], + "score": 1.0, + "content": "be kept private (e.g., see industrial implementations", + "type": "text" + }, + { + "bbox": [ + 314, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "[23, 31, 16]). However, LDP mechanisms suffer", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "from poor performance in comparison with the central DP mechanisms [17, 35, 32]. To overcome", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "this, a new privacy framework using anonymization has been proposed in the so-called shuffled model", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 210, + 106 + ], + "score": 0.32, + "content": "\\underline { { { \\sqrt { 2 2 } } } } | 2 5 | 6 | \\dot { 2 } 6 | 5 | \\dot { \\overline { { { 1 5 } } } } | \\overline { { { \\mathrm { [ 7 ] } } } } | 8 |", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 210, + 93, + 505, + 107 + ], + "score": 1.0, + "content": ". In the shuffled model, each client sends her private message to a secure", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 507, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 507, + 119 + ], + "score": 1.0, + "content": "shuffler that randomly permutes all the received messages before forwarding them to the server.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "This model enables significantly better privacy-utility performance by amplifying DP through this", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "shuffling. Therefore, in this paper we consider the shuffle privacy framework for distributed learning.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 614, + 506, + 682 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "from poor performance in comparison with the central DP mechanisms [17, 35, 32]. To overcome", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "this, a new privacy framework using anonymization has been proposed in the so-called shuffled model", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 210, + 106 + ], + "score": 0.32, + "content": "\\underline { { { \\sqrt { 2 2 } } } } | 2 5 | 6 | \\dot { 2 } 6 | 5 | \\dot { \\overline { { { 1 5 } } } } | \\overline { { { \\mathrm { [ 7 ] } } } } | 8 |", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 93, + 505, + 107 + ], + "score": 1.0, + "content": ". In the shuffled model, each client sends her private message to a secure", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 507, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 507, + 119 + ], + "score": 1.0, + "content": "shuffler that randomly permutes all the received messages before forwarding them to the server.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "This model enables significantly better privacy-utility performance by amplifying DP through this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "shuffling. Therefore, in this paper we consider the shuffle privacy framework for distributed learning.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 149, + 156 + ], + "score": 1.0, + "content": "In solving", + "type": "text" + }, + { + "bbox": [ + 149, + 143, + 162, + 156 + ], + "score": 0.63, + "content": "( 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "using (distributed) gradient descent, each exchange leaks information about the local", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "data, but we need as many steps as possible to obtain a good model; setting up the tension between", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "privacy and performance. The goal is to obtain as many such interactions as possible for a given", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "privacy budget. This is quantified through analyzing the privacy of the composition of privacy", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "mechanisms. Abadi et al. [1] developed a framework for tighter analysis of such compositions, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 196, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 392, + 211 + ], + "score": 1.0, + "content": "this was later reformulated in terms of Rényi Differential Privacy (RDP)", + "type": "text" + }, + { + "bbox": [ + 393, + 197, + 410, + 209 + ], + "score": 0.65, + "content": "\\textcircled { 1 3 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 196, + 506, + 211 + ], + "score": 1.0, + "content": ", and mapping this back", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 172, + 223 + ], + "score": 1.0, + "content": "to DP guarantee", + "type": "text" + }, + { + "bbox": [ + 173, + 208, + 190, + 221 + ], + "score": 0.84, + "content": "\\textcircled { 1 3 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 208, + 505, + 223 + ], + "score": 1.0, + "content": ". 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The central technical question addressed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 382, + 329, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 329, + 395 + ], + "score": 1.0, + "content": "in this paper is how to analyze the RDP of an arbitrary", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 392, + 329, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 329, + 406 + ], + "score": 1.0, + "content": "discrete mechanism for the subsampled shuffle privacy", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 405, + 329, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 329, + 416 + ], + "score": 1.0, + "content": "model. This enables us to answer the overall question", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "image", + "bbox": [ + 337, + 233, + 504, + 331 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 337, + 233, + 504, + 331 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 337, + 233, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 337, + 233, + 504, + 331 + ], + "score": 0.969, + "type": "image", + "image_path": "60fe5dd4ec96e99911e5639baf65cde4fbddb5ef4c7ef331a52a02859b1dfcf1.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 337, + 233, + 504, + 247.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 337, + 247.0, + 504, + 261.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 337, + 261.0, + 504, + 275.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 337, + 275.0, + 504, + 289.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 337, + 289.0, + 504, + 303.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 337, + 303.0, + 504, + 317.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 337, + 317.0, + 504, + 331.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 336, + 337, + 505, + 404 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 335, + 337, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 335, + 337, + 506, + 348 + ], + "score": 1.0, + "content": "Figure 1: An iteration from the CLDP-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 335, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 335, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "SGD Algorithm, where 3 clients are ran-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 335, + 359, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 335, + 359, + 505, + 370 + ], + "score": 1.0, + "content": "domly chosen at each iteration. 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see Theorem", + "type": "text" + }, + { + "bbox": [ + 402, + 464, + 414, + 477 + ], + "score": 0.39, + "content": "\\bigtriangledown", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "The bound is explicit", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "and amenable to numerics, including all constants.2 Furthermore, the bounds are valid for generic", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 484, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 500 + ], + "score": 1.0, + "content": "LDP mechanisms and all parameter regimes.3 We also provide a lower bound for the RDP in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 143, + 510 + ], + "score": 1.0, + "content": "Theorem", + "type": "text" + }, + { + "bbox": [ + 143, + 496, + 154, + 510 + ], + "score": 0.5, + "content": "\\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "We prove our upper bound (Theorem 1) using the following novel analysis techniques:", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "First, we reduce the problem of computing the RDP of sub-sampled shuffle mechanisms to the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 518, + 504, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 226, + 532 + ], + "score": 1.0, + "content": "problem of computing ternary", + "type": "text" + }, + { + "bbox": [ + 227, + 518, + 280, + 531 + ], + "score": 0.6, + "content": "| \\chi | ^ { \\alpha } \\cdot \\mathrm { D P } ^ { \\bullet } | \\overline { { { 4 3 } } } | ]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 518, + 504, + 532 + ], + "score": 1.0, + "content": "of shuffle (non sub-sampled) mechanisms; 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Each client", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 335, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 335, + 370, + 444, + 383 + ], + "score": 1.0, + "content": "sends the private gradient", + "type": "text" + }, + { + "bbox": [ + 445, + 370, + 492, + 383 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { p } \\left( g _ { t } ( d _ { i } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 335, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 335, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "the shuffler that randomly permutes the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 335, + 393, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 335, + 393, + 505, + 404 + ], + "score": 1.0, + "content": "gradients before passing them to the server.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + } + ], + "index": 35.25 + }, + { + "type": "text", + "bbox": [ + 110, + 416, + 505, + 438 + ], + "lines": [], + "index": 42.5, + "bbox_fs": [ + 107, + 413, + 506, + 439 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 442, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "We analyze the RDP of subsampled mechanisms in the shuffle framework by developing a novel", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 237, + 467 + ], + "score": 1.0, + "content": "bound applicable to any discrete", + "type": "text" + }, + { + "bbox": [ + 237, + 455, + 246, + 464 + ], + "score": 0.87, + "content": "\\epsilon _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 452, + 441, + 467 + ], + "score": 1.0, + "content": "-LDP mechanism as a function of the RDP order", + "type": "text" + }, + { + "bbox": [ + 442, + 454, + 449, + 464 + ], + "score": 0.7, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 452, + 506, + 467 + ], + "score": 1.0, + "content": ", subsampling", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 124, + 477 + ], + "score": 1.0, + "content": "rate", + "type": "text" + }, + { + "bbox": [ + 124, + 466, + 132, + 476 + ], + "score": 0.79, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 464, + 216, + 477 + ], + "score": 1.0, + "content": ", the LDP parameter", + "type": "text" + }, + { + "bbox": [ + 216, + 466, + 226, + 475 + ], + "score": 0.84, + "content": "\\epsilon _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 464, + 337, + 477 + ], + "score": 1.0, + "content": ", and the number of clients", + "type": "text" + }, + { + "bbox": [ + 337, + 466, + 344, + 474 + ], + "score": 0.61, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 464, + 402, + 477 + ], + "score": 1.0, + "content": "; see Theorem", + "type": "text" + }, + { + "bbox": [ + 402, + 464, + 414, + 477 + ], + "score": 0.39, + "content": "\\bigtriangledown", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "The bound is explicit", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "and amenable to numerics, including all constants.2 Furthermore, the bounds are valid for generic", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 484, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 500 + ], + "score": 1.0, + "content": "LDP mechanisms and all parameter regimes.3 We also provide a lower bound for the RDP in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 143, + 510 + ], + "score": 1.0, + "content": "Theorem", + "type": "text" + }, + { + "bbox": [ + 143, + 496, + 154, + 510 + ], + "score": 0.5, + "content": "\\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "We prove our upper bound (Theorem 1) using the following novel analysis techniques:", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "First, we reduce the problem of computing the RDP of sub-sampled shuffle mechanisms to the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 518, + 504, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 226, + 532 + ], + "score": 1.0, + "content": "problem of computing ternary", + "type": "text" + }, + { + "bbox": [ + 227, + 518, + 280, + 531 + ], + "score": 0.6, + "content": "| \\chi | ^ { \\alpha } \\cdot \\mathrm { D P } ^ { \\bullet } | \\overline { { { 4 3 } } } | ]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 518, + 504, + 532 + ], + "score": 1.0, + "content": "of shuffle (non sub-sampled) mechanisms; 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See Section 4 and the supplementary material for more results.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 506, + 147 + ], + "score": 1.0, + "content": "Related work: We give a more complete literature review in Appendix A, and focus here on the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 342, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 342, + 157 + ], + "score": 1.0, + "content": "works that are closest to the results presented in this paper.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 320, + 172 + ], + "score": 1.0, + "content": "Private optimization in the shuffled model: Recently,", + "type": "text" + }, + { + "bbox": [ + 320, + 159, + 338, + 171 + ], + "score": 0.47, + "content": "\\pmb { \\mathbb { Z } } 1 \\mathbf { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 159, + 356, + 172 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 159, + 389, + 171 + ], + "score": 0.35, + "content": " { \\mathbb { P } } ^ { \\geq \\sum { \\left. \\left. 2 8 \\right. \\right. } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "have proposed differentially", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "private SGD algorithms for federated learning, where at each iteration, each client applies an LDP", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "mechanism on the gradients with the existence of a secure shuffler between the clients and the central", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "server. However, the privacy analyses in these works developed approximate DP using advanced", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 250, + 215 + ], + "score": 1.0, + "content": "composition theorems for DP (e.g.,", + "type": "text" + }, + { + "bbox": [ + 251, + 203, + 285, + 215 + ], + "score": 0.57, + "content": "\\pm \\pm \\pm \\pm ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 203, + 475, + 215 + ], + "score": 1.0, + "content": "), which are known to be loose for composition", + "type": "text" + }, + { + "bbox": [ + 475, + 203, + 488, + 215 + ], + "score": 0.51, + "content": "\\textcircled { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 203, + 505, + 215 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 228 + ], + "score": 1.0, + "content": "the best of our knowledge, analyzing the private optimization framework using RDP and subsampling", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 275, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 275, + 238 + ], + "score": 1.0, + "content": "in the shuffled model is new to this paper.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 243, + 254 + ], + "score": 1.0, + "content": "Subsampled RDP: The works [38,", + "type": "text" + }, + { + "bbox": [ + 244, + 244, + 257, + 252 + ], + "score": 1.0, + "content": "43,", + "type": "text" + }, + { + "bbox": [ + 258, + 243, + 272, + 252 + ], + "score": 1.0, + "content": "45]", + "type": "text" + }, + { + "bbox": [ + 273, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "have studied the RDP of subsampled mechanisms without", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "shuffling. They demonstrated that this provides a tighter bound on the total privacy loss than the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "bound that can be obtained using the standard strong composition theorems. The RDP analysis of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "subsampled mechanisms in the shuffled privacy framework has not been studied before,4 and is new to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 382, + 299 + ], + "score": 1.0, + "content": "this paper. The RDP of the shuffled model was very recently studied in", + "type": "text" + }, + { + "bbox": [ + 382, + 285, + 400, + 297 + ], + "score": 0.35, + "content": "\\left[ \\left[ 2 9 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 285, + 506, + 299 + ], + "score": 1.0, + "content": ", but without incorporating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "subsampling, which poses new technical challenges, as directly bounding the RDP of subsampled", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "shuffle mechanisms is non-trivial. 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Among these, the local and central differen-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 375, + 448 + ], + "score": 1.0, + "content": "tial privacy definitions are standard and we defer them to Appendix", + "type": "text" + }, + { + "bbox": [ + 375, + 434, + 387, + 448 + ], + "score": 0.82, + "content": "\\begin{array} { l } { \\mathbf { B } . } \\\\ { . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "The other privacy definitions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 445, + 454, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 199, + 459 + ], + "score": 1.0, + "content": "(Rényi DP and ternary", + "type": "text" + }, + { + "bbox": [ + 199, + 446, + 217, + 458 + ], + "score": 0.76, + "content": "| \\chi | ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 445, + 454, + 459 + ], + "score": 1.0, + "content": "-DP) are relatively less standard and we define them below.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 104, + 460, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 209, + 476 + ], + "score": 1.0, + "content": "We say that two datasets", + "type": "text" + }, + { + "bbox": [ + 209, + 462, + 312, + 474 + ], + "score": 0.92, + "content": "\\mathcal { D } = \\{ d _ { 1 } , \\ldots , d _ { n } \\} \\in \\mathcal { X } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 460, + 331, + 476 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 331, + 462, + 437, + 474 + ], + "score": 0.92, + "content": "\\mathcal { D } ^ { \\prime } = \\{ d _ { 1 } ^ { \\prime } , \\ldots , d _ { n } ^ { \\prime } \\} \\in \\mathcal { X } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 460, + 506, + 476 + ], + "score": 1.0, + "content": "are neighboring", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 471, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 171, + 487 + ], + "score": 1.0, + "content": "(and denoted by", + "type": "text" + }, + { + "bbox": [ + 172, + 473, + 205, + 483 + ], + "score": 0.86, + "content": "\\mathcal { D } \\sim \\mathcal { D } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 471, + 405, + 487 + ], + "score": 1.0, + "content": ") if they differ in one data point, i.e., there exists an", + "type": "text" + }, + { + "bbox": [ + 406, + 474, + 433, + 485 + ], + "score": 0.9, + "content": "i \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 471, + 472, + 487 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 473, + 473, + 504, + 485 + ], + "score": 0.92, + "content": "d _ { i } \\neq d _ { i } ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 482, + 293, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 162, + 498 + ], + "score": 1.0, + "content": "and for every", + "type": "text" + }, + { + "bbox": [ + 162, + 484, + 217, + 496 + ], + "score": 0.9, + "content": "j \\in [ n ] , j \\neq i", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 482, + 256, + 498 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 256, + 484, + 289, + 498 + ], + "score": 0.92, + "content": "d _ { j } = d _ { j } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 482, + 293, + 498 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 159, + 514 + ], + "score": 1.0, + "content": "Definition 1", + "type": "text" + }, + { + "bbox": [ + 159, + 502, + 183, + 514 + ], + "score": 0.84, + "content": "( \\lambda , \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 501, + 316, + 514 + ], + "score": 1.0, + "content": "-RDP (Rényi Differential Privacy)", + "type": "text" + }, + { + "bbox": [ + 316, + 501, + 335, + 513 + ], + "score": 0.77, + "content": "\\pmb { \\mathbb { B } } \\pmb { \\mathbb { Z } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 501, + 446, + 514 + ], + "score": 1.0, + "content": "). 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Our main", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "contribution in this work is to present a stronger privacy analysis of the CLDP-SGD algorithm by", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 586, + 339, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 339, + 597 + ], + "score": 1.0, + "content": "characterizing the RDP of the sub-sampled shuffle model.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 59 + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 193, + 627 + ], + "lines": [ + { + "bbox": [ + 104, + 611, + 195, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 195, + 629 + ], + "score": 1.0, + "content": "3 Main Results", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 61 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "score": 1.0, + "content": "In this section, we present our main results. 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We", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 287, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 287, + 130 + ], + "score": 1.0, + "content": "define the subsampled-shuffle mechanism as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 131, + 402, + 144 + ], + "lines": [ + { + "bbox": [ + 209, + 131, + 402, + 144 + ], + "spans": [ + { + "bbox": [ + 209, + 131, + 402, + 144 + ], + "score": 0.92, + "content": "\\mathcal { M } \\left( \\mathcal { D } \\right) : = \\mathcal { H } _ { k } \\circ \\operatorname { s a m p } _ { k } ^ { n } \\left( \\mathcal { R } \\left( d _ { 1 } \\right) , \\dotsc , \\mathcal { R } \\left( d _ { n } \\right) \\right) .", + "type": "interline_equation", + "image_path": "0223bb31f1664a464c00b8082a8b37d0849c1e33f21bbc166f164cecae74c6d9.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 209, + 131, + 402, + 144 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 506, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 506, + 159 + ], + "score": 1.0, + "content": "Observe that each iteration of Algorithm 1 can be represented as an output of the subsampled shuffle", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 154, + 170 + ], + "score": 1.0, + "content": "mechanism", + "type": "text" + }, + { + "bbox": [ + 154, + 158, + 167, + 167 + ], + "score": 0.83, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 156, + 338, + 170 + ], + "score": 1.0, + "content": ". 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It will be useful to define the following notation. 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The term", + "type": "text" + }, + { + "bbox": [ + 486, + 399, + 495, + 408 + ], + "score": 0.75, + "content": "\\Upsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 393, + 509, + 415 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 99, + 404, + 347, + 441 + ], + "spans": [ + { + "bbox": [ + 99, + 404, + 142, + 441 + ], + "score": 1.0, + "content": "given by", + "type": "text" + }, + { + "bbox": [ + 142, + 411, + 345, + 437 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Upsilon = \\left( \\left( 1 + \\gamma \\frac { e ^ { 2 \\epsilon _ { 0 } } - 1 } { e ^ { \\epsilon _ { 0 } } } \\right) ^ { \\lambda } - 1 - \\lambda \\gamma \\frac { e ^ { 2 \\epsilon _ { 0 } } - 1 } { e ^ { \\epsilon _ { 0 } } } \\right) e ^ { - \\frac { k - 1 } { 8 e ^ { \\epsilon _ { 0 } } } } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 417, + 347, + 430 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 504, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 257, + 451 + ], + "score": 1.0, + "content": "Theorem 2 (Lower Bound). 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"content": "\\mathcal { M } \\left( \\mathcal { D } \\right) : = \\mathcal { H } _ { k } \\circ \\operatorname { s a m p } _ { k } ^ { n } \\left( \\mathcal { R } \\left( d _ { 1 } \\right) , \\dotsc , \\mathcal { R } \\left( d _ { n } \\right) \\right) .", + "type": "interline_equation", + "image_path": "0223bb31f1664a464c00b8082a8b37d0849c1e33f21bbc166f164cecae74c6d9.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 209, + 131, + 402, + 144 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 506, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 506, + 159 + ], + "score": 1.0, + "content": "Observe that each iteration of Algorithm 1 can be represented as an output of the subsampled shuffle", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 154, + 170 + ], + 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For any", + "type": "text" + }, + { + "bbox": [ + 258, + 331, + 346, + 343 + ], + "score": 0.38, + "content": "n \\in \\mathbb { N } , k \\leq n , \\epsilon _ { 0 } \\geq 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 330, + 414, + 344 + ], + "score": 1.0, + "content": "and any integer", + "type": "text" + }, + { + "bbox": [ + 414, + 331, + 439, + 342 + ], + "score": 0.9, + "content": "\\lambda \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 330, + 505, + 344 + ], + "score": 1.0, + "content": ", the RDP of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 393, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 231, + 356 + ], + "score": 1.0, + "content": "subsampled shuffle mechanism", + "type": "text" + }, + { + "bbox": [ + 232, + 343, + 245, + 353 + ], + "score": 0.79, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 342, + 291, + 356 + ], + "score": 1.0, + "content": "(defined in", + "type": "text" + }, + { + "bbox": [ + 291, + 342, + 304, + 355 + ], + "score": 0.7, + "content": "\\textcircled { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 342, + 393, + 356 + ], + "score": 1.0, + "content": ") is upper-bounded by", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 330, + 505, + 356 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 111, + 356, + 502, + 394 + ], + "spans": [ + { + "bbox": [ + 111, + 356, + 502, + 394 + ], + "score": 0.92, + "content": "\\epsilon ( \\lambda ) \\leq \\frac { 1 } { \\lambda - 1 } \\log \\left( 1 + 4 \\binom { \\lambda } { 2 } \\gamma ^ { 2 } \\frac { \\left( e ^ { \\epsilon _ { 0 } } - 1 \\right) ^ { 2 } } { \\bar { k } e ^ { \\epsilon _ { 0 } } } + \\sum _ { j = 3 } ^ { \\lambda } \\binom { \\lambda } { j } \\gamma ^ { j } j \\Gamma \\left( j / 2 \\right) \\left( \\frac { 2 \\left( e ^ { 2 \\epsilon _ { 0 } } - 1 \\right) ^ { 2 } } { \\bar { k } e ^ { 2 \\epsilon _ { 0 } } } \\right) ^ { j / 2 } + \\Upsilon \\right) ,", + "type": "inline_equation", + "image_path": "4bfb2327302568b1542701d5242fd3e403af3ec3ffa64153b1f839a71247e934.jpg" + } + ], + "index": 22 + }, + { + "bbox": [ + 103, + 393, + 509, + 415 + ], + "spans": [ + { + "bbox": [ + 103, + 393, + 133, + 415 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 397, + 228, + 411 + ], + "score": 0.76, + "content": "\\begin{array} { r } { \\overline { { k } } = \\lfloor \\frac { k - 1 } { 2 e ^ { \\epsilon _ { 0 } } } \\rfloor + 1 , \\gamma = \\frac { k } { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 393, + 249, + 415 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 250, + 397, + 349, + 411 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\Gamma \\left( z \\right) = \\int _ { 0 } ^ { \\infty } x ^ { z - 1 } e ^ { - x } d x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 393, + 486, + 415 + ], + "score": 1.0, + "content": "is the Gamma function. 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See Appendix F for a complete proof of Theorem 3.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 410, + 513 + ], + "score": 1.0, + "content": "Remark 1. Note that our convergence bound is affected by the variance of the", + "type": "text" + }, + { + "bbox": [ + 410, + 501, + 420, + 511 + ], + "score": 0.83, + "content": "\\epsilon _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 501, + 489, + 513 + ], + "score": 1.0, + "content": "-LDP mechanism", + "type": "text" + }, + { + "bbox": [ + 489, + 501, + 503, + 513 + ], + "score": 0.88, + "content": "\\mathcal { R } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 501, + 506, + 513 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 182, + 524 + ], + "score": 1.0, + "content": "For example, when", + "type": "text" + }, + { + "bbox": [ + 183, + 512, + 190, + 523 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 510, + 199, + 524 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 199, + 512, + 207, + 521 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 510, + 327, + 524 + ], + "score": 1.0, + "content": "-Lipschitz continuous w.r.t. the", + "type": "text" + }, + { + "bbox": [ + 328, + 512, + 337, + 522 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 510, + 489, + 524 + ], + "score": 1.0, + "content": "-norm, we can use the LDP mechanism", + "type": "text" + }, + { + "bbox": [ + 490, + 513, + 504, + 523 + ], + "score": 0.84, + "content": "\\mathcal { R } _ { 2 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 102, + 518, + 509, + 540 + ], + "spans": [ + { + "bbox": [ + 102, + 518, + 155, + 540 + ], + "score": 1.0, + "content": "proposed in", + "type": "text" + }, + { + "bbox": [ + 155, + 524, + 173, + 536 + ], + "score": 0.55, + "content": "\\boxed { 1 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 518, + 243, + 540 + ], + "score": 1.0, + "content": "that has variance", + "type": "text" + }, + { + "bbox": [ + 243, + 523, + 348, + 538 + ], + "score": 0.92, + "content": "\\begin{array} { r } { G _ { 2 } ^ { 2 } ( L ) = 1 4 L ^ { 2 } d \\big ( \\frac { e ^ { \\epsilon _ { 0 } } + 1 } { e ^ { \\epsilon _ { 0 } } - 1 } \\big ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 518, + 392, + 540 + ], + "score": 1.0, + "content": "; 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Convergence: If we run", + "type": "text" + }, + { + "bbox": [ + 217, + 384, + 239, + 397 + ], + "score": 0.87, + "content": "\\mathcal { A } _ { \\mathrm { c l d p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 376, + 259, + 403 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 259, + 383, + 300, + 399 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\eta _ { t } = \\frac { D } { G \\sqrt { t } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 376, + 331, + 403 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 332, + 379, + 465, + 398 + ], + "score": 0.76, + "content": "\\begin{array} { r } { G ^ { 2 } = \\operatorname* { m a x } \\lbrace d ^ { 1 - \\frac { 2 } { p } } , 1 \\rbrace L ^ { 2 } + \\frac { G _ { p } ^ { 2 } ( L ) } { \\gamma n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 376, + 500, + 403 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 224, + 397, + 397, + 424 + ], + "spans": [ + { + "bbox": [ + 224, + 397, + 397, + 424 + ], + "score": 0.51, + "content": "\\mathbb { E } \\left[ F \\left( \\theta _ { T } \\right) \\right] - F \\left( \\theta ^ { * } \\right) \\leq \\mathcal { O } \\left( \\frac { D G \\log ( T ) } { \\sqrt { T } } \\right) .", + "type": "inline_equation", + "image_path": "479bd820ec52982df6644d125494f311f86e15a806071a38862fface8445d2bd.jpg" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 100, + 376, + 500, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 431, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 224, + 444 + ], + "score": 1.0, + "content": "The proof outline of Theorem", + "type": "text" + }, + { + "bbox": [ + 225, + 431, + 234, + 444 + ], + "score": 0.6, + "content": "3", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 431, + 327, + 444 + ], + "score": 1.0, + "content": "is as follows: Note that", + "type": "text" + }, + { + "bbox": [ + 327, + 432, + 349, + 444 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { \\mathrm { c l d p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "is an iterative algorithm, where in each", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 369, + 455 + ], + "score": 1.0, + "content": "iteration we use the subsampled shuffle mechanism as defined in", + "type": "text" + }, + { + "bbox": [ + 369, + 442, + 382, + 455 + ], + "score": 0.88, + "content": "\\textcircled{3}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 442, + 506, + 455 + ], + "score": 1.0, + "content": ", for which we have computed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 242, + 466 + ], + "score": 1.0, + "content": "the RDP guarantees in Theorem", + "type": "text" + }, + { + "bbox": [ + 242, + 453, + 253, + 467 + ], + "score": 0.34, + "content": "\\bigtriangledown", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 453, + 394, + 466 + ], + "score": 1.0, + "content": "Now, for the privacy analysis of", + "type": "text" + }, + { + "bbox": [ + 394, + 454, + 416, + 465 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { \\mathrm { c l d p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 453, + 506, + 466 + ], + "score": 1.0, + "content": ", we use the adaptive", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 219, + 476 + ], + "score": 1.0, + "content": "composition theorem from", + "type": "text" + }, + { + "bbox": [ + 220, + 464, + 237, + 475 + ], + "score": 0.76, + "content": "\\textcircled { 1 3 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "Proposition 1] and then use the RDP to DP conversion given in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 137, + 488 + ], + "score": 1.0, + "content": "Lemma", + "type": "text" + }, + { + "bbox": [ + 138, + 474, + 148, + 488 + ], + "score": 0.44, + "content": "^ { 1 . }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "For the convergence analysis, we use a standard non-private SGD convergence result and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 486, + 482, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 482, + 500 + ], + "score": 1.0, + "content": "compute the required parameters for that. See Appendix F for a complete proof of Theorem 3.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 431, + 506, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 410, + 513 + ], + "score": 1.0, + "content": "Remark 1. Note that our convergence bound is affected by the variance of the", + "type": "text" + }, + { + "bbox": [ + 410, + 501, + 420, + 511 + ], + "score": 0.83, + "content": "\\epsilon _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 501, + 489, + 513 + ], + "score": 1.0, + "content": "-LDP mechanism", + "type": "text" + }, + { + "bbox": [ + 489, + 501, + 503, + 513 + ], + "score": 0.88, + "content": "\\mathcal { R } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 501, + 506, + 513 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 182, + 524 + ], + "score": 1.0, + "content": "For example, when", + "type": "text" + }, + { + "bbox": [ + 183, + 512, + 190, + 523 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 510, + 199, + 524 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 199, + 512, + 207, + 521 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 510, + 327, + 524 + ], + "score": 1.0, + "content": "-Lipschitz continuous w.r.t. the", + "type": "text" + }, + { + "bbox": [ + 328, + 512, + 337, + 522 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 510, + 489, + 524 + ], + "score": 1.0, + "content": "-norm, we can use the LDP mechanism", + "type": "text" + }, + { + "bbox": [ + 490, + 513, + 504, + 523 + ], + "score": 0.84, + "content": "\\mathcal { R } _ { 2 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 102, + 518, + 509, + 540 + ], + "spans": [ + { + "bbox": [ + 102, + 518, + 155, + 540 + ], + "score": 1.0, + "content": "proposed in", + "type": "text" + }, + { + "bbox": [ + 155, + 524, + 173, + 536 + ], + "score": 0.55, + "content": "\\boxed { 1 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 518, + 243, + 540 + ], + "score": 1.0, + "content": "that has variance", + "type": "text" + }, + { + "bbox": [ + 243, + 523, + 348, + 538 + ], + "score": 0.92, + "content": "\\begin{array} { r } { G _ { 2 } ^ { 2 } ( L ) = 1 4 L ^ { 2 } d \\big ( \\frac { e ^ { \\epsilon _ { 0 } } + 1 } { e ^ { \\epsilon _ { 0 } } - 1 } \\big ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 518, + 392, + 540 + ], + "score": 1.0, + "content": "; and when", + "type": "text" + }, + { + "bbox": [ + 393, + 525, + 400, + 537 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 518, + 410, + 540 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 410, + 525, + 418, + 535 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 518, + 509, + 540 + ], + "score": 1.0, + "content": "-Lipschitz continuous", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 142, + 550 + ], + "score": 1.0, + "content": "w.r.t. the", + "type": "text" + }, + { + "bbox": [ + 142, + 537, + 153, + 548 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 537, + 189, + 550 + ], + "score": 1.0, + "content": "-norm or", + "type": "text" + }, + { + "bbox": [ + 189, + 537, + 203, + 548 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 537, + 365, + 550 + ], + "score": 1.0, + "content": "-norm, we can use the LDP mechanisms", + "type": "text" + }, + { + "bbox": [ + 366, + 537, + 384, + 548 + ], + "score": 0.9, + "content": "\\mathcal { R } _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 537, + 396, + 550 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 396, + 538, + 410, + 548 + ], + "score": 0.88, + "content": "\\mathcal { R } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 537, + 506, + 550 + ], + "score": 1.0, + "content": ", respectively, proposed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 543, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 107, + 543, + 117, + 569 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 550, + 135, + 563 + ], + "score": 0.76, + "content": "\\lVert 2 7 \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 543, + 216, + 569 + ], + "score": 1.0, + "content": "1 that have variances", + "type": "text" + }, + { + "bbox": [ + 216, + 548, + 319, + 564 + ], + "score": 0.93, + "content": "\\begin{array} { r } { G _ { \\infty } ^ { 2 } ( L ) = L ^ { 2 } d ^ { 2 } \\big ( \\frac { e ^ { \\epsilon _ { 0 } } + 1 } { e ^ { \\epsilon _ { 0 } } - 1 } \\big ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 545, + 439, + 568 + ], + "score": 1.0, + "content": "2 and G21(L) = L2d \u0000 e✏0 +1e✏0 \u00001 \u00002 ,", + "type": "text" + }, + { + "bbox": [ + 433, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "respectively. By", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 205, + 577 + ], + "score": 1.0, + "content": "plugging these variances", + "type": "text" + }, + { + "bbox": [ + 205, + 564, + 233, + 578 + ], + "score": 0.93, + "content": "G _ { p } ^ { 2 } ( L )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 563, + 252, + 577 + ], + "score": 1.0, + "content": "(for", + "type": "text" + }, + { + "bbox": [ + 252, + 565, + 302, + 576 + ], + "score": 0.9, + "content": "p = 1 , 2 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 563, + 358, + 577 + ], + "score": 1.0, + "content": "into Theorem", + "type": "text" + }, + { + "bbox": [ + 358, + 563, + 369, + 575 + ], + "score": 0.34, + "content": "3 .", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "we get the convergence rate of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 574, + 402, + 588 + ], + "spans": [ + { + "bbox": [ + 107, + 576, + 114, + 585 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 574, + 293, + 588 + ], + "score": 1.0, + "content": "-Lipschitz continuous loss function w.r.t. the", + "type": "text" + }, + { + "bbox": [ + 293, + 576, + 303, + 587 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 574, + 345, + 588 + ], + "score": 1.0, + "content": "-norm (for", + "type": "text" + }, + { + "bbox": [ + 346, + 576, + 395, + 587 + ], + "score": 0.87, + "content": "p \\equiv \\infty , 2 , 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 574, + 402, + 588 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 102, + 501, + 509, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 261, + 603 + ], + "score": 1.0, + "content": "Remark 2. The privacy parameter in", + "type": "text" + }, + { + "bbox": [ + 261, + 588, + 274, + 602 + ], + "score": 0.88, + "content": "( 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "is not in a closed form expression and could be obtained", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "by solving an optimization problem. However, we numerically compute it for several interesting", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 611, + 442, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 442, + 624 + ], + "score": 1.0, + "content": "regimes of parameters in our numerical experiments; see Section 4 for more details.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 588, + 506, + 624 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 637, + 220, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 221, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 221, + 653 + ], + "score": 1.0, + "content": "4 Numerical Results", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "In this section, we present numerical experiments to show the performance of our bounds on RDP", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "score": 1.0, + "content": "of the subsampled shuffle mechanism and its usage for getting approximate DP of Algorithm 1 for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 684, + 245, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 245, + 695 + ], + "score": 1.0, + "content": "training machine learning models.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 661, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "Composition of a sequence of subsampled shuffle models: In Figure 2, we plot several bounds", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 189, + 723 + ], + "score": 1.0, + "content": "on the approximate", + "type": "text" + }, + { + "bbox": [ + 189, + 711, + 211, + 723 + ], + "score": 0.91, + "content": "( \\epsilon , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 710, + 316, + 723 + ], + "score": 1.0, + "content": "-DP for a composition of", + "type": "text" + }, + { + "bbox": [ + 316, + 712, + 324, + 721 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 710, + 380, + 723 + ], + "score": 1.0, + "content": "mechanisms", + "type": "text" + }, + { + "bbox": [ + 380, + 711, + 444, + 723 + ], + "score": 0.91, + "content": "( \\mathcal { M } _ { 1 } , \\ldots , \\mathcal { M } _ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 710, + 477, + 723 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 477, + 711, + 494, + 722 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 71, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 259, + 86 + ], + "score": 1.0, + "content": "a subsampled shuffle mechanism for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 259, + 73, + 290, + 85 + ], + "score": 0.92, + "content": "t \\in [ T ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 290, + 72, + 464, + 86 + ], + "score": 1.0, + "content": ". In all our experiments reported in Figure", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 465, + 71, + 476, + 86 + ], + "score": 0.73, + "content": "2 .", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 477, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "we fix", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 146, + 94 + ], + "score": 0.91, + "content": "\\delta = 1 0 ^ { - 8 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 146, + 81, + 506, + 97 + ], + "score": 1.0, + "content": ". We observe that our new bound on the RDP of the subsampled shuffle mechanism achieves", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 247, + 107 + ], + "score": 1.0, + "content": "a significant saving in total privacy", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 247, + 97, + 253, + 104 + ], + "score": 0.51, + "content": "\\epsilon", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 253, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "compared to the state-of-the-art. 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We observe in Figure", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 356, + 137, + 371, + 151 + ], + "score": 0.34, + "content": "\\boxed { 2 \\mathrm { b } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 371, + 137, + 465, + 150 + ], + "score": 1.0, + "content": "that the bound given in", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 466, + 137, + 483, + 149 + ], + "score": 0.81, + "content": "\\pmb { \\mathbb { Z 4 } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 483, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "with", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 232, + 161 + ], + "score": 1.0, + "content": "the strong composition theorem", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 233, + 149, + 250, + 160 + ], + "score": 0.7, + "content": "\\pm", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 251, + 149, + 418, + 161 + ], + "score": 1.0, + "content": "behaves better than the bound on the RDP", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 418, + 149, + 436, + 160 + ], + "score": 0.77, + "content": "\\pmb { \\bigtriangledown }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 436, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "with subsampled", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 157, + 172 + ], + "score": 1.0, + "content": "RDP bound", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 158, + 159, + 175, + 171 + ], + "score": 0.81, + "content": "\\boxed { \\boxed { 4 3 } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 176, + 158, + 440, + 172 + ], + "score": 1.0, + "content": "when the number of subsampled clients per iteration is equal to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 440, + 160, + 502, + 171 + ], + "score": 0.93, + "content": "k = \\gamma n = \\mathrm { \\bar { 1 0 ^ { 4 } } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 502, + 158, + 506, + 172 + ], + "score": 1.0, + "content": ";", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 169, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 309, + 183 + ], + "score": 1.0, + "content": "however, our bound beats both of them.6 In Figure", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 310, + 169, + 325, + 184 + ], + "score": 0.52, + "content": "\\boxed { 2 \\mathrm { c } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 325, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "we fix the number of subsampled clients per", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 448, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 164, + 194 + ], + "score": 1.0, + "content": "iteration to be", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 164, + 181, + 223, + 194 + ], + "score": 0.93, + "content": "k = \\gamma n = 1 0 ^ { 3 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 224, + 181, + 382, + 194 + ], + "score": 1.0, + "content": ", and hence, the subsampling parameter", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 382, + 184, + 390, + 194 + ], + "score": 0.79, + "content": "\\gamma", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 390, + 181, + 437, + 194 + ], + "score": 1.0, + "content": "varies with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 437, + 184, + 444, + 192 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 444, + 181, + 448, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 699, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 71, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 259, + 86 + ], + "score": 1.0, + "content": "a subsampled shuffle mechanism for", + "type": "text" + }, + { + "bbox": [ + 259, + 73, + 290, + 85 + ], + "score": 0.92, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 72, + 464, + 86 + ], + "score": 1.0, + "content": ". 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For our privacy analysis, the total privacy budget is computed by optimizing over RDP order", + "type": "text" + }, + { + "bbox": [ + 497, + 474, + 504, + 483 + ], + "score": 0.72, + "content": "\\lambda", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 482, + 507, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 267, + 499 + ], + "score": 1.0, + "content": "using our upper bound given in Theorem", + "type": "text" + }, + { + "bbox": [ + 267, + 483, + 278, + 497 + ], + "score": 0.53, + "content": "^ { 1 . }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 482, + 371, + 499 + ], + "score": 1.0, + "content": "For privacy analysis of", + "type": "text" + }, + { + "bbox": [ + 371, + 483, + 388, + 496 + ], + "score": 0.84, + "content": "\\dot { \\lVert 2 4 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 482, + 507, + 499 + ], + "score": 1.0, + "content": ", we first compute the privacy", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 302, + 509 + ], + "score": 1.0, + "content": "amplification by shuffling numerically given in", + "type": "text" + }, + { + "bbox": [ + 302, + 495, + 320, + 506 + ], + "score": 0.82, + "content": "\\lVert 2 4 \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "; then we compute its privacy obtained when", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 212, + 519 + ], + "score": 1.0, + "content": "amplified via subsampling", + "type": "text" + }, + { + "bbox": [ + 213, + 505, + 230, + 518 + ], + "score": 0.74, + "content": "\\lVert \\overline { { 4 2 } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 505, + 434, + 519 + ], + "score": 1.0, + "content": "; and finally we use the strong composition theorem", + "type": "text" + }, + { + "bbox": [ + 435, + 505, + 452, + 518 + ], + "score": 0.85, + "content": "\\overleftarrow { \\mathbb { B } 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "to obtain the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 517, + 220, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 210, + 529 + ], + "score": 1.0, + "content": "central privacy parameter", + "type": "text" + }, + { + "bbox": [ + 210, + 519, + 216, + 527 + ], + "score": 0.36, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 517, + 220, + 529 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 280, + 547 + ], + "score": 1.0, + "content": "We observe that we achieve an accuracy of", + "type": "text" + }, + { + "bbox": [ + 280, + 533, + 327, + 545 + ], + "score": 0.88, + "content": "8 0 \\% ( \\pm 1 . 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 532, + 448, + 547 + ], + "score": 1.0, + "content": "with a total privacy budget of", + "type": "text" + }, + { + "bbox": [ + 448, + 534, + 480, + 544 + ], + "score": 0.89, + "content": "\\epsilon = 1 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 250, + 558 + ], + "score": 1.0, + "content": "our new privacy analysis, whereas,", + "type": "text" + }, + { + "bbox": [ + 251, + 544, + 268, + 556 + ], + "score": 0.79, + "content": "\\dot { \\lVert 2 4 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 543, + 389, + 558 + ], + "score": 1.0, + "content": "achieves an accuracy of only", + "type": "text" + }, + { + "bbox": [ + 389, + 544, + 444, + 556 + ], + "score": 0.91, + "content": "7 0 . 7 \\% ( \\pm 2 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "with the same", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 180, + 568 + ], + "score": 1.0, + "content": "privacy budget of", + "type": "text" + }, + { + "bbox": [ + 180, + 556, + 212, + 566 + ], + "score": 0.9, + "content": "\\epsilon = 1 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "using the standard composition theorems. 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see Appendix G for more details on this, and also for more numerical comparisons.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 71, + 505, + 194 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 104, + 71, + 506, + 194 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Distributed private learning: We numerically evaluate the proposed privacy-learning performance", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "on training machine learning models. 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Adapting that result to our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 261, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 261, + 308 + ], + "score": 1.0, + "content": "setting, we have the following lemma.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 180, + 324 + ], + "score": 1.0, + "content": "Lemma 2 (From", + "type": "text" + }, + { + "bbox": [ + 180, + 311, + 186, + 322 + ], + "score": 0.81, + "content": "\\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 309, + 221, + 324 + ], + "score": 1.0, + "content": "-ternary-", + "type": "text" + }, + { + "bbox": [ + 221, + 310, + 239, + 322 + ], + "score": 0.88, + "content": "| \\chi | ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 309, + 482, + 324 + ], + "score": 1.0, + "content": "-DP to subsampled RDP). 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For any", + "type": "text" + }, + { + "bbox": [ + 241, + 321, + 295, + 333 + ], + "score": 0.91, + "content": "\\lambda \\geq 2 , k \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 321, + 468, + 334 + ], + "score": 1.0, + "content": ", RDP of the subsampled shuffle mechanism", + "type": "text" + }, + { + "bbox": [ + 469, + 322, + 481, + 331 + ], + "score": 0.76, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "(with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 332, + 481, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 203, + 350 + ], + "score": 1.0, + "content": "subsampling parameter", + "type": "text" + }, + { + "bbox": [ + 203, + 334, + 241, + 347 + ], + "score": 0.91, + "content": "\\gamma = k / n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 332, + 306, + 350 + ], + "score": 1.0, + "content": ") is bounded by:", + "type": "text" + }, + { + "bbox": [ + 306, + 332, + 476, + 348 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\epsilon ( \\lambda ) \\leq \\frac { 1 } { \\lambda - 1 } \\log \\left( 1 + \\sum _ { \\alpha = 2 } ^ { \\lambda } \\binom { \\lambda } { \\alpha } \\gamma ^ { \\alpha } \\zeta ( \\alpha ) ^ { \\alpha } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 332, + 481, + 350 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 506, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 261, + 370 + ], + "score": 1.0, + "content": "Lemma 2 can be seen as a corollary to", + "type": "text" + }, + { + "bbox": [ + 261, + 355, + 277, + 367 + ], + "score": 0.63, + "content": "\\boxed { \\ 4 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 354, + 507, + 370 + ], + "score": 1.0, + "content": "Proposition 16 and Eq. (9)]. However, for completeness,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 201, + 380 + ], + "score": 1.0, + "content": "we prove it in Appendix", + "type": "text" + }, + { + "bbox": [ + 201, + 366, + 221, + 381 + ], + "score": 0.34, + "content": "\\boxed { \\mathrm { E . 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 366, + 325, + 380 + ], + "score": 1.0, + "content": "Substituting the bound on", + "type": "text" + }, + { + "bbox": [ + 325, + 367, + 345, + 379 + ], + "score": 0.91, + "content": "\\zeta ( \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "from Theorem 4 into Lemma 2 together", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 377, + 462, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 400, + 390 + ], + "score": 1.0, + "content": "with some algebraic manipulation gives proves Theorem 1; see Appendix", + "type": "text" + }, + { + "bbox": [ + 401, + 378, + 418, + 390 + ], + "score": 0.47, + "content": "\\mathrm { E } . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 377, + 462, + 390 + ], + "score": 1.0, + "content": "for details.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 106, + 403, + 424, + 420 + ], + "lines": [ + { + "bbox": [ + 104, + 401, + 426, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 216, + 424 + ], + "score": 1.0, + "content": "6 Proof of Theorem", + "type": "text" + }, + { + "bbox": [ + 216, + 403, + 229, + 421 + ], + "score": 0.64, + "content": "\\mathbf { 4 } ;", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 401, + 276, + 424 + ], + "score": 1.0, + "content": "Ternary", + "type": "text" + }, + { + "bbox": [ + 276, + 405, + 297, + 419 + ], + "score": 0.86, + "content": "| \\chi | ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 401, + 426, + 424 + ], + "score": 1.0, + "content": "-DP of the Shuffle Model", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 506, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "The proof has two main steps. In the first step, we reduce the problem of deriving ternary divergence", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "for arbitrary neighboring datasets to the problem of deriving the ternary divergence for specific", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 504, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 192, + 463 + ], + "score": 1.0, + "content": "neighboring datasets,", + "type": "text" + }, + { + "bbox": [ + 192, + 451, + 251, + 461 + ], + "score": 0.92, + "content": "\\mathcal { D } \\stackrel { - } { \\sim } \\mathcal { D } ^ { \\prime } \\sim \\mathcal { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 452, + 340, + 463 + ], + "score": 1.0, + "content": ", where all elements in", + "type": "text" + }, + { + "bbox": [ + 341, + 452, + 350, + 461 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 452, + 418, + 463 + ], + "score": 1.0, + "content": "are the same and", + "type": "text" + }, + { + "bbox": [ + 419, + 451, + 448, + 462 + ], + "score": 0.73, + "content": "\\mathcal { D } ^ { \\prime } , \\mathcal { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 452, + 494, + 463 + ], + "score": 1.0, + "content": "differ from", + "type": "text" + }, + { + "bbox": [ + 495, + 452, + 504, + 461 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "in one entry. In the second step, we derive the ternary divergence for the special neighboring datasets.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 478, + 502, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 502, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 502, + 492 + ], + "score": 1.0, + "content": "The specific neighboring datasets to which we reduce our general problem has the following form:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 494, + 469, + 524 + ], + "lines": [ + { + "bbox": [ + 123, + 494, + 469, + 524 + ], + "spans": [ + { + "bbox": [ + 123, + 494, + 469, + 524 + ], + "score": 0.37, + "content": "\\begin{array} { r } { \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m } = \\{ ( \\mathcal { D } _ { m } , \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ^ { \\prime \\prime } ) : \\mathcal { D } _ { m } = ( d , \\ldots , d , d ) \\in \\mathcal { X } ^ { m } , \\mathcal { D } _ { m } ^ { \\prime } = ( d , \\ldots , d , d ^ { \\prime } ) \\in \\mathcal { X } ^ { m } , \\mathrm { ~ a ~ n ~ d ~ } \\mathcal { X } = 1 , \\ldots , d , 1 \\} , } \\\\ { \\mathcal { D } _ { m } ^ { \\prime \\prime } = ( d , \\ldots , d , d ^ { \\prime \\prime } ) \\in \\mathcal { X } ^ { m } , \\mathrm { ~ w h e r e ~ } d , d ^ { \\prime } , d ^ { \\prime \\prime } \\in \\mathcal { X } \\} } \\end{array}", + "type": "interline_equation", + "image_path": "64302209a85420d5699588a5bf97c3c2506407f83fcdb28ccaf3a2355c83b02b.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 123, + 494, + 469, + 504.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 123, + 504.0, + 469, + 514.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 123, + 514.0, + 469, + 524.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 507, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 273, + 541 + ], + "score": 1.0, + "content": "Consider arbitrary neighboring datasets", + "type": "text" + }, + { + "bbox": [ + 274, + 528, + 374, + 540 + ], + "score": 0.9, + "content": "\\mathcal { D } = \\left( d _ { 1 } , \\ldots , d _ { k - 1 } , d _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 527, + 379, + 541 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 379, + 528, + 483, + 541 + ], + "score": 0.9, + "content": "\\mathcal { D } ^ { \\prime } = ( d _ { 1 } , \\ldots , d _ { k - 1 } , d _ { k } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 527, + 506, + 541 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 536, + 507, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 208, + 551 + ], + "score": 0.91, + "content": "\\mathcal { D } ^ { \\prime \\prime } = ( d _ { 1 } , \\dots , d _ { k - 1 } ^ { - } , d _ { k } ^ { \\overline { { \\prime } } \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 536, + 260, + 553 + ], + "score": 1.0, + "content": ", each having", + "type": "text" + }, + { + "bbox": [ + 261, + 540, + 267, + 549 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 536, + 339, + 553 + ], + "score": 1.0, + "content": "elements. For any", + "type": "text" + }, + { + "bbox": [ + 340, + 540, + 419, + 551 + ], + "score": 0.92, + "content": "m \\in \\{ 0 , \\ldots , k - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 536, + 507, + 553 + ], + "score": 1.0, + "content": "\u0000, we define new neigh-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 100, + 544, + 502, + 590 + ], + "spans": [ + { + "bbox": [ + 100, + 544, + 156, + 590 + ], + "score": 1.0, + "content": "boring dataeach having", + "type": "text" + }, + { + "bbox": [ + 203, + 547, + 311, + 572 + ], + "score": 1.0, + "content": "= (d00k , . . . , d00k , dk), D0(k)m+1", + "type": "text" + }, + { + "bbox": [ + 305, + 547, + 441, + 570 + ], + "score": 1.0, + "content": "= (d00k , . . . , d00k , d0k), and D00(k)m+1", + "type": "text" + }, + { + "bbox": [ + 411, + 550, + 502, + 566 + ], + "score": 0.92, + "content": "\\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } = ( d _ { k } ^ { \\prime \\prime } , \\dots , d _ { k } ^ { \\prime \\prime } )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 157, + 566, + 407, + 581 + ], + "spans": [ + { + "bbox": [ + 157, + 569, + 184, + 579 + ], + "score": 0.89, + "content": "m + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 566, + 407, + 581 + ], + "score": 0.91, + "content": "( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } , \\mathcal { D } _ { m + 1 } ^ { \\prime ( k ) } , \\mathcal { D } _ { m + 1 } ^ { ( k ) } ) \\in \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 348, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 347, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 347, + 598 + ], + "score": 1.0, + "content": "The first step of the proof is given in the following theorem.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 379, + 611 + ], + "lines": [ + { + "bbox": [ + 104, + 595, + 382, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 303, + 617 + ], + "score": 1.0, + "content": "Theorem 5 (Reduction to the Special Case). Let", + "type": "text" + }, + { + "bbox": [ + 304, + 598, + 336, + 612 + ], + "score": 0.93, + "content": "\\begin{array} { r } { q = \\frac { 1 } { e ^ { \\epsilon _ { 0 } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 595, + 382, + 617 + ], + "score": 1.0, + "content": ". 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Adapting that result to our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 261, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 261, + 308 + ], + "score": 1.0, + "content": "setting, we have the following lemma.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 262, + 506, + 308 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 180, + 324 + ], + "score": 1.0, + "content": "Lemma 2 (From", + "type": "text" + }, + { + "bbox": [ + 180, + 311, + 186, + 322 + ], + "score": 0.81, + "content": "\\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 309, + 221, + 324 + ], + "score": 1.0, + "content": "-ternary-", + "type": "text" + }, + { + "bbox": [ + 221, + 310, + 239, + 322 + ], + "score": 0.88, + "content": "| \\chi | ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 309, + 482, + 324 + ], + "score": 1.0, + "content": "-DP to subsampled RDP). Suppose the shuffle mechanism", + "type": "text" + }, + { + "bbox": [ + 482, + 311, + 504, + 321 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { s h }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 131, + 334 + ], + "score": 1.0, + "content": "obeys", + "type": "text" + }, + { + "bbox": [ + 131, + 322, + 137, + 333 + ], + "score": 0.82, + "content": "\\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 321, + 171, + 334 + ], + "score": 1.0, + "content": "-ternary-", + "type": "text" + }, + { + "bbox": [ + 172, + 322, + 190, + 333 + ], + "score": 0.81, + "content": "| \\chi | ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 321, + 241, + 334 + ], + "score": 1.0, + "content": "-DP. For any", + "type": "text" + }, + { + "bbox": [ + 241, + 321, + 295, + 333 + ], + "score": 0.91, + "content": "\\lambda \\geq 2 , k \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 321, + 468, + 334 + ], + "score": 1.0, + "content": ", RDP of the subsampled shuffle mechanism", + "type": "text" + }, + { + "bbox": [ + 469, + 322, + 481, + 331 + ], + "score": 0.76, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "(with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 332, + 481, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 203, + 350 + ], + "score": 1.0, + "content": "subsampling parameter", + "type": "text" + }, + { + "bbox": [ + 203, + 334, + 241, + 347 + ], + "score": 0.91, + "content": "\\gamma = k / n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 332, + 306, + 350 + ], + "score": 1.0, + "content": ") is bounded by:", + "type": "text" + }, + { + "bbox": [ + 306, + 332, + 476, + 348 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\epsilon ( \\lambda ) \\leq \\frac { 1 } { \\lambda - 1 } \\log \\left( 1 + \\sum _ { \\alpha = 2 } ^ { \\lambda } \\binom { \\lambda } { \\alpha } \\gamma ^ { \\alpha } \\zeta ( \\alpha ) ^ { \\alpha } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 332, + 481, + 350 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 309, + 505, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 506, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 261, + 370 + ], + "score": 1.0, + "content": "Lemma 2 can be seen as a corollary to", + "type": "text" + }, + { + "bbox": [ + 261, + 355, + 277, + 367 + ], + "score": 0.63, + "content": "\\boxed { \\ 4 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 354, + 507, + 370 + ], + "score": 1.0, + "content": "Proposition 16 and Eq. (9)]. However, for completeness,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 201, + 380 + ], + "score": 1.0, + "content": "we prove it in Appendix", + "type": "text" + }, + { + "bbox": [ + 201, + 366, + 221, + 381 + ], + "score": 0.34, + "content": "\\boxed { \\mathrm { E . 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 366, + 325, + 380 + ], + "score": 1.0, + "content": "Substituting the bound on", + "type": "text" + }, + { + "bbox": [ + 325, + 367, + 345, + 379 + ], + "score": 0.91, + "content": "\\zeta ( \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "from Theorem 4 into Lemma 2 together", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 377, + 462, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 400, + 390 + ], + "score": 1.0, + "content": "with some algebraic manipulation gives proves Theorem 1; see Appendix", + "type": "text" + }, + { + "bbox": [ + 401, + 378, + 418, + 390 + ], + "score": 0.47, + "content": "\\mathrm { E } . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 377, + 462, + 390 + ], + "score": 1.0, + "content": "for details.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 354, + 507, + 390 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 403, + 424, + 420 + ], + "lines": [ + { + "bbox": [ + 104, + 401, + 426, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 216, + 424 + ], + "score": 1.0, + "content": "6 Proof of Theorem", + "type": "text" + }, + { + "bbox": [ + 216, + 403, + 229, + 421 + ], + "score": 0.64, + "content": "\\mathbf { 4 } ;", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 401, + 276, + 424 + ], + "score": 1.0, + "content": "Ternary", + "type": "text" + }, + { + "bbox": [ + 276, + 405, + 297, + 419 + ], + "score": 0.86, + "content": "| \\chi | ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 401, + 426, + 424 + ], + "score": 1.0, + "content": "-DP of the Shuffle Model", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 506, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "The proof has two main steps. In the first step, we reduce the problem of deriving ternary divergence", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "for arbitrary neighboring datasets to the problem of deriving the ternary divergence for specific", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 504, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 192, + 463 + ], + "score": 1.0, + "content": "neighboring datasets,", + "type": "text" + }, + { + "bbox": [ + 192, + 451, + 251, + 461 + ], + "score": 0.92, + "content": "\\mathcal { D } \\stackrel { - } { \\sim } \\mathcal { D } ^ { \\prime } \\sim \\mathcal { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 452, + 340, + 463 + ], + "score": 1.0, + "content": ", where all elements in", + "type": "text" + }, + { + "bbox": [ + 341, + 452, + 350, + 461 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 452, + 418, + 463 + ], + "score": 1.0, + "content": "are the same and", + "type": "text" + }, + { + "bbox": [ + 419, + 451, + 448, + 462 + ], + "score": 0.73, + "content": "\\mathcal { D } ^ { \\prime } , \\mathcal { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 452, + 494, + 463 + ], + "score": 1.0, + "content": "differ from", + "type": "text" + }, + { + "bbox": [ + 495, + 452, + 504, + 461 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "in one entry. In the second step, we derive the ternary divergence for the special neighboring datasets.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 429, + 506, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 478, + 502, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 502, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 502, + 492 + ], + "score": 1.0, + "content": "The specific neighboring datasets to which we reduce our general problem has the following form:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 477, + 502, + 492 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 494, + 469, + 524 + ], + "lines": [ + { + "bbox": [ + 123, + 494, + 469, + 524 + ], + "spans": [ + { + "bbox": [ + 123, + 494, + 469, + 524 + ], + "score": 0.37, + "content": "\\begin{array} { r } { \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m } = \\{ ( \\mathcal { D } _ { m } , \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ^ { \\prime \\prime } ) : \\mathcal { D } _ { m } = ( d , \\ldots , d , d ) \\in \\mathcal { X } ^ { m } , \\mathcal { D } _ { m } ^ { \\prime } = ( d , \\ldots , d , d ^ { \\prime } ) \\in \\mathcal { X } ^ { m } , \\mathrm { ~ a ~ n ~ d ~ } \\mathcal { X } = 1 , \\ldots , d , 1 \\} , } \\\\ { \\mathcal { D } _ { m } ^ { \\prime \\prime } = ( d , \\ldots , d , d ^ { \\prime \\prime } ) \\in \\mathcal { X } ^ { m } , \\mathrm { ~ w h e r e ~ } d , d ^ { \\prime } , d ^ { \\prime \\prime } \\in \\mathcal { X } \\} } \\end{array}", + "type": "interline_equation", + "image_path": "64302209a85420d5699588a5bf97c3c2506407f83fcdb28ccaf3a2355c83b02b.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 123, + 494, + 469, + 504.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 123, + 504.0, + 469, + 514.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 123, + 514.0, + 469, + 524.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 507, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 273, + 541 + ], + "score": 1.0, + "content": "Consider arbitrary neighboring datasets", + "type": "text" + }, + { + "bbox": [ + 274, + 528, + 374, + 540 + ], + "score": 0.9, + "content": "\\mathcal { D } = \\left( d _ { 1 } , \\ldots , d _ { k - 1 } , d _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 527, + 379, + 541 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 379, + 528, + 483, + 541 + ], + "score": 0.9, + "content": "\\mathcal { D } ^ { \\prime } = ( d _ { 1 } , \\ldots , d _ { k - 1 } , d _ { k } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 527, + 506, + 541 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 536, + 507, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 208, + 551 + ], + "score": 0.91, + "content": "\\mathcal { D } ^ { \\prime \\prime } = ( d _ { 1 } , \\dots , d _ { k - 1 } ^ { - } , d _ { k } ^ { \\overline { { \\prime } } \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 536, + 260, + 553 + ], + "score": 1.0, + "content": ", each having", + "type": "text" + }, + { + "bbox": [ + 261, + 540, + 267, + 549 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 536, + 339, + 553 + ], + "score": 1.0, + "content": "elements. For any", + "type": "text" + }, + { + "bbox": [ + 340, + 540, + 419, + 551 + ], + "score": 0.92, + "content": "m \\in \\{ 0 , \\ldots , k - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 536, + 507, + 553 + ], + "score": 1.0, + "content": "\u0000, we define new neigh-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 100, + 544, + 502, + 590 + ], + "spans": [ + { + "bbox": [ + 100, + 544, + 156, + 590 + ], + "score": 1.0, + "content": "boring dataeach having", + "type": "text" + }, + { + "bbox": [ + 203, + 547, + 311, + 572 + ], + "score": 1.0, + "content": "= (d00k , . . . , d00k , dk), D0(k)m+1", + "type": "text" + }, + { + "bbox": [ + 305, + 547, + 441, + 570 + ], + "score": 1.0, + "content": "= (d00k , . . . , d00k , d0k), and D00(k)m+1", + "type": "text" + }, + { + "bbox": [ + 411, + 550, + 502, + 566 + ], + "score": 0.92, + "content": "\\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } = ( d _ { k } ^ { \\prime \\prime } , \\dots , d _ { k } ^ { \\prime \\prime } )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 157, + 566, + 407, + 581 + ], + "spans": [ + { + "bbox": [ + 157, + 569, + 184, + 579 + ], + "score": 0.89, + "content": "m + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 566, + 407, + 581 + ], + "score": 0.91, + "content": "( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } , \\mathcal { D } _ { m + 1 } ^ { \\prime ( k ) } , \\mathcal { D } _ { m + 1 } ^ { ( k ) } ) \\in \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 100, + 527, + 507, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 348, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 347, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 347, + 598 + ], + "score": 1.0, + "content": "The first step of the proof is given in the following theorem.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 583, + 347, + 598 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 379, + 611 + ], + "lines": [ + { + "bbox": [ + 104, + 595, + 382, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 303, + 617 + ], + "score": 1.0, + "content": "Theorem 5 (Reduction to the Special Case). Let", + "type": "text" + }, + { + "bbox": [ + 304, + 598, + 336, + 612 + ], + "score": 0.93, + "content": "\\begin{array} { r } { q = \\frac { 1 } { e ^ { \\epsilon _ { 0 } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 595, + 382, + 617 + ], + "score": 1.0, + "content": ". We have:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 595, + 382, + 617 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 615, + 480, + 681 + ], + "lines": [ + { + "bbox": [ + 118, + 615, + 480, + 681 + ], + "spans": [ + { + "bbox": [ + 118, + 615, + 480, + 681 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { h \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime \\prime } ) } \\left[ \\left| \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } ) ( h ) - \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime } ) ( h ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime \\prime } ) ( h ) } \\right| ^ { \\alpha } \\right] } \\\\ & { \\qquad \\leq \\mathbb { E } _ { m \\sim \\mathrm { B i n } ( k - 1 , q ) } \\left[ \\mathbb { E } _ { h \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) } \\left[ \\left| \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { ( k ) } ) ( h ) - \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime ( k ) } ) ( h ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) ( h ) } \\right| ^ { \\alpha } \\right] \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "6165b501b24104cf860f31af8810369280ef2911bf671a46948162972cfadbaf.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 118, + 615, + 480, + 637.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 118, + 637.0, + 480, + 659.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 118, + 659.0, + 480, + 681.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 504, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "We know (by Chernoff bound) that the binomial r.v. is concentrated around its mean, which", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 276, + 713 + ], + "score": 1.0, + "content": "implies that the terms in the RHS of", + "type": "text" + }, + { + "bbox": [ + 276, + 699, + 290, + 712 + ], + "score": 0.83, + "content": "\\textcircled { 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 699, + 377, + 713 + ], + "score": 1.0, + "content": "that correspond to", + "type": "text" + }, + { + "bbox": [ + 377, + 700, + 483, + 712 + ], + "score": 0.9, + "content": "m \\ < \\ ( 1 - \\tau ) q ( k - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "(we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 149, + 724 + ], + "score": 1.0, + "content": "will take", + "type": "text" + }, + { + "bbox": [ + 149, + 711, + 195, + 723 + ], + "score": 0.91, + "content": "\\tau ~ = ~ 1 / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 709, + 469, + 724 + ], + "score": 1.0, + "content": "will contribute in a negligible amount. Then we show that", + "type": "text" + }, + { + "bbox": [ + 469, + 711, + 505, + 722 + ], + "score": 0.84, + "content": "E _ { m } : =", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 71, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 71, + 317, + 97 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\boldsymbol { h } \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) } \\left[ \\left| \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { ( k ) } ) ( \\boldsymbol { h } ) - \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime ( k ) } ) ( \\boldsymbol { h } ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m + 1 } ^ { \\prime \\prime ( k ) } ) ( \\underline { { h } } ) } \\right| ^ { \\alpha } \\right]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 317, + 77, + 440, + 90 + ], + "score": 1.0, + "content": "is a non-increasing function of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 440, + 80, + 450, + 88 + ], + "score": 0.73, + "content": "m", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 450, + 77, + 507, + 90 + ], + "score": 1.0, + "content": ". 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By this chain of arguments, we have reduced the problem to the one involving the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 507, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 491, + 412 + ], + "score": 1.0, + "content": "computation of ternary divergence only for the special form of neighboring datasets (as in Theorem", + "type": "text" + }, + { + "bbox": [ + 487, + 399, + 507, + 412 + ], + "score": 1.0, + "content": "6),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 409, + 372, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 198, + 423 + ], + "score": 1.0, + "content": "which proves Theorem", + "type": "text" + }, + { + "bbox": [ + 198, + 409, + 209, + 424 + ], + "score": 0.39, + "content": "\\bigtriangledown", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 410, + 372, + 423 + ], + "score": 1.0, + "content": "See Appendix C.1 for a complete proof.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 506, + 569 + ], + "lines": [ + { + "bbox": [ + 104, + 429, + 495, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 204, + 450 + ], + "score": 1.0, + "content": "Proof sketch of Theorem", + "type": "text" + }, + { + "bbox": [ + 205, + 432, + 215, + 446 + ], + "score": 0.5, + "content": "6 .", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 429, + 259, + 450 + ], + "score": 1.0, + "content": "Consider", + "type": "text" + }, + { + "bbox": [ + 259, + 433, + 360, + 446 + ], + "score": 0.92, + "content": "( \\mathcal { D } _ { m } ^ { \\prime \\prime } , \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ) \\in \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 429, + 495, + 450 + ], + "score": 1.0, + "content": "as in the statement of Theorem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 222, + 457 + ], + "score": 1.0, + "content": "First we observe that for any", + "type": "text" + }, + { + "bbox": [ + 222, + 445, + 248, + 455 + ], + "score": 0.89, + "content": "\\alpha \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 443, + 357, + 457 + ], + "score": 1.0, + "content": "and any three distributions", + "type": "text" + }, + { + "bbox": [ + 357, + 447, + 382, + 456 + ], + "score": 0.89, + "content": "p , q , r", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "over the same domain, we can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 455, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 129, + 472 + ], + "score": 1.0, + "content": "write", + "type": "text" + }, + { + "bbox": [ + 129, + 455, + 351, + 475 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { E } _ { r } \\left[ \\left| \\frac { p - q } { r } \\right| ^ { \\alpha } \\right] \\leq 2 ^ { \\alpha - 1 } \\left( \\mathbb { E } _ { r } \\left[ \\left| \\frac { p } { r } - 1 \\right| ^ { \\alpha } \\right] + \\mathbb { E } _ { r } \\left[ \\left| \\frac { q } { r } - 1 \\right| ^ { \\alpha } \\right] \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 455, + 506, + 472 + ], + "score": 1.0, + "content": ". This is a straight-forward application", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 472, + 507, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 214, + 488 + ], + "score": 1.0, + "content": "of the standard inequality", + "type": "text" + }, + { + "bbox": [ + 214, + 474, + 338, + 486 + ], + "score": 0.9, + "content": "| x + y | ^ { \\alpha } \\leq 2 ^ { \\alpha - 1 } ( | x | ^ { \\alpha } + | y | ^ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 472, + 420, + 488 + ], + "score": 1.0, + "content": "which holds for all", + "type": "text" + }, + { + "bbox": [ + 420, + 475, + 457, + 486 + ], + "score": 0.91, + "content": "x , y \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 472, + 476, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 475, + 503, + 486 + ], + "score": 0.9, + "content": "\\alpha \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 472, + 507, + 488 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 484, + 507, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 172, + 498 + ], + "score": 1.0, + "content": "Now, by taking", + "type": "text" + }, + { + "bbox": [ + 172, + 486, + 237, + 497 + ], + "score": 0.86, + "content": "p = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 484, + 241, + 498 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 242, + 486, + 306, + 497 + ], + "score": 0.9, + "content": "q = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 484, + 328, + 498 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 329, + 485, + 393, + 497 + ], + "score": 0.93, + "content": "r = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 484, + 507, + 498 + ], + "score": 1.0, + "content": ", we reduce the problem of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 198, + 509 + ], + "score": 1.0, + "content": "computing the ternary", + "type": "text" + }, + { + "bbox": [ + 198, + 497, + 216, + 508 + ], + "score": 0.88, + "content": "| \\chi | ^ { \\alpha } .", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "-divergence (which we need to bound) to the problem of computing the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 507, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 213, + 520 + ], + "score": 1.0, + "content": "Pearson-Vajda divergence", + "type": "text" + }, + { + "bbox": [ + 213, + 507, + 230, + 519 + ], + "score": 0.54, + "content": "| | \\overline { { 4 3 } } | |", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 506, + 375, + 520 + ], + "score": 1.0, + "content": ", which we can write in terms of the", + "type": "text" + }, + { + "bbox": [ + 375, + 509, + 383, + 517 + ], + "score": 0.81, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 506, + 507, + 520 + ], + "score": 1.0, + "content": "-th absolute moment of the r.v.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 516, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 521, + 165, + 533 + ], + "score": 0.92, + "content": "X : \\mathcal { A } _ { B } ^ { m } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 516, + 213, + 540 + ], + "score": 1.0, + "content": ", defined as", + "type": "text" + }, + { + "bbox": [ + 213, + 518, + 332, + 535 + ], + "score": 0.91, + "content": "\\begin{array} { r } { X ( \\pmb { h } ) : = \\big ( \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime } ) ( \\pmb { h } ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ) ( \\pmb { h } ) } - 1 \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 516, + 361, + 540 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 362, + 521, + 398, + 533 + ], + "score": 0.92, + "content": "\\pmb { h } \\in \\mathcal { A } _ { B } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 516, + 430, + 540 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 431, + 520, + 504, + 533 + ], + "score": 0.91, + "content": "\\mathcal { D } ^ { \\prime } \\in \\{ \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ^ { \\prime \\prime } \\} )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 218, + 547 + ], + "score": 1.0, + "content": "and distributed according to", + "type": "text" + }, + { + "bbox": [ + 218, + 534, + 316, + 546 + ], + "score": 0.9, + "content": "X ( \\pmb { h } ) \\sim \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ) ( \\pmb { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 533, + 330, + 547 + ], + "score": 1.0, + "content": ". 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By this chain of arguments, we have reduced the problem to the one involving the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 507, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 491, + 412 + ], + "score": 1.0, + "content": "computation of ternary divergence only for the special form of neighboring datasets (as in Theorem", + "type": "text" + }, + { + "bbox": [ + 487, + 399, + 507, + 412 + ], + "score": 1.0, + "content": "6),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 409, + 372, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 198, + 423 + ], + "score": 1.0, + "content": "which proves Theorem", + "type": "text" + }, + { + "bbox": [ + 198, + 409, + 209, + 424 + ], + "score": 0.39, + "content": "\\bigtriangledown", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 410, + 372, + 423 + ], + "score": 1.0, + "content": "See Appendix C.1 for a complete proof.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 18.5, + "bbox_fs": [ + 103, + 243, + 508, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 506, + 569 + ], + "lines": [ + { + "bbox": [ + 104, + 429, + 495, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 204, + 450 + ], + "score": 1.0, + "content": "Proof sketch of Theorem", + "type": "text" + }, + { + "bbox": [ + 205, + 432, + 215, + 446 + ], + "score": 0.5, + "content": "6 .", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 429, + 259, + 450 + ], + "score": 1.0, + "content": "Consider", + "type": "text" + }, + { + "bbox": [ + 259, + 433, + 360, + 446 + ], + "score": 0.92, + "content": "( \\mathcal { D } _ { m } ^ { \\prime \\prime } , \\mathcal { D } _ { m } ^ { \\prime } , \\mathcal { D } _ { m } ) \\in \\mathcal { D } _ { \\mathrm { s a m e } } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 429, + 495, + 450 + ], + "score": 1.0, + "content": "as in the statement of Theorem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 222, + 457 + ], + "score": 1.0, + "content": "First we observe that for any", + "type": "text" + }, + { + "bbox": [ + 222, + 445, + 248, + 455 + ], + "score": 0.89, + "content": "\\alpha \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 443, + 357, + 457 + ], + "score": 1.0, + "content": "and any three distributions", + "type": "text" + }, + { + "bbox": [ + 357, + 447, + 382, + 456 + ], + "score": 0.89, + "content": "p , q , r", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "over the same domain, we can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 455, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 129, + 472 + ], + "score": 1.0, + "content": "write", + "type": "text" + }, + { + "bbox": [ + 129, + 455, + 351, + 475 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { E } _ { r } \\left[ \\left| \\frac { p - q } { r } \\right| ^ { \\alpha } \\right] \\leq 2 ^ { \\alpha - 1 } \\left( \\mathbb { E } _ { r } \\left[ \\left| \\frac { p } { r } - 1 \\right| ^ { \\alpha } \\right] + \\mathbb { E } _ { r } \\left[ \\left| \\frac { q } { r } - 1 \\right| ^ { \\alpha } \\right] \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 455, + 506, + 472 + ], + "score": 1.0, + "content": ". This is a straight-forward application", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 472, + 507, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 214, + 488 + ], + "score": 1.0, + "content": "of the standard inequality", + "type": "text" + }, + { + "bbox": [ + 214, + 474, + 338, + 486 + ], + "score": 0.9, + "content": "| x + y | ^ { \\alpha } \\leq 2 ^ { \\alpha - 1 } ( | x | ^ { \\alpha } + | y | ^ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 472, + 420, + 488 + ], + "score": 1.0, + "content": "which holds for all", + "type": "text" + }, + { + "bbox": [ + 420, + 475, + 457, + 486 + ], + "score": 0.91, + "content": "x , y \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 472, + 476, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 475, + 503, + 486 + ], + "score": 0.9, + "content": "\\alpha \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 472, + 507, + 488 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 484, + 507, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 172, + 498 + ], + "score": 1.0, + "content": "Now, by taking", + "type": "text" + }, + { + "bbox": [ + 172, + 486, + 237, + 497 + ], + "score": 0.86, + "content": "p = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 484, + 241, + 498 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 242, + 486, + 306, + 497 + ], + "score": 0.9, + "content": "q = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 484, + 328, + 498 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 329, + 485, + 393, + 497 + ], + "score": 0.93, + "content": "r = \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 484, + 507, + 498 + ], + "score": 1.0, + "content": ", we reduce the problem of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 198, + 509 + ], + "score": 1.0, + "content": "computing the ternary", + "type": "text" + }, + { + "bbox": [ + 198, + 497, + 216, + 508 + ], + "score": 0.88, + "content": "| \\chi | ^ { \\alpha } .", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "-divergence (which we need to bound) to the problem of computing the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 507, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 213, + 520 + ], + "score": 1.0, + "content": "Pearson-Vajda divergence", + "type": "text" + }, + { + "bbox": [ + 213, + 507, + 230, + 519 + ], + "score": 0.54, + "content": "| | \\overline { { 4 3 } } | |", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 506, + 375, + 520 + ], + "score": 1.0, + "content": ", which we can write in terms of the", + "type": "text" + }, + { + "bbox": [ + 375, + 509, + 383, + 517 + ], + "score": 0.81, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 506, + 507, + 520 + ], + "score": 1.0, + "content": "-th absolute moment of the r.v.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 516, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 521, + 165, + 533 + ], + "score": 0.92, + "content": "X : \\mathcal { A } _ { B } ^ { m } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 516, + 213, + 540 + ], + "score": 1.0, + "content": ", defined as", + "type": "text" + }, + { + "bbox": [ + 213, + 518, + 332, + 535 + ], + "score": 0.91, + "content": "\\begin{array} { r } { X ( \\pmb { h } ) : = \\big ( \\frac { \\mathcal { M } _ { s h } ( \\mathcal { D } ^ { \\prime } ) ( \\pmb { h } ) } { \\mathcal { M } _ { s h } ( \\mathcal { D } _ { m } ) ( \\pmb { h } ) } - 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See Appendix C.3 for a complete proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 558, + 504, + 565 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 429, + 507, + 569 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 582, + 180, + 596 + ], + "lines": [ + { + "bbox": [ + 104, + 580, + 182, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 182, + 599 + ], + "score": 1.0, + "content": "7 Discussion", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "In this paper, we analyzed the Rényi differential privacy of the subsampled shuffle model by bounding", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 154, + 631 + ], + "score": 1.0, + "content": "the ternary", + "type": "text" + }, + { + "bbox": [ + 155, + 618, + 173, + 630 + ], + "score": 0.83, + "content": "| \\chi | ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "-DP of the shuffle model. We numerically demonstrated the importance of our", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "proposed bound, where we obtain a significant improvement over using the state-of-the-art in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "practical regimes. 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Another direction to explore would", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "be to analyze the RDP of the subsampled shuffle model for different sub-sampling techniques such as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 412, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 192, + 723 + ], + "score": 1.0, + "content": "Poisson subsampling", + "type": "text" + }, + { + "bbox": [ + 193, + 710, + 210, + 722 + ], + "score": 0.59, + "content": "\\underline { { \\mathbb { G 5 } } } ] |", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 710, + 284, + 723 + ], + "score": 1.0, + "content": ", random check-in", + "type": "text" + }, + { + "bbox": [ + 285, + 711, + 297, + 722 + ], + "score": 0.82, + "content": "\\textcircled { 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 710, + 393, + 723 + ], + "score": 1.0, + "content": ", or client self-sampling", + "type": "text" + }, + { + "bbox": [ + 393, + 710, + 410, + 722 + ], + "score": 0.73, + "content": "| \\overline { { 3 0 } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 710, + 412, + 723 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 677, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 506, + 86 + ], + "score": 1.0, + "content": "Societal Impact. Collaborative learning comes with significant societal risks of privacy violations,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "which is the main topic addressed in this paper. However, such learning is only as good as the data", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "used for training, and if the data is not unbiased, this could lead to significant issues related to fairness", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "and could also lead to societally undesirable outcomes. Such an issue is exacerbated when privacy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "is guaranteed on the data used for training, making a-priori fairness checks on data infeasible. This", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "can be ameliorated by properly testing models finally obtained against fairness criteria and rejecting", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "models that fail the test. This paper did not consider the issue of robustness to security, and this could", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "also be an important societal issue in collaborative learning, where a small subset of users could insert", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "malicious inputs to disrupt the learning process or worse bias the learned model covertly. This could", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "also lead to negative outcomes. This issue of robustness to malicious participants has been studied in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "several papers, and incorporating this into the framework of the paper is an important future research", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 193, + 133, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 133, + 206 + ], + "score": 1.0, + "content": "topic.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 219, + 196, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 198, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 198, + 235 + ], + "score": 1.0, + "content": "Acknowledgment", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 257 + ], + "score": 1.0, + "content": "This work was supported in part by NSF grants #2007714 and #1955632 and a Google Faculty", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 304, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 304, + 267 + ], + "score": 1.0, + "content": "research award and an Amazon Research Award.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 281, + 163, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 165, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 165, + 296 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 110, + 294, + 506, + 725 + ], + "lines": [ + { + "bbox": [ + 109, + 298, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 109, + 298, + 506, + 315 + ], + "score": 1.0, + "content": "[1] M. 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N. Baldock∗ Google Research, Brain Team rjnbaldock@gmail.com + +Hartmut Maennel Google Research, Brain Team hartmutm@google.com + +Behnam Neyshabur Google Research, Blueshift Team neyshabur@google.com + +# Abstract + +Existing work on understanding deep learning often employs measures that compress all data-dependent information into a few numbers. In this work, we adopt a perspective based on the role of individual examples. We introduce a measure of the computational difficulty of making a prediction for a given input: the (effective) prediction depth. Our extensive investigation reveals surprising yet simple relationships between the prediction depth of a given input and the model’s uncertainty, confidence, accuracy and speed of learning for that data point. We further categorize difficult examples into three interpretable groups, demonstrate how these groups are processed differently inside deep models and showcase how this understanding allows us to improve prediction accuracy. Insights from our study lead to a coherent view of a number of separately reported phenomena in the literature: early layers generalize while later layers memorize; early layers converge faster and networks learn easy data and simple functions first. + +# 1 Introduction + +Much of the existing work on understanding deep learning “integrates out” the data, viewing the inductive bias of the model, or the properties of the optimizer as central to the success of the approach. Examples of such work include studies of eigenvalues of the Hessian and the geometry of the loss landscape (Ghorbani et al., 2019; Yao et al., 2020; Sagun et al., 2016; Li et al., 2018; Pennington and Bahri, 2017; Sagun et al., 2018), studies of margin and effective generalization measures (Long and Sedghi, 2019; Unterthiner et al., 2020; Jiang et al., 2020, 2018; Kawaguchi et al., 2017) and mean-field studies of stochastic optimization (Smith et al., 2021; Stephan et al., 2017; Smith and Le, 2018). However, in practice, we are rarely concerned with only the average behavior of a model. + +One pathway to understanding the principles that govern how deep models process data is to study the properties of deep models for data points with different “amounts” or “types” of example difficulty. There are a number of definitions of example difficulty in the literature (E.g. see Carlini et al. (2019); Hooker et al. (2019); Lalor et al. (2018); Agarwal and Hooker (2020)). Two are particularly relevant to this work. Firstly, the probability of predicting the ground truth label for an example, when that example is omitted from the training set (Jiang et al., 2021), which represents a statistical view of example difficulty. Secondly, the difficulty of learning an example, parameterized by the earliest training iteration after which the model predicts the ground truth class for that example in all subsequent iterations (Toneva et al., 2019). This measure represents a learning view of example difficulty 2. + +These notions suffer from two fundamental limitations. While early-exit strategies in computer vision (Teerapittayanon et al., 2016; Huang et al., 2018) and NLP (Dehghani et al., 2018; Liu et al., 2020b; Schwartz et al., 2020; Xin et al., 2020) suggest predictions for easier examples require less computation, the above example difficulty notions do not encapsulate the processing of data inside a given converged model. Moreover, existing notions of example difficulty (E.g. Carlini et al. (2019)) provide a one-dimensional view of difficulty which can not distinguish between examples that are difficult for different reasons. + +In this paper, we take a significant step towards resolving the above shortcomings. To take the processing of the data into account we propose a new measure of example difficulty, the prediction depth, which is determined from the hidden embeddings. To escape the one-dimensional view of difficulty, we introduce three distinct difficulty types by relating the hidden embeddings for an input to high-level concepts about example difficulty: “Does this example look mislabeled?”; “Is classifying this example only easy if the label is given?”; “Is this example ambiguous both with and without its label?”. Furthermore, we show how this enhanced notion of example difficulty can unify our understanding of several seemingly unrelated phenomena in deep learning. We hope that the results presented in this work will aid the development of models that capture heteroscedastic uncertainty, our understanding of how deep networks respond to distributional shift, and the advancement of curriculum learning approaches and machine learning fairness. These connections are discussed in Section 5. + +Contributions Our main contributions are as follows: + +• We introduce a measure of computational example difficulty: the prediction depth (PD). The prediction depth, illustrated in Figure 1, represents the number of hidden layers after which the network’s final prediction is already (effectively) determined (Section 2). +We show that the prediction depth is larger for examples that visually appear to be more difficult, and that prediction depth is consistent between architectures and random seeds (Section 2.2). +• Our empirical investigation reveals that prediction depth appears to establish a linear lower bound on the consistency of a prediction. We further show that predictions are on average more accurate for validation points with small prediction depths (Section 3.1). +We demonstrate that final predictions for data points that converge earlier during training are typically determined in earlier layers which establishes a correspondence between the training history of the network and the processing of data in the hidden layers (Section 3.2). +• We show that both the adversarial input margin and the output margin are larger for examples with smaller prediction depths. We further design an intervention to reduce the output margin of a network and show that this leads to predictions being made only in the latest hidden layers (Section 3.3). +We identify three extreme forms of example difficulty by considering the prediction depth in the training and validation splits independently and demonstrate how a simple algorithm that uses the hidden embeddings in one middle layer to make predictions can lead to dramatic improvements in accuracy for inputs that strongly exhibit a specific form of example difficulty (Section 4). +We use our results to present a coherent picture of deep learning that unifies four seemingly unrelated deep learning phenomena: early layers generalize while later layers memorize; networks converge from input layer towards output layer; easy examples are learned first and networks present simpler functions earlier in training (Section 5). + +Experimental Setup: To ensure that our results are robust to the choice of architectures and datasets, we report empirical findings for ResNet18 (He et al., 2016), VGG16 (Simonyan and Zisserman, 2015) and MLP architectures trained on CIFAR10, CIFAR100 (Krizhevsky et al., 2009), Fashion MNIST (FMNIST) (Xiao et al., 2017) and SVHN (Netzer et al., 2011) datasets. All models were trained using SGD with momentum. Our MLP comprises 7 hidden layers of width 2048 with ReLU activations. Details of the datasets, architectures, and hyperparameters used can be found in Appendix A. + +Related Work: Our work uses hidden layer probes to determine example difficulty. We have discussed how our study relates to prior work on example difficulty. Hidden layer probes have also been used to study deep learning. Deep k-NN methods (Papernot and McDaniel, 2018) determine their predictions and estimate their own uncertainties by comparing the hidden embeddings of an input to those of the training set. Cohen et al. (2018) showed that SVM, $\mathbf { k }$ -Nearest Neighbors $\mathbf { k }$ -NN) + +and logistic regression probes achieve similar accuracies. However, they did not study the processing of individual data points nor did they relate the $\mathbf { k }$ -NN accuracy to notions of example difficulty. Alain and Bengio (2017) used linear classifier probes in the hidden layers to interrogate deep models and demonstrated that linear separability of the embeddings increases monotonically with depth. We provide a more detailed discussion of related work in Appendix B. + +# 2 Prediction Depth: a Computational View of Example Difficulty + +We discussed the statistical and learning views of example difficulty in Section 1. In this section, we introduce a computational view of example difficulty parametrized by the prediction depth as defined in Section 2.1. This computational view asserts that, for “easy” examples, a deep model’s final prediction is effectively made after only a few layers, while more layers are used for “difficult” examples. + +# 2.1 Definition + +Asserting that the final prediction is effectively determined in earlier layers of a model, before the output, we estimate the depth at which a prediction is made for a given input as follows 3: + +1. We construct k-NN classifier probes from the embeddings of the training set after particular layers of the network, including the input and the final softmax. The placement of $\mathbf { k }$ -NN probes is described in Appendix A.5. We use $k = 3 0$ in the $\mathbf { k }$ -NN probes. Appendix A.4 establishes that the k-NN accuracies we report are insensitive to $k$ over a wide range. 2. A prediction is defined to be made at a depth $L = l$ if the $\mathbf { k }$ -NN classification after layer $L = l - 1$ is different from the network’s final classification, but the classifications of k-NN probes after every layer $L \geq l$ are all equal to the final classification of the network. Data points consistently classified by all $\mathbf { k }$ -NN probes are determined to be (effectively) predicted in layer 0 (the input) 4. + +It is worth noting that the prediction depth can be calculated for all data points: both in the training and validation splits. This leads to two notions of computational difficulty: + +• The difficulty of predicting the (given) class for an input (in the training split) • The difficulty of making a prediction for an input, unseen in advance (from the validation split) + +We examine both notions of computational difficulty in this paper and use the distinction between them to describe different forms of example difficulty in Section 4. + +# 2.2 Prediction depth is a meaningful and robust notion of example difficulty + +In this section we show that prediction depth agrees with intuitive notions of example difficulty and that it is consistent between different training runs and similar architectures. + +Prediction depth is higher for examples and datasets that seem more difficult If prediction depth is a sensible measure of example difficulty then we would expect the following sanity checks to be observed: + +1. Individual data points that are visually confusing or mislabeled should have larger prediction depths as compared to images that are clear examples of their class. +2. Data points from tasks that are intuitively simpler should have lower prediction depths on average. + +Figure 1 shows that the prediction depth passes both of these sanity checks. Appendix C.1 presents additional images, providing further evidence for this claim. + +Prediction depth is consistent across random seeds and similar architectures Figure 2 shows that the prediction depth is highly consistent between different architectures and random seeds for all datasets. Perfect agreement is not expected as different deep learning algorithms have different inductive biases which affects the perceived difficulty of examples. We observe stronger correlation between prediction depth for ResNet18 and VGG16, than between VGG16 and MLP. This may be explained by the fact that ResNet18 and VGG16 are both convolutional networks and we expect their inductive biases to be more similar to one another than to MLP. + +![](images/41ae6d0999f3d52ac3d9fed1621b96ae037b526605cd4d18c4e702c118a942c4.jpg) +Figure 1: Deep models use fewer layers to (effectively) determine the prediction for easy examples and more layers for hard examples. Left: A cartoon illustrating the definition of prediction depth (given in Section 2.1). Also shown are training examples from CIFAR100 (“Clock”) and SVHN (“Digit 8”). The examples shown are predicted at the input (first layer) or softmax (last layer) of ResNet18. The examples predicted in the input are visually typical (“easy”), while those predicted in the softmax are mislabeled and/or visually confusing (“hard” examples). To find the prediction depth, we build $\mathbf { k }$ -NN classifiers from the embeddings of the training set in different layers of the model. The prediction depth corresponds to the earliest layer at which the predictions of all subsequent k-NN classifiers converge to a fixed label. Right: Probability of prediction depth in ResNet18 models for four datasets (training split). We see that the four distributions have different characteristic prediction depths. Ranking the mean prediction depths of these datasets in ascending order, we observe: Fashion MNIST (smallest), SVHN (second), CIFAR10 (third), and CIFAR100 (largest). This order aligns with how one might intuitively rank the difficulties of these classification tasks. + +![](images/23c98a880da3f394cd13a1b9baa89a6f129ff8653082737c21aec2cb3c228e0f.jpg) +Figure 2: Consistency of prediction depth between architectures and random seeds. Left: The panel shows the correlation coefficient between prediction depths in different architectures, for both train and validation splits in four datasets. Diagonal comparisons between an architecture and itself show the correlation for the same architecture trained with different random seeds. Right: Histograms comparing the mean value of prediction depth obtained for each data point in the training set of CIFAR10 from an ensemble of 250 trained models. In this plot, for visual simplicity, we rescale prediction depth to the interval $[ 0 , 1 ]$ for each network. Similar results for all other datasets are presented in Appendix C.2. + +# 3 Deep Learning Phenomena Through the Lens of Prediction Depth + +In this section, we explore how the prediction depth can be used to better understand three important aspects of Deep Learning: accuracy and consistency of a prediction; the order in which data is learned and the simplicity of the learned function (as measured by the margin) in the vicinity of a data point. + +# 3.1 Depth of a prediction gives a linear lower bound on its consistency + +Adopting a statistical view of example difficulty, Jiang et al. (2021) identified example difficulty with the expected accuracy of the learning algorithm for a given input, averaged over models trained on different random subsets of the training set with different random seeds. In this section, we clarify the relationship between the prediction depth and the expected accuracy by disentangling the accuracy from the sensitivity of predictions to the particular training split and random seed. Following Jiang et al. (2021), we measure the expected accuracy using the consistency score. + +Consistency score $\hat { C }$ : The frequency of classifying an example correctly when it is omitted from the training set. An empirical estimator of the consistency score for a validation point $( x , y )$ + +![](images/67ee1b181de37aba816ac42fe368a058ecc0e63537f5c06a4103f90e7f6c3a63.jpg) +Figure 3: Consistency score vs. prediction depth in the validation split (left) can be understood as the superposition of two simple functions (middle and right). We trained 250 ResNet18 models on CIFAR10, with $9 0 { : } 1 0 \%$ random train:validation splits as described in Appendix A. These histograms compare the frequency of correct predictions to the average prediction depth for a data point when it occurs in the validation split. The density of data points is indicated by the color bar, which follows a log scale. The average prediction depth forms two, surprisingly simple, linear bounds on the consistency score (see Section 3.1 for a full description.) This Figure is reproduced for all datasets and architectures in Appendix C.3, illustrating the consistency of this result. + +is given by (Jiang et al., 2021): + +$$ +\hat { C } _ { A , S } ( x , y ) = \hat { \mathbb { E } } _ { \tilde { S } \sim { \cal S } \setminus \{ ( x , y ) \} } ^ { r } \left[ \delta _ { y _ { A } , y } \right] +$$ + +where $A$ is a deep learning algorithm (architecture, loss and optimizer), $y$ is the ground truth class for $x$ , $\tilde { \cal S }$ is a random subset of $n$ points sampled from a training dataset $s$ excluding $( x , y )$ , $y _ { A }$ is the predicted class of $x$ for $A$ trained with data $\tilde { \cal S }$ , $\delta$ is the Kronecker delta and $\hat { \mathbb { E } } ^ { r }$ denotes empirical averaging with $r$ i.i.d. samples of such subsets $\tilde { \cal S }$ . + +Figure 3 (left panel) shows the relationship between consistency score and prediction depth. This plot indicates a surprising piecewise linear boundary which is symmetric around consistency score $\frac { 1 } { 2 }$ This suggests the existence of a missing concept that could simplify the picture. We next show that the missing concept is the notion of a consensus class which is defined below. + +Consensus class ${ \hat { y } } _ { A }$ : The consensus class of $x$ is defined as the predicted class for input $x$ by a majority voting ensemble of $r$ models each of which is trained on a randomly chosen subset ${ \tilde { \cal S } } \stackrel { n } { \sim } { \cal S } \backslash \{ ( x , y ) \} \stackrel { 5 } { \sim }$ . + +Figure 3 (middle and right) shows how conditioning on whether consensus class matches the ground truth can change the relationship between consistency score and the prediction depth. For points where the consensus class matches the ground truth (middle) we see that the prediction depth forms a, surprisingly simple, linear lower bound on the consistency score. For points where the consensus class differs from the ground truth (right) at low prediction depth the consistency score is bounded from above by a line that reflects the bound from the middle plot in $\begin{array} { r } { \hat { C } = \frac { 1 } { 2 } } \end{array}$ , suggesting that such points are repeatedly mislabeled with a wrong class label. At high prediction depth, the consistency score is low, which suggests highly inconsistent predictions and low accuracy. This result suggests a simple hypothesis: that predictions with low prediction depth are consistent with the consensus class, whether that matches the ground truth class or not, while predictions made in later layers depend strongly on the specific training split and random seed used for training and initialization. We measure consistency with the consensus class using the consensus-consistency score. + +Consensus-consistency score $C ^ { * }$ : The fraction of models in an ensemble that predict the ensemble’s consensus class ${ \hat { y } } _ { A } \left( x \right)$ for an unseen input $x$ . + +$$ +C _ { A , S } ^ { * } ( x ) = \hat { \mathbb { E } } _ { \tilde { S } \sim \tilde { S } \backslash \{ ( x , y ) \} } ^ { r } \left[ \delta _ { y _ { A } , \hat { y } _ { A } ( x ) } \right] +$$ + +where the notation is the same as in (1) 6. + +Figure 4 (left) establishes that our simple hypothesis is indeed correct: the prediction depth forms a linear lower bound on the consensus-consistency score for all data points, irrespective of whether the consensus class matches or differs from the ground truth. Interestingly, Figure 4 (middle and right) shows how the prediction depth in a single model, can be used to estimate both of these quantities. That is, predictions of data points with lower prediction depth are both more likely to be consistent and more likely to be correct. + +![](images/8a5ae0a88a3e513d4097c5f1dcee15b7f3d5e2e68ceae4e1b0a5b33e9d9f4a7a.jpg) +Figure 4: Left: Prediction depth provides us with a linear lower bound on consensus-consistency. Results for CIFAR100 with ResNet18. We train 250 models $( 9 0 { : } 1 0 \%$ random train:validation splits) and compare the average prediction depth when a point occurs in the validation set, to the consensus-consistency of the corresponding predictions. Predictions made for points with low mean prediction depths are highly consistent. Conversely, predictions for points with high mean prediction depths are typically more sensitive to the particular training split and random seed used during training. This left plot shows the result for CIFAR100 with ResNet18. The density of data points is indicated by the color bar, which follows a log scale. Middle: Prediction depth in one model predicts the consensus-consistency of an ensemble that does not include that model. For each dataset we train 25 ResNet18 models with the full training set (see Appendix A). The consensus-consistency of each test point is obtained from 24 of the models, while the prediction depth is obtained from the remaining 1 model. We see that prediction depth in one model predicts the consensus-consistency of a separate ensemble: a measure of the uncertainty of the prediction. The size of each marker in the middle and right plots shows the fraction of the dataset with each prediction depth. Reaffirming the second sanity check in Section 2.2, and in agreement with Figure 1 (right), intuitively simpler datasets (Fashion MNIST and SVHN) have low average prediction depths, while CIFAR100 (intuitively the hardest dataset) has the largest average prediction depth. Right: Prediction depth predicts accuracy. For each dataset we train 250 ResNet18 models $9 0 { : } 1 0 \%$ random train:validation splits). Each time a point appears in the validation split we record the prediction depth and whether the prediction was correct. Predictions made in earlier layers are more likely to be correct. Consistency of these plots is demonstrated for all datasets and architectures in Appendix C.3 where we also describe the relationship between the prediction depth and the entropy of the predictions for an ensemble. + +# 3.2 The prediction depth of an input is correlated with its learning difficulty + +In Section 3.1, we describe the relationship between the prediction depth, which represents a computational view of example difficulty and the consistency and consensus-consistency scores, which represent a statistical view. In this section we compare prediction depth to a learning view of example difficulty. We measure the difficulty of learning an example by the speed at which the model’s prediction converges for that input during training. The following definition is adapted from Toneva et al. (2019): + +Iteration learned A data point is said to be learned by a classifier at training iteration $t = \tau$ if the predicted class at iteration $t = \tau - 1$ is different from the final prediction of the converged network and the predictions at all iterations $t \geq \tau$ are equal to the final prediction of the converged network. Data points consistently classified after all training steps and at the moment of initialization, are said to be learned in step $t = 0$ 7 . + +Figure 5 (left plot) shows the positive correlation between the prediction depth and the iteration learned, for all four datasets in VGG16. Consistent results are presented for all architectures and datasets, in both the validation and training splits in Appendix C.4. As a result of the reported correlation, we anticipate that many of the data points correctly classified by the k-NN probe in a particular layer should also be correctly classified by the network at a corresponding interval of training steps. If this is correct then we would expect there to be a visual correspondence between the training learning curve (which shows how the accuracy of the network changes during training) and the accuracy of the $\mathbf { k }$ -NN probes as data passes from input, through the network, towards the output layer. We call the series of $\mathbf { k }$ -NN probe accuracies the inference learning curve. + +![](images/a02ab6322e5cd8cebfe9ac42455d2b4efbdada80864f46deb3a0b8efc92674dd.jpg) +Figure 5: Left: Data points with small prediction depths are on average learned before data points with higher prediction depths. We train 250 VGG16 models for each dataset, using a $9 0 { : } 1 0 \%$ random train:validation split as described in Appendix A. Each time an input appears in the validation split we record the prediction depth and the iteration learned in that model. This plot shows the average iteration learned for data points at each prediction depth. Marker size shows the fraction of the dataset with each prediction depth. The Pearson correlation coefficients for the four data sets are as follows. CIFAR100: 0.83. CIFAR10: 0.7. Fashion MNIST: 0.79. SVHN: 0.77. Middle and right: The training learning curve (middle) shares several important features with the inference learning curve (right). Blue, yellow and green curves represent different components of the CIFAR10 training split, in which we have randomized (and fixed) $40 \%$ of the labels, and red curves show the test split. The middle and right plots show results from 5 random seeds. The inference learning curve (right) is the sequence of k-NN probe accuracy values for each split. All three plots show results for VGG16. The hyperparameters used are given in Appendix A. + +![](images/cd8a095f06b0d86716987a5edc4dae02cc92e33acd326dbeae942c6b16d73e2a.jpg) +Figure 6: Left and Middle: Test examples with smaller prediction depths, on average, have larger output and input margins. We train 25 VGG16 models with different random seeds on CIFAR10 (see Appendix A for details) and compare the mean prediction depth of each test point in these 25 runs to its mean output and input margins (log scales). Correlation coefficients are $- 0 . 7 0$ (output margin) and $- 0 . 6 9$ (input margin). The density of data points is indicated by the color bar, which follows a log scale. Although the prediction depth could be at most 14, no data point has an average prediction depth greater than 12. Right: An intervention that does not encourage large output margin ( $^ { * } O$ -Hinge”) results, as predicted, in models where the predictions are effectively determined in higher layers in the network compared to the standard training $( \ ^ { \ast } C E ^ { \prime \prime } )$ . + +To test this hypothesis we train a model on a training split where a subset of labels are corrupted and compare the training and inference learning curves on four splits of the data: unchanged training data; mislabeled training data; the original labels of the mislabeled training data and the test split. In Figure 5 (middle and right plots) we see that many of the important features of the training learning curve are indeed present in the inference learning curve. During training (middle), mislabeled data are initially processed as though they are a member of their original class (before they were mislabeled) (Liu et al., 2020a). After an initial period of learning, the network begins to learn the new (random) labels that have been assigned to those data points, so the orange curve moves upwards, and the green curve downwards. At this point, a maximum is observed in the training accuracy (Arpit et al., 2017). In the right plot we see that these same phenomena occur in the inference learning curve. + +# 3.3 Deep models exhibit larger margins for inputs with lower prediction depth + +It is reported in the literature that deep networks learn functions of increasing complexity during training (Hu et al., 2020; Kalimeris et al., 2019). We frame this observation differently: the learned function is “locally simpler” in the vicinity of data points with smaller prediction depths, and these points are typically learned earlier in training (Section 3.2). + +Two known measures of the simplicity of a learned function are the output margin (the difference between the largest and second-largest logits) and the adversarial input margin (the smallest norm required for an adversarial perturbation in the input to change the model’s class prediction). We estimate the adversarial input margin, $\gamma$ , with a linear approximation (Jiang et al., 2018): for an input x with predicted class i, γ ' minj6=i |zi−zj ||∇x(zi−zj )| where $z _ { j }$ is the logit returned by the network for class $j$ . Figure 6 (left and middle plots) show that data points with smaller prediction depths have both larger input and output margins on average and that variances of the input and output margins decrease as the prediction depth increases. + +![](images/6adf476b71571c6b4cc9499c21b8fe2583b9b43fa207152bd6831ec0fc9ca860.jpg) +Figure 7: The prediction depth can be the same, or very different for the same input when it occurs in the train and validation splits. Corners of this plot correspond to different forms of example difficulty. (See Section 4 for discussion.) We train 250 ResNet18 models on CIFAR10 with random $9 0 { : } 1 0 \%$ train:validation splits as described in Appendix A. These histograms compare average prediction depth for each data point when it occurs in the validation split vs the training split. This behavior is consistently reproduced for all datasets and architectures in Appendix C.6. Below we show extreme (not hand-chosen) images of “Birds” that appear closest to the corners of this plot. The consensus class is given above each image (tiebreaks favor the class “Bird”.) + +To illustrate the strength of the relationship between the prediction depth and output margin, we demonstrate that reducing the output margin of the learned function results in a model that clusters the data only in the latest layers: such a solution has a very high average prediction depth. We do not minimize the output margin directly but rather use a loss and an optimizer that do not encourage high output margin. Naturally there are many unknowns that may contribute to this effect. We simply report the intervention and the outcome. + +The intervention is performed as follows: we construct a loss function that does not promote confidence: a zero-margin hinge loss ( $^ { 6 6 } 0$ -Hinge”), and optimize the network using full-batch gradient descent with momentum and very small learning rate. For an input $x$ with label $i$ the 0-Hinge loss is given by $\begin{array} { r } { l ( x ) = \sum _ { j \neq i } \operatorname* { m a x } ( \dot { 0 } , z _ { i } - z _ { j } ) } \end{array}$ where $z _ { j }$ represents the logit for class $j$ . The form of this intervention is justified in Appendix A.7. As a control, we additionally train a model in the standard fashion using the cross-entropy loss and SGD with momentum and large initial learning rate. Since full-batch gradients are computationally expensive, we train on a subset of CIFAR10 (see Appendix A.7, where we also give the hyperparameters and learning curves.). The output margin obtained with the intervention is 5 orders of magnitude smaller than in the control experiment: $2 . 0 \times 1 0 ^ { - 4 } \pm 2 . 0 \times 1 0 ^ { - 4 }$ for the 0-Hinge loss and $1 . { \overline { { 6 } } } \times 1 0 ^ { 1 } \pm 0 . 5 0 \times 1 0 ^ { 1 }$ for cross-entropy loss. Figure 6 (right) compares the accuracies of the $\mathbf { k }$ -NN probes resulting from these training approaches. The 0-Hinge loss training achieves only a marginal improvement in accuracy (red) over an untrained network (purple), and the training split is accurately clustered only in the latest layers. This confirms the predicted behavior: the intervention leads to a model that exhibits both very small average output margins and very late clustering of the data. Very late clustering of the data implies high prediction depths since the $\mathbf { k }$ -NN probe classifications change in the latest layers for many data points. + +# 4 Beyond a One-Dimensional Picture of Example Difficulty + +In this section we transcend the one-dimensional picture of example difficulty by identifying different underlying reasons behind the difficulty of an example, in a way that is general to different architectures and datasets. + +Figure 7 shows that the prediction depth can be different when an input occurs in the training split vs. the validation split. Thus, there are two axes of example difficulty: + +1. Difficulty of making a prediction when an input is in the validation set 2. Difficulty of finding commonalities during training with other examples of the same ground truth class + +Both axes have a range from “clear” to “ambiguous”. In Section 3.1 we show that predictions made for validation points with later prediction depths are often inconsistent, with low consensusconsistency. Conversely, a low prediction depth typically indicates an input with high consensusconsistency. For Axis 1 we will identify validation points with low prediction depths as “clear” and those with high prediction depths as “ambiguous”. We will additionally identify a low or high prediction depth in the training split with examples that are respectively “clear” and “ambiguous” on Axis 2. By making combinations of low/high values of $( \mathrm { P D } _ { \mathrm { V a l . } }$ , $\mathrm { P D } _ { \mathrm { T r a i n } _ { . } }$ ) we obtain four extremes of example difficulty: + +![](images/14930e4f6df689d31626c023acd5f7f776fdcf3f0a5998ae876b3c3fa70cabe4.jpg) +Figure 8: Average k-NN probe confidence (solid lines) and accuracy (dotted lines) for the ground truth class (left) and consensus class (right), in the validation split for examples exhibiting extreme forms of difficulty. Mean values for 100 examples with each form of difficulty, identified as the 100 examples closest to the corners in Figure 7 (left). This result is for CIFAR10 with ResNet18: similar plots for all datasets and architectures are shown in Appendix C.7. See Section 4 for the discussion of the result and how it can be used to improve prediction accuracy. + +Easy examples: (Low $\mathrm { P D } _ { \mathrm { V a l . } }$ , Low $\mathrm { P D } _ { \mathrm { T r a i n } }$ ). Such examples are often visually typical members of their class and the predicted label nearly always matches the ground truth. +Looks like a different class: (Low $\mathrm { P D } _ { \mathrm { V a l . } }$ , High $\mathrm { P D } _ { \mathrm { T r a i n } }$ ). In the validation set, there is a clear (and nearly always incorrect) classification for such an input, but it is difficult to connect such inputs to other examples of their ground truth class during training. Mislabeled examples are of this kind, as are visually confusing images which at first appear to show something else. +Ambiguous unless the label is given: (High $\mathrm { P D } _ { \mathrm { V a l . } }$ , Low $\mathrm { P D } _ { \mathrm { T r a i n } }$ ). These examples are difficult to connect to their predicted class in the validation split but easy to connect to their ground truth class during training. These points may, for example, visually resemble both their own class and another class. They are likely to be misclassified. +Ambiguous: (High $\mathrm { P D } _ { \mathrm { V a l . } }$ , High $\mathrm { P D } _ { \mathrm { T r a i n } }$ ). These examples may be corrupted or show an example of a rare sub-class. Predictions for these inputs can depend strongly on the random seed used for training and initialization. + +In Figure 7 we visualize CIFAR10 “Bird” images with the extreme forms of example difficulty for ResNet18, as identified using the prediction depth in the training and validation splits. In the full dataset (left panel) we see that the prediction depth can be very different in the training and validation splits: the two prediction depths are typically similar for points where the consensus class is equal to the ground truth (right panel), but can be very different when the consensus class is different from the ground truth (middle panel). This behavior is consistently reproduced for all datasets and architectures in Appendix C.6. + +Looking at these examples of the class “Bird” with different difficulty types, we observe that ResNet18 finds small garden birds easiest, while birds in flight against a blue background “look like airplanes”, ostriches are “ambiguous without their label” and the “ambiguous” examples are either unclear photographs or examples of rare sub-groups that don’t appear frequently in the data. We found the consensus-consistency of inputs that are “Ambiguous” or “Ambiguous without its label” to be significantly lower than those of examples that are “Easy” or “Look like a different class”. + +In order to better understand how networks process examples with different, extreme forms of example difficulty, Fig. 8 examines how the k-NN confidence (fraction of votes) and accuracy of the ground truth class and of the consensus class progress, as validation points pass through the network. “Easy” examples are classified as their consensus class (which is equal to their ground truth class) in all k-NN probes and the confidence in the consensus class steadily increases as data points proceed through the hidden layers. Examples that “look like a different class” are also processed as members of their consensus class, similarly to “easy” examples. However, unlike “easy” examples, their consensus classes do not match their ground truth classes. Examples that are “ambiguous without their labels” are initially processed as members of their ground truth classes with intermediate confidence, but in later layers become mistaken for their consensus class. “Ambiguous” examples are processed with low confidence and accuracy in the early layers, for both ground truth and consensus classes. In later layers “ambiguous” examples are recognized, with intermediate confidence and accuracy, as members of the consensus class, which matches the ground truth class for a sizeable fraction of “ambiguous” examples. + +Improving the prediction accuracy Can the prediction accuracy be improved using our understanding of how each class of difficult examples are processed by deep models? Figure 8 suggest that $\mathbf { k }$ -NN probes in intermediate layers may be more accurate than the full deep model for examples that are “ambiguous without their label” (data points closest to the lower right corner of Figure 7). In order to test this hypothesis, we compare the accuracy of the $\mathbf { k }$ -NN probe in layer 4 to the full model’s prediction for the 100 examples closest to the lower right corner of Figure 8. We obtain a striking improvement in accuracy from $2 5 \%$ to $98 \%$ for these examples. This showcases how insights from this study can be directly used to improve prediction accuracy. + +# 5 Discussion + +Summary We have introduced a notion of example difficulty called the prediction depth, which uses the processing of data inside the network to score the difficulty of an example. We have shown how the prediction depth is related to the accuracy and uncertainty of a prediction, the adversarial input margin and the output margin of the learned solution, and that data points that are easier according to the prediction depth are also typically learned earlier in training. We have also shown that the difficulty of an example can be both similar, or very different depending on whether an input appears in the validation split or the training split, and described four extremes of example difficulty. For data points that are “ambiguous without their label”, we have demonstrated how returning the $\mathbf { k }$ -NN prediction in a middle layer can lead to impressive increases in model accuracy: for CIFAR10 in ResNet18 we obtained an increase in accuracy from $2 5 \%$ to $98 \%$ for the inputs that are most “ambiguous without their label”. + +Connecting known phenomena In the literature, the following phenomena are separately reported from different experimental paradigms: + +1. Early layers generalize while later layers memorize (Stephenson et al., 2021). +2. Model layers converge from input layer towards output layer (Raghu et al., 2017; Morcos et al., 2018). +3. Deep models learn easy data (Jiang et al., 2021; Toneva et al., 2019) and simple functions first (Hu et al., 2020; Kalimeris et al., 2019). + +Following this paper, a coherent and closely related picture emerges: + +1. Predictions made in early layers are more likely to be consistent than those made in later layers. Consistent predictions are likely to be correct and the expected accuracy of inconsistent predictions is naturally low (Section 3.1). +2. Data points learned early in training typically have smaller prediction depths than those learned later during training (Section 3.2). +3. On average, deep neural networks exhibit wider input and output margins (common measures of “local simplicity”) in the vicinity of data with smaller prediction depths (Section 3.3). + +Pertinence of example difficulty to topics in machine learning Curriculum Learning attempts to treat hard examples differently from easy examples during training. Robustness to distribution shifts that change the relative frequencies of common and rare subgroups in the test set (which we have shown can have different forms of example difficulty) is important for ML Fairness. Methods developed to address heteroscedastic uncertainty typically address example difficulty as a onedimensional quantity. We expand upon the relevance of our work to these three topics in Appendix D. + +Limitations We believe that the results we report stem from a deep model’s representation, which is hierarchical by construction. We expect that the same results will therefore apply in larger models, larger datasets, and tasks other than image classification, but testing this remains as further work. Although we demonstrate that returning the results of a hidden k-NN can yield dramatic increases in accuracy for examples that are “ambiguous without their label”, we otherwise do not explore ways to practically apply the insights we present. In particular, we expressly do not claim that all that is required for good accuracy is to reduce the prediction depth: freezing later layers of the network would not be expected to result in good generalization. + +# Funding Transparency Statement + +This research was funded by, and undertaken at, Google. All calculations were performed using Google’s computer infrastructure. + +# Acknowledgment + +We would like to thank Hanie Sedghi, Ilya Tolstikhin, Ibrahim Alabdulmohsin, Daniel Keysers and Julian Eisenschlos for valuable discussions on the topic and Arthur Baldock for proofreading the manuscript. + +# References + +Agarwal, C. and Hooker, S. (2020). Estimating example difficulty using variance of gradients. In ICML, Workshop on Human Interpretability in Machine Learning (WHI). +Alain, G. and Bengio, Y. (2017). Understanding intermediate layers using linear classifier probes. 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Weak and strong gradient directions: Explaining memorization, generalization, and hardness of examples at scale. arXiv preprint arXiv:2003.07422. \ No newline at end of file diff --git a/parse/train/fmgYOUahK9/fmgYOUahK9_content_list.json b/parse/train/fmgYOUahK9/fmgYOUahK9_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..70e833b2c0576f8a94039410fa0eb80498dd5cf4 --- /dev/null +++ b/parse/train/fmgYOUahK9/fmgYOUahK9_content_list.json @@ -0,0 +1,1098 @@ +[ + { + "type": "text", + "text": "Deep Learning Through the Lens of Example Difficulty ", + "text_level": 1, + "bbox": [ + 225, + 122, + 774, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Robert J. N. Baldock∗ Google Research, Brain Team rjnbaldock@gmail.com ", + "bbox": [ + 254, + 226, + 455, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hartmut Maennel Google Research, Brain Team hartmutm@google.com ", + "bbox": [ + 542, + 226, + 743, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Behnam Neyshabur Google Research, Blueshift Team neyshabur@google.com ", + "bbox": [ + 387, + 289, + 609, + 332 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 367, + 535, + 382 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Existing work on understanding deep learning often employs measures that compress all data-dependent information into a few numbers. In this work, we adopt a perspective based on the role of individual examples. We introduce a measure of the computational difficulty of making a prediction for a given input: the (effective) prediction depth. Our extensive investigation reveals surprising yet simple relationships between the prediction depth of a given input and the model’s uncertainty, confidence, accuracy and speed of learning for that data point. We further categorize difficult examples into three interpretable groups, demonstrate how these groups are processed differently inside deep models and showcase how this understanding allows us to improve prediction accuracy. Insights from our study lead to a coherent view of a number of separately reported phenomena in the literature: early layers generalize while later layers memorize; early layers converge faster and networks learn easy data and simple functions first. ", + "bbox": [ + 233, + 392, + 767, + 571 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 176, + 590, + 310, + 607 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Much of the existing work on understanding deep learning “integrates out” the data, viewing the inductive bias of the model, or the properties of the optimizer as central to the success of the approach. Examples of such work include studies of eigenvalues of the Hessian and the geometry of the loss landscape (Ghorbani et al., 2019; Yao et al., 2020; Sagun et al., 2016; Li et al., 2018; Pennington and Bahri, 2017; Sagun et al., 2018), studies of margin and effective generalization measures (Long and Sedghi, 2019; Unterthiner et al., 2020; Jiang et al., 2020, 2018; Kawaguchi et al., 2017) and mean-field studies of stochastic optimization (Smith et al., 2021; Stephan et al., 2017; Smith and Le, 2018). However, in practice, we are rarely concerned with only the average behavior of a model. ", + "bbox": [ + 174, + 616, + 826, + 727 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One pathway to understanding the principles that govern how deep models process data is to study the properties of deep models for data points with different “amounts” or “types” of example difficulty. There are a number of definitions of example difficulty in the literature (E.g. see Carlini et al. (2019); Hooker et al. (2019); Lalor et al. (2018); Agarwal and Hooker (2020)). Two are particularly relevant to this work. Firstly, the probability of predicting the ground truth label for an example, when that example is omitted from the training set (Jiang et al., 2021), which represents a statistical view of example difficulty. Secondly, the difficulty of learning an example, parameterized by the earliest training iteration after which the model predicts the ground truth class for that example in all subsequent iterations (Toneva et al., 2019). This measure represents a learning view of example difficulty 2. ", + "bbox": [ + 174, + 728, + 826, + 864 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "These notions suffer from two fundamental limitations. While early-exit strategies in computer vision (Teerapittayanon et al., 2016; Huang et al., 2018) and NLP (Dehghani et al., 2018; Liu et al., 2020b; Schwartz et al., 2020; Xin et al., 2020) suggest predictions for easier examples require less computation, the above example difficulty notions do not encapsulate the processing of data inside a given converged model. Moreover, existing notions of example difficulty (E.g. Carlini et al. (2019)) provide a one-dimensional view of difficulty which can not distinguish between examples that are difficult for different reasons. ", + "bbox": [ + 174, + 92, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we take a significant step towards resolving the above shortcomings. To take the processing of the data into account we propose a new measure of example difficulty, the prediction depth, which is determined from the hidden embeddings. To escape the one-dimensional view of difficulty, we introduce three distinct difficulty types by relating the hidden embeddings for an input to high-level concepts about example difficulty: “Does this example look mislabeled?”; “Is classifying this example only easy if the label is given?”; “Is this example ambiguous both with and without its label?”. Furthermore, we show how this enhanced notion of example difficulty can unify our understanding of several seemingly unrelated phenomena in deep learning. We hope that the results presented in this work will aid the development of models that capture heteroscedastic uncertainty, our understanding of how deep networks respond to distributional shift, and the advancement of curriculum learning approaches and machine learning fairness. These connections are discussed in Section 5. ", + "bbox": [ + 174, + 189, + 825, + 354 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions Our main contributions are as follows: ", + "bbox": [ + 176, + 364, + 540, + 378 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We introduce a measure of computational example difficulty: the prediction depth (PD). The prediction depth, illustrated in Figure 1, represents the number of hidden layers after which the network’s final prediction is already (effectively) determined (Section 2). \nWe show that the prediction depth is larger for examples that visually appear to be more difficult, and that prediction depth is consistent between architectures and random seeds (Section 2.2). \n• Our empirical investigation reveals that prediction depth appears to establish a linear lower bound on the consistency of a prediction. We further show that predictions are on average more accurate for validation points with small prediction depths (Section 3.1). \nWe demonstrate that final predictions for data points that converge earlier during training are typically determined in earlier layers which establishes a correspondence between the training history of the network and the processing of data in the hidden layers (Section 3.2). \n• We show that both the adversarial input margin and the output margin are larger for examples with smaller prediction depths. We further design an intervention to reduce the output margin of a network and show that this leads to predictions being made only in the latest hidden layers (Section 3.3). \nWe identify three extreme forms of example difficulty by considering the prediction depth in the training and validation splits independently and demonstrate how a simple algorithm that uses the hidden embeddings in one middle layer to make predictions can lead to dramatic improvements in accuracy for inputs that strongly exhibit a specific form of example difficulty (Section 4). \nWe use our results to present a coherent picture of deep learning that unifies four seemingly unrelated deep learning phenomena: early layers generalize while later layers memorize; networks converge from input layer towards output layer; easy examples are learned first and networks present simpler functions earlier in training (Section 5). ", + "bbox": [ + 217, + 385, + 825, + 753 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Experimental Setup: To ensure that our results are robust to the choice of architectures and datasets, we report empirical findings for ResNet18 (He et al., 2016), VGG16 (Simonyan and Zisserman, 2015) and MLP architectures trained on CIFAR10, CIFAR100 (Krizhevsky et al., 2009), Fashion MNIST (FMNIST) (Xiao et al., 2017) and SVHN (Netzer et al., 2011) datasets. All models were trained using SGD with momentum. Our MLP comprises 7 hidden layers of width 2048 with ReLU activations. Details of the datasets, architectures, and hyperparameters used can be found in Appendix A. ", + "bbox": [ + 174, + 758, + 825, + 842 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Related Work: Our work uses hidden layer probes to determine example difficulty. We have discussed how our study relates to prior work on example difficulty. Hidden layer probes have also been used to study deep learning. Deep k-NN methods (Papernot and McDaniel, 2018) determine their predictions and estimate their own uncertainties by comparing the hidden embeddings of an input to those of the training set. Cohen et al. (2018) showed that SVM, $\\mathbf { k }$ -Nearest Neighbors $\\mathbf { k }$ -NN) ", + "bbox": [ + 174, + 842, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "and logistic regression probes achieve similar accuracies. However, they did not study the processing of individual data points nor did they relate the $\\mathbf { k }$ -NN accuracy to notions of example difficulty. Alain and Bengio (2017) used linear classifier probes in the hidden layers to interrogate deep models and demonstrated that linear separability of the embeddings increases monotonically with depth. We provide a more detailed discussion of related work in Appendix B. ", + "bbox": [ + 174, + 90, + 825, + 161 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Prediction Depth: a Computational View of Example Difficulty ", + "text_level": 1, + "bbox": [ + 174, + 174, + 730, + 191 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We discussed the statistical and learning views of example difficulty in Section 1. In this section, we introduce a computational view of example difficulty parametrized by the prediction depth as defined in Section 2.1. This computational view asserts that, for “easy” examples, a deep model’s final prediction is effectively made after only a few layers, while more layers are used for “difficult” examples. ", + "bbox": [ + 173, + 199, + 826, + 268 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Definition ", + "text_level": 1, + "bbox": [ + 174, + 280, + 282, + 295 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Asserting that the final prediction is effectively determined in earlier layers of a model, before the output, we estimate the depth at which a prediction is made for a given input as follows 3: ", + "bbox": [ + 176, + 299, + 823, + 328 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. We construct k-NN classifier probes from the embeddings of the training set after particular layers of the network, including the input and the final softmax. The placement of $\\mathbf { k }$ -NN probes is described in Appendix A.5. We use $k = 3 0$ in the $\\mathbf { k }$ -NN probes. Appendix A.4 establishes that the k-NN accuracies we report are insensitive to $k$ over a wide range. 2. A prediction is defined to be made at a depth $L = l$ if the $\\mathbf { k }$ -NN classification after layer $L = l - 1$ is different from the network’s final classification, but the classifications of k-NN probes after every layer $L \\geq l$ are all equal to the final classification of the network. Data points consistently classified by all $\\mathbf { k }$ -NN probes are determined to be (effectively) predicted in layer 0 (the input) 4. ", + "bbox": [ + 210, + 333, + 825, + 460 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "It is worth noting that the prediction depth can be calculated for all data points: both in the training and validation splits. This leads to two notions of computational difficulty: ", + "bbox": [ + 171, + 465, + 823, + 493 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• The difficulty of predicting the (given) class for an input (in the training split) • The difficulty of making a prediction for an input, unseen in advance (from the validation split) ", + "bbox": [ + 218, + 498, + 825, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We examine both notions of computational difficulty in this paper and use the distinction between them to describe different forms of example difficulty in Section 4. ", + "bbox": [ + 173, + 549, + 823, + 579 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Prediction depth is a meaningful and robust notion of example difficulty ", + "text_level": 1, + "bbox": [ + 176, + 588, + 714, + 603 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section we show that prediction depth agrees with intuitive notions of example difficulty and that it is consistent between different training runs and similar architectures. ", + "bbox": [ + 174, + 608, + 823, + 637 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Prediction depth is higher for examples and datasets that seem more difficult If prediction depth is a sensible measure of example difficulty then we would expect the following sanity checks to be observed: ", + "bbox": [ + 176, + 645, + 825, + 686 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. Individual data points that are visually confusing or mislabeled should have larger prediction depths as compared to images that are clear examples of their class. \n2. Data points from tasks that are intuitively simpler should have lower prediction depths on average. ", + "bbox": [ + 212, + 693, + 825, + 751 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 shows that the prediction depth passes both of these sanity checks. Appendix C.1 presents additional images, providing further evidence for this claim. ", + "bbox": [ + 171, + 756, + 823, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Prediction depth is consistent across random seeds and similar architectures Figure 2 shows that the prediction depth is highly consistent between different architectures and random seeds for all datasets. Perfect agreement is not expected as different deep learning algorithms have different inductive biases which affects the perceived difficulty of examples. We observe stronger correlation between prediction depth for ResNet18 and VGG16, than between VGG16 and MLP. This may be explained by the fact that ResNet18 and VGG16 are both convolutional networks and we expect their inductive biases to be more similar to one another than to MLP. ", + "bbox": [ + 173, + 792, + 825, + 849 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/41ae6d0999f3d52ac3d9fed1621b96ae037b526605cd4d18c4e702c118a942c4.jpg", + "image_caption": [ + "Figure 1: Deep models use fewer layers to (effectively) determine the prediction for easy examples and more layers for hard examples. Left: A cartoon illustrating the definition of prediction depth (given in Section 2.1). Also shown are training examples from CIFAR100 (“Clock”) and SVHN (“Digit 8”). The examples shown are predicted at the input (first layer) or softmax (last layer) of ResNet18. The examples predicted in the input are visually typical (“easy”), while those predicted in the softmax are mislabeled and/or visually confusing (“hard” examples). To find the prediction depth, we build $\\mathbf { k }$ -NN classifiers from the embeddings of the training set in different layers of the model. The prediction depth corresponds to the earliest layer at which the predictions of all subsequent k-NN classifiers converge to a fixed label. Right: Probability of prediction depth in ResNet18 models for four datasets (training split). We see that the four distributions have different characteristic prediction depths. Ranking the mean prediction depths of these datasets in ascending order, we observe: Fashion MNIST (smallest), SVHN (second), CIFAR10 (third), and CIFAR100 (largest). This order aligns with how one might intuitively rank the difficulties of these classification tasks. " + ], + "image_footnote": [], + "bbox": [ + 178, + 88, + 821, + 228 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/23c98a880da3f394cd13a1b9baa89a6f129ff8653082737c21aec2cb3c228e0f.jpg", + "image_caption": [ + "Figure 2: Consistency of prediction depth between architectures and random seeds. Left: The panel shows the correlation coefficient between prediction depths in different architectures, for both train and validation splits in four datasets. Diagonal comparisons between an architecture and itself show the correlation for the same architecture trained with different random seeds. Right: Histograms comparing the mean value of prediction depth obtained for each data point in the training set of CIFAR10 from an ensemble of 250 trained models. In this plot, for visual simplicity, we rescale prediction depth to the interval $[ 0 , 1 ]$ for each network. Similar results for all other datasets are presented in Appendix C.2. " + ], + "image_footnote": [], + "bbox": [ + 178, + 402, + 820, + 526 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 637, + 825, + 679 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Deep Learning Phenomena Through the Lens of Prediction Depth ", + "text_level": 1, + "bbox": [ + 173, + 693, + 754, + 710 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we explore how the prediction depth can be used to better understand three important aspects of Deep Learning: accuracy and consistency of a prediction; the order in which data is learned and the simplicity of the learned function (as measured by the margin) in the vicinity of a data point. ", + "bbox": [ + 174, + 718, + 825, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 Depth of a prediction gives a linear lower bound on its consistency ", + "text_level": 1, + "bbox": [ + 174, + 771, + 671, + 786 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Adopting a statistical view of example difficulty, Jiang et al. (2021) identified example difficulty with the expected accuracy of the learning algorithm for a given input, averaged over models trained on different random subsets of the training set with different random seeds. In this section, we clarify the relationship between the prediction depth and the expected accuracy by disentangling the accuracy from the sensitivity of predictions to the particular training split and random seed. Following Jiang et al. (2021), we measure the expected accuracy using the consistency score. ", + "bbox": [ + 174, + 791, + 825, + 875 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Consistency score $\\hat { C }$ : The frequency of classifying an example correctly when it is omitted from the training set. An empirical estimator of the consistency score for a validation point $( x , y )$ ", + "bbox": [ + 178, + 882, + 821, + 911 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/67ee1b181de37aba816ac42fe368a058ecc0e63537f5c06a4103f90e7f6c3a63.jpg", + "image_caption": [ + "Figure 3: Consistency score vs. prediction depth in the validation split (left) can be understood as the superposition of two simple functions (middle and right). We trained 250 ResNet18 models on CIFAR10, with $9 0 { : } 1 0 \\%$ random train:validation splits as described in Appendix A. These histograms compare the frequency of correct predictions to the average prediction depth for a data point when it occurs in the validation split. The density of data points is indicated by the color bar, which follows a log scale. The average prediction depth forms two, surprisingly simple, linear bounds on the consistency score (see Section 3.1 for a full description.) This Figure is reproduced for all datasets and architectures in Appendix C.3, illustrating the consistency of this result. " + ], + "image_footnote": [], + "bbox": [ + 176, + 90, + 821, + 207 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "is given by (Jiang et al., 2021): ", + "bbox": [ + 232, + 335, + 436, + 349 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/5730aeecf83fec515bfe03f509fd7fa6e15a2f6c15a6a194a6df8b4a4d27036c.jpg", + "text": "$$\n\\hat { C } _ { A , S } ( x , y ) = \\hat { \\mathbb { E } } _ { \\tilde { S } \\sim { \\cal S } \\setminus \\{ ( x , y ) \\} } ^ { r } \\left[ \\delta _ { y _ { A } , y } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 411, + 357, + 645, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $A$ is a deep learning algorithm (architecture, loss and optimizer), $y$ is the ground truth class for $x$ , $\\tilde { \\cal S }$ is a random subset of $n$ points sampled from a training dataset $s$ excluding $( x , y )$ , $y _ { A }$ is the predicted class of $x$ for $A$ trained with data $\\tilde { \\cal S }$ , $\\delta$ is the Kronecker delta and $\\hat { \\mathbb { E } } ^ { r }$ denotes empirical averaging with $r$ i.i.d. samples of such subsets $\\tilde { \\cal S }$ . ", + "bbox": [ + 230, + 386, + 823, + 449 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 3 (left panel) shows the relationship between consistency score and prediction depth. This plot indicates a surprising piecewise linear boundary which is symmetric around consistency score $\\frac { 1 } { 2 }$ This suggests the existence of a missing concept that could simplify the picture. We next show that the missing concept is the notion of a consensus class which is defined below. ", + "bbox": [ + 176, + 455, + 825, + 511 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Consensus class ${ \\hat { y } } _ { A }$ : The consensus class of $x$ is defined as the predicted class for input $x$ by a majority voting ensemble of $r$ models each of which is trained on a randomly chosen subset ${ \\tilde { \\cal S } } \\stackrel { n } { \\sim } { \\cal S } \\backslash \\{ ( x , y ) \\} \\stackrel { 5 } { \\sim }$ . ", + "bbox": [ + 174, + 517, + 825, + 564 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 3 (middle and right) shows how conditioning on whether consensus class matches the ground truth can change the relationship between consistency score and the prediction depth. For points where the consensus class matches the ground truth (middle) we see that the prediction depth forms a, surprisingly simple, linear lower bound on the consistency score. For points where the consensus class differs from the ground truth (right) at low prediction depth the consistency score is bounded from above by a line that reflects the bound from the middle plot in $\\begin{array} { r } { \\hat { C } = \\frac { 1 } { 2 } } \\end{array}$ , suggesting that such points are repeatedly mislabeled with a wrong class label. At high prediction depth, the consistency score is low, which suggests highly inconsistent predictions and low accuracy. This result suggests a simple hypothesis: that predictions with low prediction depth are consistent with the consensus class, whether that matches the ground truth class or not, while predictions made in later layers depend strongly on the specific training split and random seed used for training and initialization. We measure consistency with the consensus class using the consensus-consistency score. ", + "bbox": [ + 173, + 569, + 825, + 738 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Consensus-consistency score $C ^ { * }$ : The fraction of models in an ensemble that predict the ensemble’s consensus class ${ \\hat { y } } _ { A } \\left( x \\right)$ for an unseen input $x$ . ", + "bbox": [ + 171, + 743, + 823, + 772 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6a1a098c8a6a2978c52b97c9e07e0fa9551c7045cde12caea458017c72e5794f.jpg", + "text": "$$\nC _ { A , S } ^ { * } ( x ) = \\hat { \\mathbb { E } } _ { \\tilde { S } \\sim \\tilde { S } \\backslash \\{ ( x , y ) \\} } ^ { r } \\left[ \\delta _ { y _ { A } , \\hat { y } _ { A } ( x ) } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 403, + 779, + 651, + 804 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the notation is the same as in (1) 6. ", + "bbox": [ + 233, + 811, + 503, + 825 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 4 (left) establishes that our simple hypothesis is indeed correct: the prediction depth forms a linear lower bound on the consensus-consistency score for all data points, irrespective of whether the consensus class matches or differs from the ground truth. Interestingly, Figure 4 (middle and right) shows how the prediction depth in a single model, can be used to estimate both of these quantities. That is, predictions of data points with lower prediction depth are both more likely to be consistent and more likely to be correct. ", + "bbox": [ + 174, + 832, + 821, + 862 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8a5ae0a88a3e513d4097c5f1dcee15b7f3d5e2e68ceae4e1b0a5b33e9d9f4a7a.jpg", + "image_caption": [ + "Figure 4: Left: Prediction depth provides us with a linear lower bound on consensus-consistency. Results for CIFAR100 with ResNet18. We train 250 models $( 9 0 { : } 1 0 \\%$ random train:validation splits) and compare the average prediction depth when a point occurs in the validation set, to the consensus-consistency of the corresponding predictions. Predictions made for points with low mean prediction depths are highly consistent. Conversely, predictions for points with high mean prediction depths are typically more sensitive to the particular training split and random seed used during training. This left plot shows the result for CIFAR100 with ResNet18. The density of data points is indicated by the color bar, which follows a log scale. Middle: Prediction depth in one model predicts the consensus-consistency of an ensemble that does not include that model. For each dataset we train 25 ResNet18 models with the full training set (see Appendix A). The consensus-consistency of each test point is obtained from 24 of the models, while the prediction depth is obtained from the remaining 1 model. We see that prediction depth in one model predicts the consensus-consistency of a separate ensemble: a measure of the uncertainty of the prediction. The size of each marker in the middle and right plots shows the fraction of the dataset with each prediction depth. Reaffirming the second sanity check in Section 2.2, and in agreement with Figure 1 (right), intuitively simpler datasets (Fashion MNIST and SVHN) have low average prediction depths, while CIFAR100 (intuitively the hardest dataset) has the largest average prediction depth. Right: Prediction depth predicts accuracy. For each dataset we train 250 ResNet18 models $9 0 { : } 1 0 \\%$ random train:validation splits). Each time a point appears in the validation split we record the prediction depth and whether the prediction was correct. Predictions made in earlier layers are more likely to be correct. Consistency of these plots is demonstrated for all datasets and architectures in Appendix C.3 where we also describe the relationship between the prediction depth and the entropy of the predictions for an ensemble. " + ], + "image_footnote": [], + "bbox": [ + 178, + 93, + 821, + 208 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 484, + 825, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 The prediction depth of an input is correlated with its learning difficulty ", + "text_level": 1, + "bbox": [ + 174, + 551, + 714, + 566 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Section 3.1, we describe the relationship between the prediction depth, which represents a computational view of example difficulty and the consistency and consensus-consistency scores, which represent a statistical view. In this section we compare prediction depth to a learning view of example difficulty. We measure the difficulty of learning an example by the speed at which the model’s prediction converges for that input during training. The following definition is adapted from Toneva et al. (2019): ", + "bbox": [ + 174, + 571, + 825, + 655 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Iteration learned A data point is said to be learned by a classifier at training iteration $t = \\tau$ if the predicted class at iteration $t = \\tau - 1$ is different from the final prediction of the converged network and the predictions at all iterations $t \\geq \\tau$ are equal to the final prediction of the converged network. Data points consistently classified after all training steps and at the moment of initialization, are said to be learned in step $t = 0$ 7 . ", + "bbox": [ + 174, + 661, + 825, + 731 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 5 (left plot) shows the positive correlation between the prediction depth and the iteration learned, for all four datasets in VGG16. Consistent results are presented for all architectures and datasets, in both the validation and training splits in Appendix C.4. As a result of the reported correlation, we anticipate that many of the data points correctly classified by the k-NN probe in a particular layer should also be correctly classified by the network at a corresponding interval of training steps. If this is correct then we would expect there to be a visual correspondence between the training learning curve (which shows how the accuracy of the network changes during training) and the accuracy of the $\\mathbf { k }$ -NN probes as data passes from input, through the network, towards the output layer. We call the series of $\\mathbf { k }$ -NN probe accuracies the inference learning curve. ", + "bbox": [ + 173, + 736, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/a02ab6322e5cd8cebfe9ac42455d2b4efbdada80864f46deb3a0b8efc92674dd.jpg", + "image_caption": [ + "Figure 5: Left: Data points with small prediction depths are on average learned before data points with higher prediction depths. We train 250 VGG16 models for each dataset, using a $9 0 { : } 1 0 \\%$ random train:validation split as described in Appendix A. Each time an input appears in the validation split we record the prediction depth and the iteration learned in that model. This plot shows the average iteration learned for data points at each prediction depth. Marker size shows the fraction of the dataset with each prediction depth. The Pearson correlation coefficients for the four data sets are as follows. CIFAR100: 0.83. CIFAR10: 0.7. Fashion MNIST: 0.79. SVHN: 0.77. Middle and right: The training learning curve (middle) shares several important features with the inference learning curve (right). Blue, yellow and green curves represent different components of the CIFAR10 training split, in which we have randomized (and fixed) $40 \\%$ of the labels, and red curves show the test split. The middle and right plots show results from 5 random seeds. The inference learning curve (right) is the sequence of k-NN probe accuracy values for each split. All three plots show results for VGG16. The hyperparameters used are given in Appendix A. " + ], + "image_footnote": [], + "bbox": [ + 183, + 90, + 812, + 185 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/cd8a095f06b0d86716987a5edc4dae02cc92e33acd326dbeae942c6b16d73e2a.jpg", + "image_caption": [ + "Figure 6: Left and Middle: Test examples with smaller prediction depths, on average, have larger output and input margins. We train 25 VGG16 models with different random seeds on CIFAR10 (see Appendix A for details) and compare the mean prediction depth of each test point in these 25 runs to its mean output and input margins (log scales). Correlation coefficients are $- 0 . 7 0$ (output margin) and $- 0 . 6 9$ (input margin). The density of data points is indicated by the color bar, which follows a log scale. Although the prediction depth could be at most 14, no data point has an average prediction depth greater than 12. Right: An intervention that does not encourage large output margin ( $^ { * } O$ -Hinge”) results, as predicted, in models where the predictions are effectively determined in higher layers in the network compared to the standard training $( \\ ^ { \\ast } C E ^ { \\prime \\prime } )$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 356, + 818, + 457 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To test this hypothesis we train a model on a training split where a subset of labels are corrupted and compare the training and inference learning curves on four splits of the data: unchanged training data; mislabeled training data; the original labels of the mislabeled training data and the test split. In Figure 5 (middle and right plots) we see that many of the important features of the training learning curve are indeed present in the inference learning curve. During training (middle), mislabeled data are initially processed as though they are a member of their original class (before they were mislabeled) (Liu et al., 2020a). After an initial period of learning, the network begins to learn the new (random) labels that have been assigned to those data points, so the orange curve moves upwards, and the green curve downwards. At this point, a maximum is observed in the training accuracy (Arpit et al., 2017). In the right plot we see that these same phenomena occur in the inference learning curve. ", + "bbox": [ + 173, + 574, + 825, + 727 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.3 Deep models exhibit larger margins for inputs with lower prediction depth ", + "text_level": 1, + "bbox": [ + 176, + 743, + 728, + 758 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "It is reported in the literature that deep networks learn functions of increasing complexity during training (Hu et al., 2020; Kalimeris et al., 2019). We frame this observation differently: the learned function is “locally simpler” in the vicinity of data points with smaller prediction depths, and these points are typically learned earlier in training (Section 3.2). ", + "bbox": [ + 174, + 765, + 825, + 820 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Two known measures of the simplicity of a learned function are the output margin (the difference between the largest and second-largest logits) and the adversarial input margin (the smallest norm required for an adversarial perturbation in the input to change the model’s class prediction). We estimate the adversarial input margin, $\\gamma$ , with a linear approximation (Jiang et al., 2018): for an input x with predicted class i, γ ' minj6=i |zi−zj ||∇x(zi−zj )| where $z _ { j }$ is the logit returned by the network for class $j$ . Figure 6 (left and middle plots) show that data points with smaller prediction depths have both larger input and output margins on average and that variances of the input and output margins decrease as the prediction depth increases. ", + "bbox": [ + 174, + 821, + 825, + 912 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/6adf476b71571c6b4cc9499c21b8fe2583b9b43fa207152bd6831ec0fc9ca860.jpg", + "image_caption": [ + "Figure 7: The prediction depth can be the same, or very different for the same input when it occurs in the train and validation splits. Corners of this plot correspond to different forms of example difficulty. (See Section 4 for discussion.) We train 250 ResNet18 models on CIFAR10 with random $9 0 { : } 1 0 \\%$ train:validation splits as described in Appendix A. These histograms compare average prediction depth for each data point when it occurs in the validation split vs the training split. This behavior is consistently reproduced for all datasets and architectures in Appendix C.6. Below we show extreme (not hand-chosen) images of “Birds” that appear closest to the corners of this plot. The consensus class is given above each image (tiebreaks favor the class “Bird”.) " + ], + "image_footnote": [], + "bbox": [ + 179, + 89, + 820, + 262 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 378, + 823, + 406 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To illustrate the strength of the relationship between the prediction depth and output margin, we demonstrate that reducing the output margin of the learned function results in a model that clusters the data only in the latest layers: such a solution has a very high average prediction depth. We do not minimize the output margin directly but rather use a loss and an optimizer that do not encourage high output margin. Naturally there are many unknowns that may contribute to this effect. We simply report the intervention and the outcome. ", + "bbox": [ + 174, + 407, + 825, + 489 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The intervention is performed as follows: we construct a loss function that does not promote confidence: a zero-margin hinge loss ( $^ { 6 6 } 0$ -Hinge”), and optimize the network using full-batch gradient descent with momentum and very small learning rate. For an input $x$ with label $i$ the 0-Hinge loss is given by $\\begin{array} { r } { l ( x ) = \\sum _ { j \\neq i } \\operatorname* { m a x } ( \\dot { 0 } , z _ { i } - z _ { j } ) } \\end{array}$ where $z _ { j }$ represents the logit for class $j$ . The form of this intervention is justified in Appendix A.7. As a control, we additionally train a model in the standard fashion using the cross-entropy loss and SGD with momentum and large initial learning rate. Since full-batch gradients are computationally expensive, we train on a subset of CIFAR10 (see Appendix A.7, where we also give the hyperparameters and learning curves.). The output margin obtained with the intervention is 5 orders of magnitude smaller than in the control experiment: $2 . 0 \\times 1 0 ^ { - 4 } \\pm 2 . 0 \\times 1 0 ^ { - 4 }$ for the 0-Hinge loss and $1 . { \\overline { { 6 } } } \\times 1 0 ^ { 1 } \\pm 0 . 5 0 \\times 1 0 ^ { 1 }$ for cross-entropy loss. Figure 6 (right) compares the accuracies of the $\\mathbf { k }$ -NN probes resulting from these training approaches. The 0-Hinge loss training achieves only a marginal improvement in accuracy (red) over an untrained network (purple), and the training split is accurately clustered only in the latest layers. This confirms the predicted behavior: the intervention leads to a model that exhibits both very small average output margins and very late clustering of the data. Very late clustering of the data implies high prediction depths since the $\\mathbf { k }$ -NN probe classifications change in the latest layers for many data points. ", + "bbox": [ + 173, + 492, + 826, + 714 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 Beyond a One-Dimensional Picture of Example Difficulty ", + "text_level": 1, + "bbox": [ + 174, + 728, + 683, + 746 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section we transcend the one-dimensional picture of example difficulty by identifying different underlying reasons behind the difficulty of an example, in a way that is general to different architectures and datasets. ", + "bbox": [ + 178, + 755, + 823, + 796 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 7 shows that the prediction depth can be different when an input occurs in the training split vs. the validation split. Thus, there are two axes of example difficulty: ", + "bbox": [ + 173, + 797, + 821, + 825 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "1. Difficulty of making a prediction when an input is in the validation set 2. Difficulty of finding commonalities during training with other examples of the same ground truth class ", + "bbox": [ + 212, + 832, + 825, + 877 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Both axes have a range from “clear” to “ambiguous”. In Section 3.1 we show that predictions made for validation points with later prediction depths are often inconsistent, with low consensusconsistency. Conversely, a low prediction depth typically indicates an input with high consensusconsistency. For Axis 1 we will identify validation points with low prediction depths as “clear” and those with high prediction depths as “ambiguous”. We will additionally identify a low or high prediction depth in the training split with examples that are respectively “clear” and “ambiguous” on Axis 2. By making combinations of low/high values of $( \\mathrm { P D } _ { \\mathrm { V a l . } }$ , $\\mathrm { P D } _ { \\mathrm { T r a i n } _ { . } }$ ) we obtain four extremes of example difficulty: ", + "bbox": [ + 173, + 883, + 821, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/14930e4f6df689d31626c023acd5f7f776fdcf3f0a5998ae876b3c3fa70cabe4.jpg", + "image_caption": [ + "Figure 8: Average k-NN probe confidence (solid lines) and accuracy (dotted lines) for the ground truth class (left) and consensus class (right), in the validation split for examples exhibiting extreme forms of difficulty. Mean values for 100 examples with each form of difficulty, identified as the 100 examples closest to the corners in Figure 7 (left). This result is for CIFAR10 with ResNet18: similar plots for all datasets and architectures are shown in Appendix C.7. See Section 4 for the discussion of the result and how it can be used to improve prediction accuracy. " + ], + "image_footnote": [], + "bbox": [ + 174, + 89, + 818, + 207 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 304, + 826, + 388 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Easy examples: (Low $\\mathrm { P D } _ { \\mathrm { V a l . } }$ , Low $\\mathrm { P D } _ { \\mathrm { T r a i n } }$ ). Such examples are often visually typical members of their class and the predicted label nearly always matches the ground truth. \nLooks like a different class: (Low $\\mathrm { P D } _ { \\mathrm { V a l . } }$ , High $\\mathrm { P D } _ { \\mathrm { T r a i n } }$ ). In the validation set, there is a clear (and nearly always incorrect) classification for such an input, but it is difficult to connect such inputs to other examples of their ground truth class during training. Mislabeled examples are of this kind, as are visually confusing images which at first appear to show something else. \nAmbiguous unless the label is given: (High $\\mathrm { P D } _ { \\mathrm { V a l . } }$ , Low $\\mathrm { P D } _ { \\mathrm { T r a i n } }$ ). These examples are difficult to connect to their predicted class in the validation split but easy to connect to their ground truth class during training. These points may, for example, visually resemble both their own class and another class. They are likely to be misclassified. \nAmbiguous: (High $\\mathrm { P D } _ { \\mathrm { V a l . } }$ , High $\\mathrm { P D } _ { \\mathrm { T r a i n } }$ ). These examples may be corrupted or show an example of a rare sub-class. Predictions for these inputs can depend strongly on the random seed used for training and initialization. ", + "bbox": [ + 173, + 392, + 826, + 587 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Figure 7 we visualize CIFAR10 “Bird” images with the extreme forms of example difficulty for ResNet18, as identified using the prediction depth in the training and validation splits. In the full dataset (left panel) we see that the prediction depth can be very different in the training and validation splits: the two prediction depths are typically similar for points where the consensus class is equal to the ground truth (right panel), but can be very different when the consensus class is different from the ground truth (middle panel). This behavior is consistently reproduced for all datasets and architectures in Appendix C.6. ", + "bbox": [ + 174, + 592, + 825, + 688 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Looking at these examples of the class “Bird” with different difficulty types, we observe that ResNet18 finds small garden birds easiest, while birds in flight against a blue background “look like airplanes”, ostriches are “ambiguous without their label” and the “ambiguous” examples are either unclear photographs or examples of rare sub-groups that don’t appear frequently in the data. We found the consensus-consistency of inputs that are “Ambiguous” or “Ambiguous without its label” to be significantly lower than those of examples that are “Easy” or “Look like a different class”. ", + "bbox": [ + 174, + 689, + 825, + 772 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In order to better understand how networks process examples with different, extreme forms of example difficulty, Fig. 8 examines how the k-NN confidence (fraction of votes) and accuracy of the ground truth class and of the consensus class progress, as validation points pass through the network. “Easy” examples are classified as their consensus class (which is equal to their ground truth class) in all k-NN probes and the confidence in the consensus class steadily increases as data points proceed through the hidden layers. Examples that “look like a different class” are also processed as members of their consensus class, similarly to “easy” examples. However, unlike “easy” examples, their consensus classes do not match their ground truth classes. Examples that are “ambiguous without their labels” are initially processed as members of their ground truth classes with intermediate confidence, but in later layers become mistaken for their consensus class. “Ambiguous” examples are processed with low confidence and accuracy in the early layers, for both ground truth and consensus classes. In later layers “ambiguous” examples are recognized, with intermediate confidence and accuracy, as members of the consensus class, which matches the ground truth class for a sizeable fraction of “ambiguous” examples. ", + "bbox": [ + 174, + 773, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 147 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Improving the prediction accuracy Can the prediction accuracy be improved using our understanding of how each class of difficult examples are processed by deep models? Figure 8 suggest that $\\mathbf { k }$ -NN probes in intermediate layers may be more accurate than the full deep model for examples that are “ambiguous without their label” (data points closest to the lower right corner of Figure 7). In order to test this hypothesis, we compare the accuracy of the $\\mathbf { k }$ -NN probe in layer 4 to the full model’s prediction for the 100 examples closest to the lower right corner of Figure 8. We obtain a striking improvement in accuracy from $2 5 \\%$ to $98 \\%$ for these examples. This showcases how insights from this study can be directly used to improve prediction accuracy. ", + "bbox": [ + 174, + 156, + 825, + 267 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Discussion ", + "text_level": 1, + "bbox": [ + 173, + 280, + 294, + 296 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Summary We have introduced a notion of example difficulty called the prediction depth, which uses the processing of data inside the network to score the difficulty of an example. We have shown how the prediction depth is related to the accuracy and uncertainty of a prediction, the adversarial input margin and the output margin of the learned solution, and that data points that are easier according to the prediction depth are also typically learned earlier in training. We have also shown that the difficulty of an example can be both similar, or very different depending on whether an input appears in the validation split or the training split, and described four extremes of example difficulty. For data points that are “ambiguous without their label”, we have demonstrated how returning the $\\mathbf { k }$ -NN prediction in a middle layer can lead to impressive increases in model accuracy: for CIFAR10 in ResNet18 we obtained an increase in accuracy from $2 5 \\%$ to $98 \\%$ for the inputs that are most “ambiguous without their label”. ", + "bbox": [ + 173, + 305, + 825, + 457 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Connecting known phenomena In the literature, the following phenomena are separately reported from different experimental paradigms: ", + "bbox": [ + 176, + 467, + 825, + 496 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "1. Early layers generalize while later layers memorize (Stephenson et al., 2021). \n2. Model layers converge from input layer towards output layer (Raghu et al., 2017; Morcos et al., 2018). \n3. Deep models learn easy data (Jiang et al., 2021; Toneva et al., 2019) and simple functions first (Hu et al., 2020; Kalimeris et al., 2019). ", + "bbox": [ + 212, + 498, + 825, + 574 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Following this paper, a coherent and closely related picture emerges: ", + "bbox": [ + 184, + 579, + 624, + 593 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "1. Predictions made in early layers are more likely to be consistent than those made in later layers. Consistent predictions are likely to be correct and the expected accuracy of inconsistent predictions is naturally low (Section 3.1). \n2. Data points learned early in training typically have smaller prediction depths than those learned later during training (Section 3.2). \n3. On average, deep neural networks exhibit wider input and output margins (common measures of “local simplicity”) in the vicinity of data with smaller prediction depths (Section 3.3). ", + "bbox": [ + 212, + 598, + 825, + 700 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Pertinence of example difficulty to topics in machine learning Curriculum Learning attempts to treat hard examples differently from easy examples during training. Robustness to distribution shifts that change the relative frequencies of common and rare subgroups in the test set (which we have shown can have different forms of example difficulty) is important for ML Fairness. Methods developed to address heteroscedastic uncertainty typically address example difficulty as a onedimensional quantity. We expand upon the relevance of our work to these three topics in Appendix D. ", + "bbox": [ + 174, + 708, + 825, + 791 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Limitations We believe that the results we report stem from a deep model’s representation, which is hierarchical by construction. We expect that the same results will therefore apply in larger models, larger datasets, and tasks other than image classification, but testing this remains as further work. Although we demonstrate that returning the results of a hidden k-NN can yield dramatic increases in accuracy for examples that are “ambiguous without their label”, we otherwise do not explore ways to practically apply the insights we present. In particular, we expressly do not claim that all that is required for good accuracy is to reduce the prediction depth: freezing later layers of the network would not be expected to result in good generalization. ", + "bbox": [ + 174, + 800, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Funding Transparency Statement ", + "text_level": 1, + "bbox": [ + 174, + 89, + 455, + 107 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "This research was funded by, and undertaken at, Google. All calculations were performed using Google’s computer infrastructure. ", + "bbox": [ + 173, + 114, + 823, + 142 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Acknowledgment ", + "text_level": 1, + "bbox": [ + 174, + 156, + 321, + 172 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We would like to thank Hanie Sedghi, Ilya Tolstikhin, Ibrahim Alabdulmohsin, Daniel Keysers and Julian Eisenschlos for valuable discussions on the topic and Arthur Baldock for proofreading the manuscript. ", + "bbox": [ + 174, + 181, + 825, + 223 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 237, + 266, + 253 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Agarwal, C. and Hooker, S. (2020). Estimating example difficulty using variance of gradients. In ICML, Workshop on Human Interpretability in Machine Learning (WHI). \nAlain, G. and Bengio, Y. (2017). 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Our extensive investigation reveals surprising yet simple relation-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 365, + 471, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 471, + 378 + ], + "score": 1.0, + "content": "ships between the prediction depth of a given input and the model’s uncertainty,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 377, + 471, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 471, + 388 + ], + "score": 1.0, + "content": "confidence, accuracy and speed of learning for that data point. We further cate-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 388, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 399 + ], + "score": 1.0, + "content": "gorize difficult examples into three interpretable groups, demonstrate how these", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 399, + 471, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 471, + 410 + ], + "score": 1.0, + "content": "groups are processed differently inside deep models and showcase how this under-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 410, + 470, + 421 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 470, + 421 + ], + "score": 1.0, + "content": "standing allows us to improve prediction accuracy. Insights from our study lead to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 420, + 470, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 420, + 470, + 432 + ], + "score": 1.0, + "content": "a coherent view of a number of separately reported phenomena in the literature:", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 432, + 470, + 443 + ], + "spans": [ + { + "bbox": [ + 142, + 432, + 470, + 443 + ], + "score": 1.0, + "content": "early layers generalize while later layers memorize; early layers converge faster", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 442, + 366, + 454 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 366, + 454 + ], + "score": 1.0, + "content": "and networks learn easy data and simple functions first.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18, + "bbox_fs": [ + 140, + 311, + 471, + 454 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 468, + 190, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 192, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 192, + 483 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 506, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "score": 1.0, + "content": "Much of the existing work on understanding deep learning “integrates out” the data, viewing the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "inductive bias of the model, or the properties of the optimizer as central to the success of the approach.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "Examples of such work include studies of eigenvalues of the Hessian and the geometry of the loss", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "landscape (Ghorbani et al., 2019; Yao et al., 2020; Sagun et al., 2016; Li et al., 2018; Pennington", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "and Bahri, 2017; Sagun et al., 2018), studies of margin and effective generalization measures (Long", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "and Sedghi, 2019; Unterthiner et al., 2020; Jiang et al., 2020, 2018; Kawaguchi et al., 2017) and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 553, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 507, + 567 + ], + "score": 1.0, + "content": "mean-field studies of stochastic optimization (Smith et al., 2021; Stephan et al., 2017; Smith and Le,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 564, + 493, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 493, + 577 + ], + "score": 1.0, + "content": "2018). However, in practice, we are rarely concerned with only the average behavior of a model.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 487, + 507, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 506, + 685 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "One pathway to understanding the principles that govern how deep models process data is to study the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "properties of deep models for data points with different “amounts” or “types” of example difficulty.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 597, + 507, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 507, + 611 + ], + "score": 1.0, + "content": "There are a number of definitions of example difficulty in the literature (E.g. see Carlini et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "(2019); Hooker et al. (2019); Lalor et al. (2018); Agarwal and Hooker (2020)). Two are particularly", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "relevant to this work. Firstly, the probability of predicting the ground truth label for an example,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 632, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 643 + ], + "score": 1.0, + "content": "when that example is omitted from the training set (Jiang et al., 2021), which represents a statistical", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "view of example difficulty. Secondly, the difficulty of learning an example, parameterized by the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "earliest training iteration after which the model predicts the ground truth class for that example in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 662, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 677 + ], + "score": 1.0, + "content": "all subsequent iterations (Toneva et al., 2019). This measure represents a learning view of example", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 673, + 155, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 155, + 688 + ], + "score": 1.0, + "content": "difficulty 2.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 576, + 507, + 688 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "These notions suffer from two fundamental limitations. While early-exit strategies in computer", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "vision (Teerapittayanon et al., 2016; Huang et al., 2018) and NLP (Dehghani et al., 2018; Liu et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "2020b; Schwartz et al., 2020; Xin et al., 2020) suggest predictions for easier examples require less", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "computation, the above example difficulty notions do not encapsulate the processing of data inside a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "given converged model. Moreover, existing notions of example difficulty (E.g. Carlini et al. (2019))", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "provide a one-dimensional view of difficulty which can not distinguish between examples that are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 226, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 226, + 150 + ], + "score": 1.0, + "content": "difficult for different reasons.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "score": 1.0, + "content": "In this paper, we take a significant step towards resolving the above shortcomings. To take the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "score": 1.0, + "content": "processing of the data into account we propose a new measure of example difficulty, the prediction", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 172, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 506, + 184 + ], + "score": 1.0, + "content": "depth, which is determined from the hidden embeddings. To escape the one-dimensional view of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "difficulty, we introduce three distinct difficulty types by relating the hidden embeddings for an input to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 191, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 506, + 208 + ], + "score": 1.0, + "content": "high-level concepts about example difficulty: “Does this example look mislabeled?”; “Is classifying", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "this example only easy if the label is given?”; “Is this example ambiguous both with and without", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "its label?”. Furthermore, we show how this enhanced notion of example difficulty can unify our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "understanding of several seemingly unrelated phenomena in deep learning. We hope that the results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 236, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 506, + 251 + ], + "score": 1.0, + "content": "presented in this work will aid the development of models that capture heteroscedastic uncertainty,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "our understanding of how deep networks respond to distributional shift, and the advancement of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "curriculum learning approaches and machine learning fairness. These connections are discussed in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 270, + 149, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 149, + 281 + ], + "score": 1.0, + "content": "Section 5.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 108, + 289, + 331, + 300 + ], + "lines": [ + { + "bbox": [ + 107, + 288, + 332, + 301 + ], + "spans": [ + { + "bbox": [ + 107, + 288, + 332, + 301 + ], + "score": 1.0, + "content": "Contributions Our main contributions are as follows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 133, + 305, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 132, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 132, + 303, + 506, + 318 + ], + "score": 1.0, + "content": "• We introduce a measure of computational example difficulty: the prediction depth (PD). The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 142, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "prediction depth, illustrated in Figure 1, represents the number of hidden layers after which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 327, + 450, + 339 + ], + "spans": [ + { + "bbox": [ + 142, + 327, + 450, + 339 + ], + "score": 1.0, + "content": "the network’s final prediction is already (effectively) determined (Section 2).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 138, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 138, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "We show that the prediction depth is larger for examples that visually appear to be more", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 141, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "difficult, and that prediction depth is consistent between architectures and random seeds", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 363, + 198, + 374 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 198, + 374 + ], + "score": 1.0, + "content": "(Section 2.2).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 132, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 132, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "• Our empirical investigation reveals that prediction depth appears to establish a linear lower", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 140, + 385, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 140, + 385, + 506, + 401 + ], + "score": 1.0, + "content": "bound on the consistency of a prediction. We further show that predictions are on average", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "score": 1.0, + "content": "more accurate for validation points with small prediction depths (Section 3.1).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 138, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 138, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "We demonstrate that final predictions for data points that converge earlier during training", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "are typically determined in earlier layers which establishes a correspondence between the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "training history of the network and the processing of data in the hidden layers (Section 3.2).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 132, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 132, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "• We show that both the adversarial input margin and the output margin are larger for examples", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "with smaller prediction depths. 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All models were trained using", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 645, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 656 + ], + "score": 1.0, + "content": "SGD with momentum. Our MLP comprises 7 hidden layers of width 2048 with ReLU activations.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 655, + 479, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 479, + 668 + ], + "score": 1.0, + "content": "Details of the datasets, architectures, and hyperparameters used can be found in Appendix A.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Related Work: Our work uses hidden layer probes to determine example difficulty. We have", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "discussed how our study relates to prior work on example difficulty. Hidden layer probes have also", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "been used to study deep learning. Deep k-NN methods (Papernot and McDaniel, 2018) determine", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "their predictions and estimate their own uncertainties by comparing the hidden embeddings of an", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 390, + 723 + ], + "score": 1.0, + "content": "input to those of the training set. Cohen et al. (2018) showed that SVM,", + "type": "text" + }, + { + "bbox": [ + 390, + 711, + 397, + 721 + ], + "score": 0.27, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 711, + 477, + 723 + ], + "score": 1.0, + "content": "-Nearest Neighbors", + "type": "text" + }, + { + "bbox": [ + 478, + 712, + 484, + 721 + ], + "score": 0.3, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "-NN)", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "These notions suffer from two fundamental limitations. While early-exit strategies in computer", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "vision (Teerapittayanon et al., 2016; Huang et al., 2018) and NLP (Dehghani et al., 2018; Liu et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "2020b; Schwartz et al., 2020; Xin et al., 2020) suggest predictions for easier examples require less", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "computation, the above example difficulty notions do not encapsulate the processing of data inside a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "given converged model. Moreover, existing notions of example difficulty (E.g. Carlini et al. (2019))", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "provide a one-dimensional view of difficulty which can not distinguish between examples that are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 226, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 226, + 150 + ], + "score": 1.0, + "content": "difficult for different reasons.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 72, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "score": 1.0, + "content": "In this paper, we take a significant step towards resolving the above shortcomings. To take the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "score": 1.0, + "content": "processing of the data into account we propose a new measure of example difficulty, the prediction", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 172, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 506, + 184 + ], + "score": 1.0, + "content": "depth, which is determined from the hidden embeddings. To escape the one-dimensional view of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "difficulty, we introduce three distinct difficulty types by relating the hidden embeddings for an input to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 191, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 506, + 208 + ], + "score": 1.0, + "content": "high-level concepts about example difficulty: “Does this example look mislabeled?”; “Is classifying", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "this example only easy if the label is given?”; “Is this example ambiguous both with and without", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "its label?”. Furthermore, we show how this enhanced notion of example difficulty can unify our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "understanding of several seemingly unrelated phenomena in deep learning. We hope that the results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 236, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 506, + 251 + ], + "score": 1.0, + "content": "presented in this work will aid the development of models that capture heteroscedastic uncertainty,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "our understanding of how deep networks respond to distributional shift, and the advancement of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "curriculum learning approaches and machine learning fairness. These connections are discussed in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 270, + 149, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 149, + 281 + ], + "score": 1.0, + "content": "Section 5.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 150, + 506, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 289, + 331, + 300 + ], + "lines": [ + { + "bbox": [ + 107, + 288, + 332, + 301 + ], + "spans": [ + { + "bbox": [ + 107, + 288, + 332, + 301 + ], + "score": 1.0, + "content": "Contributions Our main contributions are as follows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 107, + 288, + 332, + 301 + ] + }, + { + "type": "list", + "bbox": [ + 133, + 305, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 132, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 132, + 303, + 506, + 318 + ], + "score": 1.0, + "content": "• We introduce a measure of computational example difficulty: the prediction depth (PD). The", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 142, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "prediction depth, illustrated in Figure 1, represents the number of hidden layers after which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 327, + 450, + 339 + ], + "spans": [ + { + "bbox": [ + 142, + 327, + 450, + 339 + ], + "score": 1.0, + "content": "the network’s final prediction is already (effectively) determined (Section 2).", + "type": "text" + } + ], + "index": 22, + "is_list_end_line": true + }, + { + "bbox": [ + 138, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 138, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "We show that the prediction depth is larger for examples that visually appear to be more", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 141, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "difficult, and that prediction depth is consistent between architectures and random seeds", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 363, + 198, + 374 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 198, + 374 + ], + "score": 1.0, + "content": "(Section 2.2).", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 132, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "• Our empirical investigation reveals that prediction depth appears to establish a linear lower", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 140, + 385, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 140, + 385, + 506, + 401 + ], + "score": 1.0, + "content": "bound on the consistency of a prediction. We further show that predictions are on average", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 456, + 411 + ], + "score": 1.0, + "content": "more accurate for validation points with small prediction depths (Section 3.1).", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + }, + { + "bbox": [ + 138, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 138, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "We demonstrate that final predictions for data points that converge earlier during training", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "are typically determined in earlier layers which establishes a correspondence between the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "training history of the network and the processing of data in the hidden layers (Section 3.2).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 132, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 132, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "• We show that both the adversarial input margin and the output margin are larger for examples", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "with smaller prediction depths. We further design an intervention to reduce the output margin", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 141, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "of a network and show that this leads to predictions being made only in the latest hidden", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 480, + 225, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 480, + 225, + 492 + ], + "score": 1.0, + "content": "layers (Section 3.3).", + "type": "text" + } + ], + "index": 35, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 139, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "We identify three extreme forms of example difficulty by considering the prediction depth in", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 141, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "the training and validation splits independently and demonstrate how a simple algorithm that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 142, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "uses the hidden embeddings in one middle layer to make predictions can lead to dramatic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 141, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "improvements in accuracy for inputs that strongly exhibit a specific form of example", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 538, + 229, + 550 + ], + "spans": [ + { + "bbox": [ + 141, + 538, + 229, + 550 + ], + "score": 1.0, + "content": "difficulty (Section 4).", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 138, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 138, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "We use our results to present a coherent picture of deep learning that unifies four seemingly", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 142, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "unrelated deep learning phenomena: early layers generalize while later layers memorize;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 141, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "networks converge from input layer towards output layer; easy examples are learned first", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 142, + 585, + 422, + 597 + ], + "spans": [ + { + "bbox": [ + 142, + 585, + 422, + 597 + ], + "score": 1.0, + "content": "and networks present simpler functions earlier in training (Section 5).", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + } + ], + "index": 32, + "bbox_fs": [ + 132, + 303, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "Experimental Setup: To ensure that our results are robust to the choice of architectures and datasets,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "we report empirical findings for ResNet18 (He et al., 2016), VGG16 (Simonyan and Zisserman, 2015)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "and MLP architectures trained on CIFAR10, CIFAR100 (Krizhevsky et al., 2009), Fashion MNIST", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "(FMNIST) (Xiao et al., 2017) and SVHN (Netzer et al., 2011) datasets. 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Our MLP comprises 7 hidden layers of width 2048 with ReLU activations.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 655, + 479, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 479, + 668 + ], + "score": 1.0, + "content": "Details of the datasets, architectures, and hyperparameters used can be found in Appendix A.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 600, + 506, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Related Work: Our work uses hidden layer probes to determine example difficulty. We have", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "discussed how our study relates to prior work on example difficulty. Hidden layer probes have also", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "been used to study deep learning. Deep k-NN methods (Papernot and McDaniel, 2018) determine", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "their predictions and estimate their own uncertainties by comparing the hidden embeddings of an", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 390, + 723 + ], + "score": 1.0, + "content": "input to those of the training set. Cohen et al. (2018) showed that SVM,", + "type": "text" + }, + { + "bbox": [ + 390, + 711, + 397, + 721 + ], + "score": 0.27, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 711, + 477, + 723 + ], + "score": 1.0, + "content": "-Nearest Neighbors", + "type": "text" + }, + { + "bbox": [ + 478, + 712, + 484, + 721 + ], + "score": 0.3, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "-NN)", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53, + "bbox_fs": [ + 105, + 667, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "and logistic regression probes achieve similar accuracies. 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Alain", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "and Bengio (2017) used linear classifier probes in the hidden layers to interrogate deep models and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "demonstrated that linear separability of the embeddings increases monotonically with depth. 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This leads to two notions of computational difficulty:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 367, + 506, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 134, + 395, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 133, + 394, + 453, + 408 + ], + "spans": [ + { + "bbox": [ + 133, + 394, + 453, + 408 + ], + "score": 1.0, + "content": "• The difficulty of predicting the (given) class for an input (in the training split)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 132, + 406, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 132, + 406, + 506, + 424 + ], + "score": 1.0, + "content": "• The difficulty of making a prediction for an input, unseen in advance (from the validation", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 140, + 419, + 167, + 433 + ], + "spans": [ + { + "bbox": [ + 140, + 419, + 167, + 433 + ], + "score": 1.0, + "content": "split)", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 132, + 394, + 506, + 433 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "We examine both notions of computational difficulty in this paper and use the distinction between", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 447, + 375, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 375, + 459 + ], + "score": 1.0, + "content": "them to describe different forms of example difficulty in Section 4.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 434, + 505, + 459 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 437, + 478 + ], + "lines": [ + { + "bbox": [ + 104, + 464, + 437, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 437, + 482 + ], + "score": 1.0, + "content": "2.2 Prediction depth is a meaningful and robust notion of example difficulty", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 482, + 504, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "In this section we show that prediction depth agrees with intuitive notions of example difficulty and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 411, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 411, + 505 + ], + "score": 1.0, + "content": "that it is consistent between different training runs and similar architectures.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 482, + 505, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 511, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "Prediction depth is higher for examples and datasets that seem more difficult If prediction", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "depth is a sensible measure of example difficulty then we would expect the following sanity checks", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 533, + 169, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 169, + 545 + ], + "score": 1.0, + "content": "to be observed:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 511, + 505, + 545 + ] + }, + { + "type": "list", + "bbox": [ + 130, + 549, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 130, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 130, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "1. Individual data points that are visually confusing or mislabeled should have larger prediction", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 559, + 415, + 572 + ], + "spans": [ + { + "bbox": [ + 141, + 559, + 415, + 572 + ], + "score": 1.0, + "content": "depths as compared to images that are clear examples of their class.", + "type": "text" + } + ], + "index": 37, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 128, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "2. 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Appendix C.1 presents", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 610, + 348, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 348, + 622 + ], + "score": 1.0, + "content": "additional images, providing further evidence for this claim.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 598, + 505, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "Prediction depth is consistent across random seeds and similar architectures Figure 2 shows", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 640, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 651 + ], + "score": 1.0, + "content": "that the prediction depth is highly consistent between different architectures and random seeds for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "all datasets. Perfect agreement is not expected as different deep learning algorithms have different", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "inductive biases which affects the perceived difficulty of examples. We observe stronger correlation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 505, + 504, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 504, + 517 + ], + "score": 1.0, + "content": "between prediction depth for ResNet18 and VGG16, than between VGG16 and MLP. 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Left: A cartoon illustrating the definition of prediction depth (given in Section 2.1).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "score": 1.0, + "content": "Also shown are training examples from CIFAR100 (“Clock”) and SVHN (“Digit 8”). The examples shown are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 218, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 230 + ], + "score": 1.0, + "content": "predicted at the input (first layer) or softmax (last layer) of ResNet18. 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Right: Probability of prediction depth in ResNet18", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "models for four datasets (training split). We see that the four distributions have different characteristic prediction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "score": 1.0, + "content": "depths. Ranking the mean prediction depths of these datasets in ascending order, we observe: Fashion MNIST", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "score": 1.0, + "content": "(smallest), SVHN (second), CIFAR10 (third), and CIFAR100 (largest). This order aligns with how one might", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 298, + 319, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 319, + 309 + ], + "score": 1.0, + "content": "intuitively rank the difficulties of these classification tasks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + } + ], + "index": 4.75 + }, + { + "type": "image", + "bbox": [ + 109, + 319, + 502, + 417 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 319, + 502, + 417 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 319, + 502, + 417 + ], + "spans": [ + { + "bbox": [ + 109, + 319, + 502, + 417 + ], + "score": 0.961, + "type": "image", + "image_path": "23c98a880da3f394cd13a1b9baa89a6f129ff8653082737c21aec2cb3c228e0f.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 109, + 319, + 502, + 351.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 109, + 351.6666666666667, + 502, + 384.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 109, + 384.33333333333337, + 502, + 417.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 423, + 505, + 494 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "Figure 2: Consistency of prediction depth between architectures and random seeds. Left: The panel shows the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 433, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 504, + 444 + ], + "score": 1.0, + "content": "correlation coefficient between prediction depths in different architectures, for both train and validation splits", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "in four datasets. Diagonal comparisons between an architecture and itself show the correlation for the same", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "architecture trained with different random seeds. Right: Histograms comparing the mean value of prediction", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "depth obtained for each data point in the training set of CIFAR10 from an ensemble of 250 trained models. In", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 366, + 484 + ], + "score": 1.0, + "content": "this plot, for visual simplicity, we rescale prediction depth to the interval", + "type": "text" + }, + { + "bbox": [ + 366, + 473, + 385, + 484 + ], + "score": 0.26, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "for each network. 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This may be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "explained by the fact that ResNet18 and VGG16 are both convolutional networks and we expect their", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 527, + 361, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 361, + 538 + ], + "score": 1.0, + "content": "inductive biases to be more similar to one another than to MLP.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 106, + 549, + 462, + 563 + ], + "lines": [ + { + "bbox": [ + 103, + 548, + 464, + 566 + ], + "spans": [ + { + "bbox": [ + 103, + 548, + 464, + 566 + ], + "score": 1.0, + "content": "3 Deep Learning Phenomena Through the Lens of Prediction Depth", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "In this section, we explore how the prediction depth can be used to better understand three important", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "aspects of Deep Learning: accuracy and consistency of a prediction; the order in which data is learned", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "and the simplicity of the learned function (as measured by the margin) in the vicinity of a data point.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 611, + 411, + 623 + ], + "lines": [ + { + "bbox": [ + 104, + 610, + 412, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 412, + 626 + ], + "score": 1.0, + "content": "3.1 Depth of a prediction gives a linear lower bound on its consistency", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "Adopting a statistical view of example difficulty, Jiang et al. (2021) identified example difficulty with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "the expected accuracy of the learning algorithm for a given input, averaged over models trained on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "different random subsets of the training set with different random seeds. 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Left: A cartoon illustrating the definition of prediction depth (given in Section 2.1).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "score": 1.0, + "content": "Also shown are training examples from CIFAR100 (“Clock”) and SVHN (“Digit 8”). The examples shown are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 218, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 230 + ], + "score": 1.0, + "content": "predicted at the input (first layer) or softmax (last layer) of ResNet18. 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To find the prediction depth, we build", + "type": "text" + }, + { + "bbox": [ + 288, + 239, + 294, + 247 + ], + "score": 0.29, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "-NN classifiers from the embeddings of the training set in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 248, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 259 + ], + "score": 1.0, + "content": "different layers of the model. The prediction depth corresponds to the earliest layer at which the predictions of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "all subsequent k-NN classifiers converge to a fixed label. Right: Probability of prediction depth in ResNet18", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "models for four datasets (training split). We see that the four distributions have different characteristic prediction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "score": 1.0, + "content": "depths. Ranking the mean prediction depths of these datasets in ascending order, we observe: Fashion MNIST", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "score": 1.0, + "content": "(smallest), SVHN (second), CIFAR10 (third), and CIFAR100 (largest). This order aligns with how one might", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 298, + 319, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 319, + 309 + ], + "score": 1.0, + "content": "intuitively rank the difficulties of these classification tasks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + } + ], + "index": 4.75 + }, + { + "type": "image", + "bbox": [ + 109, + 319, + 502, + 417 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 319, + 502, + 417 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 319, + 502, + 417 + ], + "spans": [ + { + "bbox": [ + 109, + 319, + 502, + 417 + ], + "score": 0.961, + "type": "image", + "image_path": "23c98a880da3f394cd13a1b9baa89a6f129ff8653082737c21aec2cb3c228e0f.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 109, + 319, + 502, + 351.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 109, + 351.6666666666667, + 502, + 384.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 109, + 384.33333333333337, + 502, + 417.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 423, + 505, + 494 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "Figure 2: Consistency of prediction depth between architectures and random seeds. Left: The panel shows the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 433, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 504, + 444 + ], + "score": 1.0, + "content": "correlation coefficient between prediction depths in different architectures, for both train and validation splits", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "in four datasets. Diagonal comparisons between an architecture and itself show the correlation for the same", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "architecture trained with different random seeds. Right: Histograms comparing the mean value of prediction", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "depth obtained for each data point in the training set of CIFAR10 from an ensemble of 250 trained models. In", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 366, + 484 + ], + "score": 1.0, + "content": "this plot, for visual simplicity, we rescale prediction depth to the interval", + "type": "text" + }, + { + "bbox": [ + 366, + 473, + 385, + 484 + ], + "score": 0.26, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "for each network. 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Interestingly, Figure 4 (middle and right)", + "type": "text", + "cross_page": true + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "score": 1.0, + "content": "shows how the prediction depth in a single model, can be used to estimate both of these quantities.", + "type": "text", + "cross_page": true + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "That is, predictions of data points with lower prediction depth are both more likely to be consistent", + "type": "text", + "cross_page": true + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 226, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 226, + 430 + ], + "score": 1.0, + "content": "and more likely to be correct.", + "type": "text", + "cross_page": true + } + ], + "index": 26 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 659, + 505, + 684 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 74, + 503, + 165 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 74, + 503, + 165 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 74, + 503, + 165 + ], + "spans": [ + { + "bbox": [ + 109, + 74, + 503, + 165 + ], + "score": 0.963, + "type": "image", + "image_path": "8a5ae0a88a3e513d4097c5f1dcee15b7f3d5e2e68ceae4e1b0a5b33e9d9f4a7a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 74, + 503, + 104.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 104.33333333333333, + 503, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 134.66666666666666, + 503, + 165.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 172, + 506, + 372 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "Figure 4: Left: Prediction depth provides us with a linear lower bound on consensus-consistency. Results", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 307, + 195 + ], + "score": 1.0, + "content": "for CIFAR100 with ResNet18. We train 250 models", + "type": "text" + }, + { + "bbox": [ + 307, + 183, + 339, + 192 + ], + "score": 0.85, + "content": "( 9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "random train:validation splits) and compare", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "the average prediction depth when a point occurs in the validation set, to the consensus-consistency of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 203, + 507, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 203, + 507, + 214 + ], + "score": 1.0, + "content": "corresponding predictions. Predictions made for points with low mean prediction depths are highly consistent.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "Conversely, predictions for points with high mean prediction depths are typically more sensitive to the particular", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 222, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 507, + 234 + ], + "score": 1.0, + "content": "training split and random seed used during training. This left plot shows the result for CIFAR100 with ResNet18.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "The density of data points is indicated by the color bar, which follows a log scale. Middle: Prediction depth", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "in one model predicts the consensus-consistency of an ensemble that does not include that model. For each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "dataset we train 25 ResNet18 models with the full training set (see Appendix A). The consensus-consistency of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "each test point is obtained from 24 of the models, while the prediction depth is obtained from the remaining 1", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "model. We see that prediction depth in one model predicts the consensus-consistency of a separate ensemble: a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 294 + ], + "score": 1.0, + "content": "measure of the uncertainty of the prediction. The size of each marker in the middle and right plots shows the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "fraction of the dataset with each prediction depth. Reaffirming the second sanity check in Section 2.2, and in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "agreement with Figure 1 (right), intuitively simpler datasets (Fashion MNIST and SVHN) have low average", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "prediction depths, while CIFAR100 (intuitively the hardest dataset) has the largest average prediction depth.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 443, + 335 + ], + "score": 1.0, + "content": "Right: Prediction depth predicts accuracy. For each dataset we train 250 ResNet18 models", + "type": "text" + }, + { + "bbox": [ + 444, + 322, + 474, + 332 + ], + "score": 0.84, + "content": "9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "random", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "train:validation splits). Each time a point appears in the validation split we record the prediction depth and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 104, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "whether the prediction was correct. Predictions made in earlier layers are more likely to be correct. Consistency", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "of these plots is demonstrated for all datasets and architectures in Appendix C.3 where we also describe the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 363, + 441, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 441, + 374 + ], + "score": 1.0, + "content": "relationship between the prediction depth and the entropy of the predictions for an ensemble.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 12.5 + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 396 + ], + "score": 1.0, + "content": "consensus class matches or differs from the ground truth. Interestingly, Figure 4 (middle and right)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "score": 1.0, + "content": "shows how the prediction depth in a single model, can be used to estimate both of these quantities.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "That is, predictions of data points with lower prediction depth are both more likely to be consistent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 226, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 226, + 430 + ], + "score": 1.0, + "content": "and more likely to be correct.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 437, + 437, + 449 + ], + "lines": [ + { + "bbox": [ + 104, + 435, + 438, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 438, + 453 + ], + "score": 1.0, + "content": "3.2 The prediction depth of an input is correlated with its learning difficulty", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "In Section 3.1, we describe the relationship between the prediction depth, which represents a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "score": 1.0, + "content": "computational view of example difficulty and the consistency and consensus-consistency scores,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "which represent a statistical view. In this section we compare prediction depth to a learning view", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "of example difficulty. We measure the difficulty of learning an example by the speed at which the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "model’s prediction converges for that input during training. The following definition is adapted", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 507, + 213, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 213, + 521 + ], + "score": 1.0, + "content": "from Toneva et al. (2019):", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 456, + 536 + ], + "score": 1.0, + "content": "Iteration learned A data point is said to be learned by a classifier at training iteration", + "type": "text" + }, + { + "bbox": [ + 457, + 525, + 480, + 534 + ], + "score": 0.89, + "content": "t = \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "if the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 140, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 140, + 534, + 249, + 548 + ], + "score": 1.0, + "content": "predicted class at iteration", + "type": "text" + }, + { + "bbox": [ + 249, + 536, + 290, + 545 + ], + "score": 0.91, + "content": "t = \\tau - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "is different from the final prediction of the converged", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 141, + 546, + 322, + 558 + ], + "score": 1.0, + "content": "network and the predictions at all iterations", + "type": "text" + }, + { + "bbox": [ + 322, + 546, + 347, + 557 + ], + "score": 0.9, + "content": "t \\geq \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "are equal to the final prediction of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 555, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 141, + 555, + 506, + 570 + ], + "score": 1.0, + "content": "converged network. Data points consistently classified after all training steps and at the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 567, + 393, + 581 + ], + "spans": [ + { + "bbox": [ + 141, + 567, + 359, + 581 + ], + "score": 1.0, + "content": "moment of initialization, are said to be learned in step", + "type": "text" + }, + { + "bbox": [ + 360, + 568, + 384, + 578 + ], + "score": 0.82, + "content": "t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 567, + 393, + 581 + ], + "score": 1.0, + "content": "7 .", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "Figure 5 (left plot) shows the positive correlation between the prediction depth and the iteration", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "learned, for all four datasets in VGG16. Consistent results are presented for all architectures and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "datasets, in both the validation and training splits in Appendix C.4. As a result of the reported", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "score": 1.0, + "content": "correlation, we anticipate that many of the data points correctly classified by the k-NN probe in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "a particular layer should also be correctly classified by the network at a corresponding interval of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "training steps. If this is correct then we would expect there to be a visual correspondence between the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "score": 1.0, + "content": "training learning curve (which shows how the accuracy of the network changes during training) and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 183, + 674 + ], + "score": 1.0, + "content": "the accuracy of the", + "type": "text" + }, + { + "bbox": [ + 184, + 661, + 191, + 671 + ], + "score": 0.37, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "-NN probes as data passes from input, through the network, towards the output", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 426, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 214, + 685 + ], + "score": 1.0, + "content": "layer. We call the series of", + "type": "text" + }, + { + "bbox": [ + 214, + 672, + 221, + 681 + ], + "score": 0.49, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 671, + 426, + 685 + ], + "score": 1.0, + "content": "-NN probe accuracies the inference learning curve.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 691, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "7Note that this definition can be applied to points in both training and validation splits. In order to compare", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "different models and datasets we rescale the iteration learned in each model so that the 95th percentile occurs at", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 712, + 231, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 231, + 722 + ], + "score": 1.0, + "content": "1.0 and network initialization at 0.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 74, + 503, + 165 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 74, + 503, + 165 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 74, + 503, + 165 + ], + "spans": [ + { + "bbox": [ + 109, + 74, + 503, + 165 + ], + "score": 0.963, + "type": "image", + "image_path": "8a5ae0a88a3e513d4097c5f1dcee15b7f3d5e2e68ceae4e1b0a5b33e9d9f4a7a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 74, + 503, + 104.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 104.33333333333333, + 503, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 134.66666666666666, + 503, + 165.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 172, + 506, + 372 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "Figure 4: Left: Prediction depth provides us with a linear lower bound on consensus-consistency. Results", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 307, + 195 + ], + "score": 1.0, + "content": "for CIFAR100 with ResNet18. We train 250 models", + "type": "text" + }, + { + "bbox": [ + 307, + 183, + 339, + 192 + ], + "score": 0.85, + "content": "( 9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "random train:validation splits) and compare", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "the average prediction depth when a point occurs in the validation set, to the consensus-consistency of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 203, + 507, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 203, + 507, + 214 + ], + "score": 1.0, + "content": "corresponding predictions. Predictions made for points with low mean prediction depths are highly consistent.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "Conversely, predictions for points with high mean prediction depths are typically more sensitive to the particular", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 222, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 507, + 234 + ], + "score": 1.0, + "content": "training split and random seed used during training. This left plot shows the result for CIFAR100 with ResNet18.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "The density of data points is indicated by the color bar, which follows a log scale. Middle: Prediction depth", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "in one model predicts the consensus-consistency of an ensemble that does not include that model. For each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "dataset we train 25 ResNet18 models with the full training set (see Appendix A). The consensus-consistency of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "each test point is obtained from 24 of the models, while the prediction depth is obtained from the remaining 1", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "model. We see that prediction depth in one model predicts the consensus-consistency of a separate ensemble: a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 294 + ], + "score": 1.0, + "content": "measure of the uncertainty of the prediction. The size of each marker in the middle and right plots shows the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "fraction of the dataset with each prediction depth. Reaffirming the second sanity check in Section 2.2, and in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "agreement with Figure 1 (right), intuitively simpler datasets (Fashion MNIST and SVHN) have low average", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "prediction depths, while CIFAR100 (intuitively the hardest dataset) has the largest average prediction depth.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 443, + 335 + ], + "score": 1.0, + "content": "Right: Prediction depth predicts accuracy. For each dataset we train 250 ResNet18 models", + "type": "text" + }, + { + "bbox": [ + 444, + 322, + 474, + 332 + ], + "score": 0.84, + "content": "9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "random", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "train:validation splits). Each time a point appears in the validation split we record the prediction depth and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 104, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "whether the prediction was correct. Predictions made in earlier layers are more likely to be correct. Consistency", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "of these plots is demonstrated for all datasets and architectures in Appendix C.3 where we also describe the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 363, + 441, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 441, + 374 + ], + "score": 1.0, + "content": "relationship between the prediction depth and the entropy of the predictions for an ensemble.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 12.5 + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 428 + ], + "lines": [], + "index": 24.5, + "bbox_fs": [ + 105, + 385, + 506, + 430 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 437, + 437, + 449 + ], + "lines": [ + { + "bbox": [ + 104, + 435, + 438, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 438, + 453 + ], + "score": 1.0, + "content": "3.2 The prediction depth of an input is correlated with its learning difficulty", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "In Section 3.1, we describe the relationship between the prediction depth, which represents a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "score": 1.0, + "content": "computational view of example difficulty and the consistency and consensus-consistency scores,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "which represent a statistical view. In this section we compare prediction depth to a learning view", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "of example difficulty. We measure the difficulty of learning an example by the speed at which the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "model’s prediction converges for that input during training. The following definition is adapted", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 507, + 213, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 213, + 521 + ], + "score": 1.0, + "content": "from Toneva et al. (2019):", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 452, + 507, + 521 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 456, + 536 + ], + "score": 1.0, + "content": "Iteration learned A data point is said to be learned by a classifier at training iteration", + "type": "text" + }, + { + "bbox": [ + 457, + 525, + 480, + 534 + ], + "score": 0.89, + "content": "t = \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "if the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 140, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 140, + 534, + 249, + 548 + ], + "score": 1.0, + "content": "predicted class at iteration", + "type": "text" + }, + { + "bbox": [ + 249, + 536, + 290, + 545 + ], + "score": 0.91, + "content": "t = \\tau - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "is different from the final prediction of the converged", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 141, + 546, + 322, + 558 + ], + "score": 1.0, + "content": "network and the predictions at all iterations", + "type": "text" + }, + { + "bbox": [ + 322, + 546, + 347, + 557 + ], + "score": 0.9, + "content": "t \\geq \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "are equal to the final prediction of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 555, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 141, + 555, + 506, + 570 + ], + "score": 1.0, + "content": "converged network. Data points consistently classified after all training steps and at the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 567, + 393, + 581 + ], + "spans": [ + { + "bbox": [ + 141, + 567, + 359, + 581 + ], + "score": 1.0, + "content": "moment of initialization, are said to be learned in step", + "type": "text" + }, + { + "bbox": [ + 360, + 568, + 384, + 578 + ], + "score": 0.82, + "content": "t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 567, + 393, + 581 + ], + "score": 1.0, + "content": "7 .", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 524, + 506, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "Figure 5 (left plot) shows the positive correlation between the prediction depth and the iteration", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "learned, for all four datasets in VGG16. Consistent results are presented for all architectures and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "datasets, in both the validation and training splits in Appendix C.4. As a result of the reported", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "score": 1.0, + "content": "correlation, we anticipate that many of the data points correctly classified by the k-NN probe in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "a particular layer should also be correctly classified by the network at a corresponding interval of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "training steps. If this is correct then we would expect there to be a visual correspondence between the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "score": 1.0, + "content": "training learning curve (which shows how the accuracy of the network changes during training) and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 183, + 674 + ], + "score": 1.0, + "content": "the accuracy of the", + "type": "text" + }, + { + "bbox": [ + 184, + 661, + 191, + 671 + ], + "score": 0.37, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "-NN probes as data passes from input, through the network, towards the output", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 426, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 214, + 685 + ], + "score": 1.0, + "content": "layer. We call the series of", + "type": "text" + }, + { + "bbox": [ + 214, + 672, + 221, + 681 + ], + "score": 0.49, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 671, + 426, + 685 + ], + "score": 1.0, + "content": "-NN probe accuracies the inference learning curve.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 584, + 506, + 685 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 72, + 497, + 147 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 72, + 497, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 72, + 497, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 72, + 497, + 147 + ], + "score": 0.967, + "type": "image", + "image_path": "a02ab6322e5cd8cebfe9ac42455d2b4efbdada80864f46deb3a0b8efc92674dd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 72, + 497, + 97.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 97.0, + 497, + 122.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 122.0, + 497, + 147.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 155, + 505, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Figure 5: Left: Data points with small prediction depths are on average learned before data points with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 390, + 176 + ], + "score": 1.0, + "content": "higher prediction depths. We train 250 VGG16 models for each dataset, using a", + "type": "text" + }, + { + "bbox": [ + 390, + 165, + 419, + 175 + ], + "score": 0.89, + "content": "9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 165, + 505, + 176 + ], + "score": 1.0, + "content": "random train:validation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "split as described in Appendix A. Each time an input appears in the validation split we record the prediction", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 185, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 506, + 196 + ], + "score": 1.0, + "content": "depth and the iteration learned in that model. This plot shows the average iteration learned for data points at", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 195, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 506, + 206 + ], + "score": 1.0, + "content": "each prediction depth. Marker size shows the fraction of the dataset with each prediction depth. The Pearson", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 216 + ], + "score": 1.0, + "content": "correlation coefficients for the four data sets are as follows. CIFAR100: 0.83. CIFAR10: 0.7. Fashion MNIST:", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "0.79. SVHN: 0.77. Middle and right: The training learning curve (middle) shares several important features", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 236 + ], + "score": 1.0, + "content": "with the inference learning curve (right). Blue, yellow and green curves represent different components of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 347, + 247 + ], + "score": 1.0, + "content": "CIFAR10 training split, in which we have randomized (and fixed)", + "type": "text" + }, + { + "bbox": [ + 347, + 235, + 365, + 245 + ], + "score": 0.87, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "of the labels, and red curves show the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "test split. The middle and right plots show results from 5 random seeds. The inference learning curve (right)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "is the sequence of k-NN probe accuracy values for each split. All three plots show results for VGG16. The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 280, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 280, + 276 + ], + "score": 1.0, + "content": "hyperparameters used are given in Appendix A.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + } + ], + "index": 4.75 + }, + { + "type": "image", + "bbox": [ + 108, + 282, + 501, + 362 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 282, + 501, + 362 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 282, + 501, + 362 + ], + "spans": [ + { + "bbox": [ + 108, + 282, + 501, + 362 + ], + "score": 0.968, + "type": "image", + "image_path": "cd8a095f06b0d86716987a5edc4dae02cc92e33acd326dbeae942c6b16d73e2a.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 108, + 282, + 501, + 308.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 108, + 308.6666666666667, + 501, + 335.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 108, + 335.33333333333337, + 501, + 362.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 369, + 505, + 449 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "Figure 6: Left and Middle: Test examples with smaller prediction depths, on average, have larger output", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "and input margins. We train 25 VGG16 models with different random seeds on CIFAR10 (see Appendix A for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "details) and compare the mean prediction depth of each test point in these 25 runs to its mean output and input", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 281, + 411 + ], + "score": 1.0, + "content": "margins (log scales). Correlation coefficients are", + "type": "text" + }, + { + "bbox": [ + 281, + 399, + 306, + 409 + ], + "score": 0.76, + "content": "- 0 . 7 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 398, + 379, + 411 + ], + "score": 1.0, + "content": "(output margin) and", + "type": "text" + }, + { + "bbox": [ + 380, + 399, + 405, + 409 + ], + "score": 0.75, + "content": "- 0 . 6 9", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "(input margin). The density", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "of data points is indicated by the color bar, which follows a log scale. Although the prediction depth could be at", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "most 14, no data point has an average prediction depth greater than 12. Right: An intervention that does not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 221, + 441 + ], + "score": 1.0, + "content": "encourage large output margin (", + "type": "text" + }, + { + "bbox": [ + 222, + 429, + 231, + 438 + ], + "score": 0.32, + "content": "^ { * } O", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "-Hinge”) results, as predicted, in models where the predictions are effectively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 419, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 388, + 450 + ], + "score": 1.0, + "content": "determined in higher layers in the network compared to the standard training", + "type": "text" + }, + { + "bbox": [ + 388, + 439, + 415, + 449 + ], + "score": 0.57, + "content": "( \\ ^ { \\ast } C E ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 438, + 419, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + } + ], + "index": 18.75 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "To test this hypothesis we train a model on a training split where a subset of labels are corrupted and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 465, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 481 + ], + "score": 1.0, + "content": "compare the training and inference learning curves on four splits of the data: unchanged training", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "data; mislabeled training data; the original labels of the mislabeled training data and the test split. In", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "Figure 5 (middle and right plots) we see that many of the important features of the training learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "curve are indeed present in the inference learning curve. During training (middle), mislabeled", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "data are initially processed as though they are a member of their original class (before they were", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "mislabeled) (Liu et al., 2020a). After an initial period of learning, the network begins to learn the new", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "(random) labels that have been assigned to those data points, so the orange curve moves upwards,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "and the green curve downwards. At this point, a maximum is observed in the training accuracy (Arpit", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 506, + 568 + ], + "score": 1.0, + "content": "et al., 2017). In the right plot we see that these same phenomena occur in the inference learning", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 566, + 134, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 134, + 577 + ], + "score": 1.0, + "content": "curve.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 108, + 589, + 446, + 601 + ], + "lines": [ + { + "bbox": [ + 104, + 588, + 447, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 447, + 603 + ], + "score": 1.0, + "content": "3.3 Deep models exhibit larger margins for inputs with lower prediction depth", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 620 + ], + "score": 1.0, + "content": "It is reported in the literature that deep networks learn functions of increasing complexity during", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "training (Hu et al., 2020; Kalimeris et al., 2019). We frame this observation differently: the learned", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "function is “locally simpler” in the vicinity of data points with smaller prediction depths, and these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 640, + 345, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 345, + 652 + ], + "score": 1.0, + "content": "points are typically learned earlier in training (Section 3.2).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 651, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "Two known measures of the simplicity of a learned function are the output margin (the difference", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "between the largest and second-largest logits) and the adversarial input margin (the smallest norm", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "required for an adversarial perturbation in the input to change the model’s class prediction). We", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 683, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 257, + 698 + ], + "score": 1.0, + "content": "estimate the adversarial input margin,", + "type": "text" + }, + { + "bbox": [ + 257, + 689, + 263, + 696 + ], + "score": 0.87, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 683, + 506, + 698 + ], + "score": 1.0, + "content": ", with a linear approximation (Jiang et al., 2018): for an input", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 696, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 305, + 713 + ], + "score": 1.0, + "content": "x with predicted class i, γ ' minj6=i |zi−zj ||∇x(zi−zj )|", + "type": "text" + }, + { + "bbox": [ + 305, + 697, + 334, + 711 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 334, + 702, + 343, + 710 + ], + "score": 0.89, + "content": "z _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 697, + 506, + 711 + ], + "score": 1.0, + "content": "is the logit returned by the network for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 128, + 723 + ], + "score": 1.0, + "content": "class", + "type": "text" + }, + { + "bbox": [ + 129, + 711, + 135, + 722 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 711, + 505, + 723 + ], + "score": 1.0, + "content": ". 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We train 250 VGG16 models for each dataset, using a", + "type": "text" + }, + { + "bbox": [ + 390, + 165, + 419, + 175 + ], + "score": 0.89, + "content": "9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 165, + 505, + 176 + ], + "score": 1.0, + "content": "random train:validation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "split as described in Appendix A. Each time an input appears in the validation split we record the prediction", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 185, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 506, + 196 + ], + "score": 1.0, + "content": "depth and the iteration learned in that model. This plot shows the average iteration learned for data points at", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 195, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 506, + 206 + ], + "score": 1.0, + "content": "each prediction depth. Marker size shows the fraction of the dataset with each prediction depth. The Pearson", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 216 + ], + "score": 1.0, + "content": "correlation coefficients for the four data sets are as follows. CIFAR100: 0.83. CIFAR10: 0.7. Fashion MNIST:", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "0.79. SVHN: 0.77. Middle and right: The training learning curve (middle) shares several important features", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 236 + ], + "score": 1.0, + "content": "with the inference learning curve (right). Blue, yellow and green curves represent different components of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 347, + 247 + ], + "score": 1.0, + "content": "CIFAR10 training split, in which we have randomized (and fixed)", + "type": "text" + }, + { + "bbox": [ + 347, + 235, + 365, + 245 + ], + "score": 0.87, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "of the labels, and red curves show the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "test split. The middle and right plots show results from 5 random seeds. The inference learning curve (right)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "is the sequence of k-NN probe accuracy values for each split. All three plots show results for VGG16. The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 280, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 280, + 276 + ], + "score": 1.0, + "content": "hyperparameters used are given in Appendix A.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + } + ], + "index": 4.75 + }, + { + "type": "image", + "bbox": [ + 108, + 282, + 501, + 362 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 282, + 501, + 362 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 282, + 501, + 362 + ], + "spans": [ + { + "bbox": [ + 108, + 282, + 501, + 362 + ], + "score": 0.968, + "type": "image", + "image_path": "cd8a095f06b0d86716987a5edc4dae02cc92e33acd326dbeae942c6b16d73e2a.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 108, + 282, + 501, + 308.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 108, + 308.6666666666667, + 501, + 335.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 108, + 335.33333333333337, + 501, + 362.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 369, + 505, + 449 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "Figure 6: Left and Middle: Test examples with smaller prediction depths, on average, have larger output", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "and input margins. We train 25 VGG16 models with different random seeds on CIFAR10 (see Appendix A for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "details) and compare the mean prediction depth of each test point in these 25 runs to its mean output and input", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 281, + 411 + ], + "score": 1.0, + "content": "margins (log scales). Correlation coefficients are", + "type": "text" + }, + { + "bbox": [ + 281, + 399, + 306, + 409 + ], + "score": 0.76, + "content": "- 0 . 7 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 398, + 379, + 411 + ], + "score": 1.0, + "content": "(output margin) and", + "type": "text" + }, + { + "bbox": [ + 380, + 399, + 405, + 409 + ], + "score": 0.75, + "content": "- 0 . 6 9", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "(input margin). The density", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "of data points is indicated by the color bar, which follows a log scale. Although the prediction depth could be at", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "most 14, no data point has an average prediction depth greater than 12. Right: An intervention that does not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 221, + 441 + ], + "score": 1.0, + "content": "encourage large output margin (", + "type": "text" + }, + { + "bbox": [ + 222, + 429, + 231, + 438 + ], + "score": 0.32, + "content": "^ { * } O", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "-Hinge”) results, as predicted, in models where the predictions are effectively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 419, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 388, + 450 + ], + "score": 1.0, + "content": "determined in higher layers in the network compared to the standard training", + "type": "text" + }, + { + "bbox": [ + 388, + 439, + 415, + 449 + ], + "score": 0.57, + "content": "( \\ ^ { \\ast } C E ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 438, + 419, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + } + ], + "index": 18.75 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "To test this hypothesis we train a model on a training split where a subset of labels are corrupted and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 465, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 481 + ], + "score": 1.0, + "content": "compare the training and inference learning curves on four splits of the data: unchanged training", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "data; mislabeled training data; the original labels of the mislabeled training data and the test split. In", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "Figure 5 (middle and right plots) we see that many of the important features of the training learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "curve are indeed present in the inference learning curve. During training (middle), mislabeled", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "data are initially processed as though they are a member of their original class (before they were", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "mislabeled) (Liu et al., 2020a). After an initial period of learning, the network begins to learn the new", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "(random) labels that have been assigned to those data points, so the orange curve moves upwards,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "and the green curve downwards. At this point, a maximum is observed in the training accuracy (Arpit", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 506, + 568 + ], + "score": 1.0, + "content": "et al., 2017). In the right plot we see that these same phenomena occur in the inference learning", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 566, + 134, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 134, + 577 + ], + "score": 1.0, + "content": "curve.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 456, + 506, + 577 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 589, + 446, + 601 + ], + "lines": [ + { + "bbox": [ + 104, + 588, + 447, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 447, + 603 + ], + "score": 1.0, + "content": "3.3 Deep models exhibit larger margins for inputs with lower prediction depth", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 620 + ], + "score": 1.0, + "content": "It is reported in the literature that deep networks learn functions of increasing complexity during", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "training (Hu et al., 2020; Kalimeris et al., 2019). We frame this observation differently: the learned", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "function is “locally simpler” in the vicinity of data points with smaller prediction depths, and these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 640, + 345, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 345, + 652 + ], + "score": 1.0, + "content": "points are typically learned earlier in training (Section 3.2).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 605, + 505, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 651, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "Two known measures of the simplicity of a learned function are the output margin (the difference", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "between the largest and second-largest logits) and the adversarial input margin (the smallest norm", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "required for an adversarial perturbation in the input to change the model’s class prediction). We", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 683, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 257, + 698 + ], + "score": 1.0, + "content": "estimate the adversarial input margin,", + "type": "text" + }, + { + "bbox": [ + 257, + 689, + 263, + 696 + ], + "score": 0.87, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 683, + 506, + 698 + ], + "score": 1.0, + "content": ", with a linear approximation (Jiang et al., 2018): for an input", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 696, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 305, + 713 + ], + "score": 1.0, + "content": "x with predicted class i, γ ' minj6=i |zi−zj ||∇x(zi−zj )|", + "type": "text" + }, + { + "bbox": [ + 305, + 697, + 334, + 711 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 334, + 702, + 343, + 710 + ], + "score": 0.89, + "content": "z _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 697, + 506, + 711 + ], + "score": 1.0, + "content": "is the logit returned by the network for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 128, + 723 + ], + "score": 1.0, + "content": "class", + "type": "text" + }, + { + "bbox": [ + 129, + 711, + 135, + 722 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 711, + 505, + 723 + ], + "score": 1.0, + "content": ". Figure 6 (left and middle plots) show that data points with smaller prediction depths have", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "both larger input and output margins on average and that variances of the input and output margins", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 311, + 277, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 277, + 324 + ], + "score": 1.0, + "content": "decrease as the prediction depth increases.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 651, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 71, + 502, + 208 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 71, + 502, + 208 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 71, + 502, + 208 + ], + "spans": [ + { + "bbox": [ + 110, + 71, + 502, + 208 + ], + "score": 0.955, + "type": "image", + "image_path": "6adf476b71571c6b4cc9499c21b8fe2583b9b43fa207152bd6831ec0fc9ca860.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 71, + 502, + 116.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 116.66666666666666, + 502, + 162.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 162.33333333333331, + 502, + 207.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 217, + 505, + 288 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 216, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 230 + ], + "score": 1.0, + "content": "Figure 7: The prediction depth can be the same, or very different for the same input when it occurs in the train", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "score": 1.0, + "content": "and validation splits. Corners of this plot correspond to different forms of example difficulty. (See Section 4", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 383, + 248 + ], + "score": 1.0, + "content": "for discussion.) We train 250 ResNet18 models on CIFAR10 with random", + "type": "text" + }, + { + "bbox": [ + 384, + 237, + 414, + 246 + ], + "score": 0.89, + "content": "9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "train:validation splits as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "described in Appendix A. These histograms compare average prediction depth for each data point when it", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "occurs in the validation split vs the training split. This behavior is consistently reproduced for all datasets and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "architectures in Appendix C.6. Below we show extreme (not hand-chosen) images of “Birds” that appear closest", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 277, + 495, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 495, + 289 + ], + "score": 1.0, + "content": "to the corners of this plot. The consensus class is given above each image (tiebreaks favor the class “Bird”.)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 504, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "both larger input and output margins on average and that variances of the input and output margins", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 311, + 277, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 277, + 324 + ], + "score": 1.0, + "content": "decrease as the prediction depth increases.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "To illustrate the strength of the relationship between the prediction depth and output margin, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 333, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 346 + ], + "score": 1.0, + "content": "demonstrate that reducing the output margin of the learned function results in a model that clusters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "the data only in the latest layers: such a solution has a very high average prediction depth. We do", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "not minimize the output margin directly but rather use a loss and an optimizer that do not encourage", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "high output margin. Naturally there are many unknowns that may contribute to this effect. We simply", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 378, + 268, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 268, + 389 + ], + "score": 1.0, + "content": "report the intervention and the outcome.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 506, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "The intervention is performed as follows: we construct a loss function that does not promote", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 401, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 257, + 412 + ], + "score": 1.0, + "content": "confidence: a zero-margin hinge loss (", + "type": "text" + }, + { + "bbox": [ + 257, + 401, + 267, + 410 + ], + "score": 0.39, + "content": "^ { 6 6 } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 401, + 505, + 412 + ], + "score": 1.0, + "content": "-Hinge”), and optimize the network using full-batch gradient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 379, + 424 + ], + "score": 1.0, + "content": "descent with momentum and very small learning rate. For an input", + "type": "text" + }, + { + "bbox": [ + 380, + 413, + 387, + 421 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 411, + 430, + 424 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 430, + 412, + 435, + 421 + ], + "score": 0.57, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "the 0-Hinge loss", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 103, + 421, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 103, + 421, + 155, + 438 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 155, + 421, + 279, + 436 + ], + "score": 0.92, + "content": "\\begin{array} { r } { l ( x ) = \\sum _ { j \\neq i } \\operatorname* { m a x } ( \\dot { 0 } , z _ { i } - z _ { j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 421, + 308, + 438 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 308, + 424, + 319, + 434 + ], + "score": 0.85, + "content": "z _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 421, + 439, + 438 + ], + "score": 1.0, + "content": "represents the logit for class", + "type": "text" + }, + { + "bbox": [ + 440, + 423, + 446, + 434 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 421, + 507, + 438 + ], + "score": 1.0, + "content": ". The form of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "this intervention is justified in Appendix A.7. As a control, we additionally train a model in the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "standard fashion using the cross-entropy loss and SGD with momentum and large initial learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "rate. Since full-batch gradients are computationally expensive, we train on a subset of CIFAR10", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "(see Appendix A.7, where we also give the hyperparameters and learning curves.). The output", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 506, + 490 + ], + "score": 1.0, + "content": "margin obtained with the intervention is 5 orders of magnitude smaller than in the control experiment:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 212, + 499 + ], + "score": 0.92, + "content": "2 . 0 \\times 1 0 ^ { - 4 } \\pm 2 . 0 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 487, + 314, + 501 + ], + "score": 1.0, + "content": "for the 0-Hinge loss and", + "type": "text" + }, + { + "bbox": [ + 315, + 488, + 412, + 499 + ], + "score": 0.92, + "content": "1 . { \\overline { { 6 } } } \\times 1 0 ^ { 1 } \\pm 0 . 5 0 \\times 1 0 ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 487, + 506, + 501 + ], + "score": 1.0, + "content": "for cross-entropy loss.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 291, + 512 + ], + "score": 1.0, + "content": "Figure 6 (right) compares the accuracies of the", + "type": "text" + }, + { + "bbox": [ + 291, + 500, + 298, + 510 + ], + "score": 0.37, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "-NN probes resulting from these training approaches.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "The 0-Hinge loss training achieves only a marginal improvement in accuracy (red) over an untrained", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "network (purple), and the training split is accurately clustered only in the latest layers. This confirms", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "the predicted behavior: the intervention leads to a model that exhibits both very small average output", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "margins and very late clustering of the data. Very late clustering of the data implies high prediction", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 553, + 474, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 172, + 568 + ], + "score": 1.0, + "content": "depths since the", + "type": "text" + }, + { + "bbox": [ + 172, + 555, + 179, + 564 + ], + "score": 0.34, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 553, + 474, + 568 + ], + "score": 1.0, + "content": "-NN probe classifications change in the latest layers for many data points.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 577, + 418, + 591 + ], + "lines": [ + { + "bbox": [ + 104, + 575, + 418, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 418, + 595 + ], + "score": 1.0, + "content": "4 Beyond a One-Dimensional Picture of Example Difficulty", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 109, + 598, + 504, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "In this section we transcend the one-dimensional picture of example difficulty by identifying dif-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "ferent underlying reasons behind the difficulty of an example, in a way that is general to different", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 619, + 214, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 214, + 632 + ], + "score": 1.0, + "content": "architectures and datasets.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 503, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "Figure 7 shows that the prediction depth can be different when an input occurs in the training split vs.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 642, + 374, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 374, + 655 + ], + "score": 1.0, + "content": "the validation split. Thus, there are two axes of example difficulty:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 130, + 659, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 130, + 659, + 425, + 671 + ], + "spans": [ + { + "bbox": [ + 130, + 659, + 425, + 671 + ], + "score": 1.0, + "content": "1. Difficulty of making a prediction when an input is in the validation set", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 128, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 128, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "2. Difficulty of finding commonalities during training with other examples of the same ground", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 684, + 186, + 695 + ], + "spans": [ + { + "bbox": [ + 141, + 684, + 186, + 695 + ], + "score": 1.0, + "content": "truth class", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Both axes have a range from “clear” to “ambiguous”. 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Corners of this plot correspond to different forms of example difficulty. (See Section 4", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 383, + 248 + ], + "score": 1.0, + "content": "for discussion.) We train 250 ResNet18 models on CIFAR10 with random", + "type": "text" + }, + { + "bbox": [ + 384, + 237, + 414, + 246 + ], + "score": 0.89, + "content": "9 0 { : } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "train:validation splits as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "described in Appendix A. These histograms compare average prediction depth for each data point when it", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "occurs in the validation split vs the training split. This behavior is consistently reproduced for all datasets and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "architectures in Appendix C.6. Below we show extreme (not hand-chosen) images of “Birds” that appear closest", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 277, + 495, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 495, + 289 + ], + "score": 1.0, + "content": "to the corners of this plot. The consensus class is given above each image (tiebreaks favor the class “Bird”.)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 504, + 322 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 105, + 299, + 505, + 324 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "To illustrate the strength of the relationship between the prediction depth and output margin, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 333, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 346 + ], + "score": 1.0, + "content": "demonstrate that reducing the output margin of the learned function results in a model that clusters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "the data only in the latest layers: such a solution has a very high average prediction depth. We do", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "not minimize the output margin directly but rather use a loss and an optimizer that do not encourage", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "high output margin. Naturally there are many unknowns that may contribute to this effect. We simply", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 378, + 268, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 268, + 389 + ], + "score": 1.0, + "content": "report the intervention and the outcome.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 322, + 506, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 506, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "The intervention is performed as follows: we construct a loss function that does not promote", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 401, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 257, + 412 + ], + "score": 1.0, + "content": "confidence: a zero-margin hinge loss (", + "type": "text" + }, + { + "bbox": [ + 257, + 401, + 267, + 410 + ], + "score": 0.39, + "content": "^ { 6 6 } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 401, + 505, + 412 + ], + "score": 1.0, + "content": "-Hinge”), and optimize the network using full-batch gradient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 379, + 424 + ], + "score": 1.0, + "content": "descent with momentum and very small learning rate. For an input", + "type": "text" + }, + { + "bbox": [ + 380, + 413, + 387, + 421 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 411, + 430, + 424 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 430, + 412, + 435, + 421 + ], + "score": 0.57, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "the 0-Hinge loss", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 103, + 421, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 103, + 421, + 155, + 438 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 155, + 421, + 279, + 436 + ], + "score": 0.92, + "content": "\\begin{array} { r } { l ( x ) = \\sum _ { j \\neq i } \\operatorname* { m a x } ( \\dot { 0 } , z _ { i } - z _ { j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 421, + 308, + 438 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 308, + 424, + 319, + 434 + ], + "score": 0.85, + "content": "z _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 421, + 439, + 438 + ], + "score": 1.0, + "content": "represents the logit for class", + "type": "text" + }, + { + "bbox": [ + 440, + 423, + 446, + 434 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 421, + 507, + 438 + ], + "score": 1.0, + "content": ". The form of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "this intervention is justified in Appendix A.7. As a control, we additionally train a model in the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "standard fashion using the cross-entropy loss and SGD with momentum and large initial learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "rate. Since full-batch gradients are computationally expensive, we train on a subset of CIFAR10", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "(see Appendix A.7, where we also give the hyperparameters and learning curves.). The output", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 506, + 490 + ], + "score": 1.0, + "content": "margin obtained with the intervention is 5 orders of magnitude smaller than in the control experiment:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 212, + 499 + ], + "score": 0.92, + "content": "2 . 0 \\times 1 0 ^ { - 4 } \\pm 2 . 0 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 487, + 314, + 501 + ], + "score": 1.0, + "content": "for the 0-Hinge loss and", + "type": "text" + }, + { + "bbox": [ + 315, + 488, + 412, + 499 + ], + "score": 0.92, + "content": "1 . { \\overline { { 6 } } } \\times 1 0 ^ { 1 } \\pm 0 . 5 0 \\times 1 0 ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 487, + 506, + 501 + ], + "score": 1.0, + "content": "for cross-entropy loss.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 291, + 512 + ], + "score": 1.0, + "content": "Figure 6 (right) compares the accuracies of the", + "type": "text" + }, + { + "bbox": [ + 291, + 500, + 298, + 510 + ], + "score": 0.37, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "-NN probes resulting from these training approaches.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "The 0-Hinge loss training achieves only a marginal improvement in accuracy (red) over an untrained", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "network (purple), and the training split is accurately clustered only in the latest layers. This confirms", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "the predicted behavior: the intervention leads to a model that exhibits both very small average output", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "margins and very late clustering of the data. Very late clustering of the data implies high prediction", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 553, + 474, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 172, + 568 + ], + "score": 1.0, + "content": "depths since the", + "type": "text" + }, + { + "bbox": [ + 172, + 555, + 179, + 564 + ], + "score": 0.34, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 553, + 474, + 568 + ], + "score": 1.0, + "content": "-NN probe classifications change in the latest layers for many data points.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 25.5, + "bbox_fs": [ + 103, + 389, + 507, + 568 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 577, + 418, + 591 + ], + "lines": [ + { + "bbox": [ + 104, + 575, + 418, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 418, + 595 + ], + "score": 1.0, + "content": "4 Beyond a One-Dimensional Picture of Example Difficulty", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 109, + 598, + 504, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "In this section we transcend the one-dimensional picture of example difficulty by identifying dif-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "ferent underlying reasons behind the difficulty of an example, in a way that is general to different", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 619, + 214, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 214, + 632 + ], + "score": 1.0, + "content": "architectures and datasets.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 597, + 506, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 503, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "Figure 7 shows that the prediction depth can be different when an input occurs in the training split vs.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 642, + 374, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 374, + 655 + ], + "score": 1.0, + "content": "the validation split. Thus, there are two axes of example difficulty:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 632, + 505, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 659, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 130, + 659, + 425, + 671 + ], + "spans": [ + { + "bbox": [ + 130, + 659, + 425, + 671 + ], + "score": 1.0, + "content": "1. Difficulty of making a prediction when an input is in the validation set", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 128, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 128, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "2. Difficulty of finding commonalities during training with other examples of the same ground", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 684, + 186, + 695 + ], + "spans": [ + { + "bbox": [ + 141, + 684, + 186, + 695 + ], + "score": 1.0, + "content": "truth class", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 128, + 659, + 505, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Both axes have a range from “clear” to “ambiguous”. In Section 3.1 we show that predictions", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "made for validation points with later prediction depths are often inconsistent, with low consensus-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 241, + 507, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 507, + 254 + ], + "score": 1.0, + "content": "consistency. Conversely, a low prediction depth typically indicates an input with high consensus-", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 253, + 507, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 507, + 265 + ], + "score": 1.0, + "content": "consistency. For Axis 1 we will identify validation points with low prediction depths as “clear”", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "and those with high prediction depths as “ambiguous”. We will additionally identify a low or high", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "prediction depth in the training split with examples that are respectively “clear” and “ambiguous” on", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 331, + 299 + ], + "score": 1.0, + "content": "Axis 2. By making combinations of low/high values of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 331, + 286, + 361, + 297 + ], + "score": 0.5, + "content": "( \\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 362, + 284, + 365, + 299 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 366, + 285, + 401, + 297 + ], + "score": 0.62, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } _ { . } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 402, + 284, + 506, + 299 + ], + "score": 1.0, + "content": ") we obtain four extremes", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 296, + 195, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 195, + 310 + ], + "score": 1.0, + "content": "of example difficulty:", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 501, + 164 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 501, + 164 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 501, + 164 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 501, + 164 + ], + "score": 0.966, + "type": "image", + "image_path": "14930e4f6df689d31626c023acd5f7f776fdcf3f0a5998ae876b3c3fa70cabe4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 501, + 102.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 102.0, + 501, + 133.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 133.0, + 501, + 164.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 171, + 505, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "Figure 8: Average k-NN probe confidence (solid lines) and accuracy (dotted lines) for the ground truth class", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 182, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 192 + ], + "score": 1.0, + "content": "(left) and consensus class (right), in the validation split for examples exhibiting extreme forms of difficulty.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "score": 1.0, + "content": "Mean values for 100 examples with each form of difficulty, identified as the 100 examples closest to the corners", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "in Figure 7 (left). This result is for CIFAR10 with ResNet18: similar plots for all datasets and architectures", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "score": 1.0, + "content": "are shown in Appendix C.7. See Section 4 for the discussion of the result and how it can be used to improve", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 220, + 182, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 182, + 234 + ], + "score": 1.0, + "content": "prediction accuracy.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 506, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 507, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 507, + 254 + ], + "score": 1.0, + "content": "consistency. Conversely, a low prediction depth typically indicates an input with high consensus-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 253, + 507, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 507, + 265 + ], + "score": 1.0, + "content": "consistency. For Axis 1 we will identify validation points with low prediction depths as “clear”", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "and those with high prediction depths as “ambiguous”. We will additionally identify a low or high", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "prediction depth in the training split with examples that are respectively “clear” and “ambiguous” on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 331, + 299 + ], + "score": 1.0, + "content": "Axis 2. By making combinations of low/high values of", + "type": "text" + }, + { + "bbox": [ + 331, + 286, + 361, + 297 + ], + "score": 0.5, + "content": "( \\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 284, + 365, + 299 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 366, + 285, + 401, + 297 + ], + "score": 0.62, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } _ { . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 284, + 506, + 299 + ], + "score": 1.0, + "content": ") we obtain four extremes", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 296, + 195, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 195, + 310 + ], + "score": 1.0, + "content": "of example difficulty:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 506, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 201, + 324 + ], + "score": 1.0, + "content": "Easy examples: (Low", + "type": "text" + }, + { + "bbox": [ + 201, + 311, + 230, + 323 + ], + "score": 0.86, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 311, + 255, + 324 + ], + "score": 1.0, + "content": ", Low", + "type": "text" + }, + { + "bbox": [ + 256, + 311, + 291, + 323 + ], + "score": 0.88, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "). Such examples are often visually typical members", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 323, + 450, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 450, + 335 + ], + "score": 1.0, + "content": "of their class and the predicted label nearly always matches the ground truth.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 336, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 252, + 351 + ], + "score": 1.0, + "content": "Looks like a different class: (Low", + "type": "text" + }, + { + "bbox": [ + 253, + 337, + 281, + 348 + ], + "score": 0.83, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 336, + 310, + 351 + ], + "score": 1.0, + "content": ", High", + "type": "text" + }, + { + "bbox": [ + 310, + 336, + 346, + 348 + ], + "score": 0.86, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 336, + 505, + 351 + ], + "score": 1.0, + "content": "). In the validation set, there is a clear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 141, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "(and nearly always incorrect) classification for such an input, but it is difficult to connect such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 141, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "inputs to other examples of their ground truth class during training. Mislabeled examples are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "of this kind, as are visually confusing images which at first appear to show something else.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 291, + 398 + ], + "score": 1.0, + "content": "Ambiguous unless the label is given: (High", + "type": "text" + }, + { + "bbox": [ + 291, + 384, + 320, + 396 + ], + "score": 0.81, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 384, + 345, + 398 + ], + "score": 1.0, + "content": ", Low", + "type": "text" + }, + { + "bbox": [ + 345, + 384, + 381, + 396 + ], + "score": 0.87, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "). These examples are difficult", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "to connect to their predicted class in the validation split but easy to connect to their ground", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 141, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "truth class during training. These points may, for example, visually resemble both their own", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 417, + 380, + 430 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 380, + 430 + ], + "score": 1.0, + "content": "class and another class. They are likely to be misclassified.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 188, + 445 + ], + "score": 1.0, + "content": "Ambiguous: (High", + "type": "text" + }, + { + "bbox": [ + 189, + 432, + 217, + 443 + ], + "score": 0.55, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 430, + 244, + 445 + ], + "score": 1.0, + "content": ", High", + "type": "text" + }, + { + "bbox": [ + 245, + 432, + 280, + 443 + ], + "score": 0.81, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "). These examples may be corrupted or show an example", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "of a rare sub-class. Predictions for these inputs can depend strongly on the random seed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 453, + 283, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 283, + 466 + ], + "score": 1.0, + "content": "used for training and initialization.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "In Figure 7 we visualize CIFAR10 “Bird” images with the extreme forms of example difficulty for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "ResNet18, as identified using the prediction depth in the training and validation splits. In the full", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "dataset (left panel) we see that the prediction depth can be very different in the training and validation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "splits: the two prediction depths are typically similar for points where the consensus class is equal", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "to the ground truth (right panel), but can be very different when the consensus class is different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "from the ground truth (middle panel). This behavior is consistently reproduced for all datasets and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 535, + 231, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 231, + 547 + ], + "score": 1.0, + "content": "architectures in Appendix C.6.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "Looking at these examples of the class “Bird” with different difficulty types, we observe that ResNet18", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "finds small garden birds easiest, while birds in flight against a blue background “look like airplanes”,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 580 + ], + "score": 1.0, + "content": "ostriches are “ambiguous without their label” and the “ambiguous” examples are either unclear", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "photographs or examples of rare sub-groups that don’t appear frequently in the data. We found", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 602 + ], + "score": 1.0, + "content": "the consensus-consistency of inputs that are “Ambiguous” or “Ambiguous without its label” to be", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 468, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 468, + 612 + ], + "score": 1.0, + "content": "significantly lower than those of examples that are “Easy” or “Look like a different class”.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 613, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "In order to better understand how networks process examples with different, extreme forms of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "score": 1.0, + "content": "example difficulty, Fig. 8 examines how the k-NN confidence (fraction of votes) and accuracy of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "the ground truth class and of the consensus class progress, as validation points pass through the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 646, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 504, + 657 + ], + "score": 1.0, + "content": "network. “Easy” examples are classified as their consensus class (which is equal to their ground truth", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "class) in all k-NN probes and the confidence in the consensus class steadily increases as data points", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "proceed through the hidden layers. Examples that “look like a different class” are also processed as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "members of their consensus class, similarly to “easy” examples. However, unlike “easy” examples,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "their consensus classes do not match their ground truth classes. Examples that are “ambiguous", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 504, + 712 + ], + "score": 1.0, + "content": "without their labels” are initially processed as members of their ground truth classes with intermediate", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "confidence, but in later layers become mistaken for their consensus class. “Ambiguous” examples are", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 501, + 164 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 501, + 164 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 501, + 164 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 501, + 164 + ], + "score": 0.966, + "type": "image", + "image_path": "14930e4f6df689d31626c023acd5f7f776fdcf3f0a5998ae876b3c3fa70cabe4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 501, + 102.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 102.0, + 501, + 133.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 133.0, + 501, + 164.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 171, + 505, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "Figure 8: Average k-NN probe confidence (solid lines) and accuracy (dotted lines) for the ground truth class", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 182, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 192 + ], + "score": 1.0, + "content": "(left) and consensus class (right), in the validation split for examples exhibiting extreme forms of difficulty.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "score": 1.0, + "content": "Mean values for 100 examples with each form of difficulty, identified as the 100 examples closest to the corners", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "in Figure 7 (left). This result is for CIFAR10 with ResNet18: similar plots for all datasets and architectures", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "score": 1.0, + "content": "are shown in Appendix C.7. See Section 4 for the discussion of the result and how it can be used to improve", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 220, + 182, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 182, + 234 + ], + "score": 1.0, + "content": "prediction accuracy.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 506, + 308 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 104, + 241, + 507, + 310 + ], + "lines_deleted": true + }, + { + "type": "list", + "bbox": [ + 106, + 311, + 506, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 201, + 324 + ], + "score": 1.0, + "content": "Easy examples: (Low", + "type": "text" + }, + { + "bbox": [ + 201, + 311, + 230, + 323 + ], + "score": 0.86, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 311, + 255, + 324 + ], + "score": 1.0, + "content": ", Low", + "type": "text" + }, + { + "bbox": [ + 256, + 311, + 291, + 323 + ], + "score": 0.88, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "). Such examples are often visually typical members", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 323, + 450, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 450, + 335 + ], + "score": 1.0, + "content": "of their class and the predicted label nearly always matches the ground truth.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 336, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 252, + 351 + ], + "score": 1.0, + "content": "Looks like a different class: (Low", + "type": "text" + }, + { + "bbox": [ + 253, + 337, + 281, + 348 + ], + "score": 0.83, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 336, + 310, + 351 + ], + "score": 1.0, + "content": ", High", + "type": "text" + }, + { + "bbox": [ + 310, + 336, + 346, + 348 + ], + "score": 0.86, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 336, + 505, + 351 + ], + "score": 1.0, + "content": "). In the validation set, there is a clear", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 141, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "(and nearly always incorrect) classification for such an input, but it is difficult to connect such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 141, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "inputs to other examples of their ground truth class during training. Mislabeled examples are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "of this kind, as are visually confusing images which at first appear to show something else.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 291, + 398 + ], + "score": 1.0, + "content": "Ambiguous unless the label is given: (High", + "type": "text" + }, + { + "bbox": [ + 291, + 384, + 320, + 396 + ], + "score": 0.81, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 384, + 345, + 398 + ], + "score": 1.0, + "content": ", Low", + "type": "text" + }, + { + "bbox": [ + 345, + 384, + 381, + 396 + ], + "score": 0.87, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "). These examples are difficult", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "to connect to their predicted class in the validation split but easy to connect to their ground", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 141, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "truth class during training. These points may, for example, visually resemble both their own", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 417, + 380, + 430 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 380, + 430 + ], + "score": 1.0, + "content": "class and another class. They are likely to be misclassified.", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 188, + 445 + ], + "score": 1.0, + "content": "Ambiguous: (High", + "type": "text" + }, + { + "bbox": [ + 189, + 432, + 217, + 443 + ], + "score": 0.55, + "content": "\\mathrm { P D } _ { \\mathrm { V a l . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 430, + 244, + 445 + ], + "score": 1.0, + "content": ", High", + "type": "text" + }, + { + "bbox": [ + 245, + 432, + 280, + 443 + ], + "score": 0.81, + "content": "\\mathrm { P D } _ { \\mathrm { T r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "). These examples may be corrupted or show an example", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "of a rare sub-class. Predictions for these inputs can depend strongly on the random seed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 453, + 283, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 283, + 466 + ], + "score": 1.0, + "content": "used for training and initialization.", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + } + ], + "index": 21, + "bbox_fs": [ + 105, + 311, + 506, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "In Figure 7 we visualize CIFAR10 “Bird” images with the extreme forms of example difficulty for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "ResNet18, as identified using the prediction depth in the training and validation splits. In the full", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "dataset (left panel) we see that the prediction depth can be very different in the training and validation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "splits: the two prediction depths are typically similar for points where the consensus class is equal", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "to the ground truth (right panel), but can be very different when the consensus class is different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "from the ground truth (middle panel). This behavior is consistently reproduced for all datasets and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 535, + 231, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 231, + 547 + ], + "score": 1.0, + "content": "architectures in Appendix C.6.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 469, + 506, + 547 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "Looking at these examples of the class “Bird” with different difficulty types, we observe that ResNet18", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "finds small garden birds easiest, while birds in flight against a blue background “look like airplanes”,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 580 + ], + "score": 1.0, + "content": "ostriches are “ambiguous without their label” and the “ambiguous” examples are either unclear", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "photographs or examples of rare sub-groups that don’t appear frequently in the data. We found", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 602 + ], + "score": 1.0, + "content": "the consensus-consistency of inputs that are “Ambiguous” or “Ambiguous without its label” to be", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 468, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 468, + 612 + ], + "score": 1.0, + "content": "significantly lower than those of examples that are “Easy” or “Look like a different class”.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 546, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 613, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "In order to better understand how networks process examples with different, extreme forms of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "score": 1.0, + "content": "example difficulty, Fig. 8 examines how the k-NN confidence (fraction of votes) and accuracy of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "the ground truth class and of the consensus class progress, as validation points pass through the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 646, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 504, + 657 + ], + "score": 1.0, + "content": "network. “Easy” examples are classified as their consensus class (which is equal to their ground truth", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "class) in all k-NN probes and the confidence in the consensus class steadily increases as data points", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "proceed through the hidden layers. Examples that “look like a different class” are also processed as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "members of their consensus class, similarly to “easy” examples. However, unlike “easy” examples,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "their consensus classes do not match their ground truth classes. Examples that are “ambiguous", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 504, + 712 + ], + "score": 1.0, + "content": "without their labels” are initially processed as members of their ground truth classes with intermediate", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "confidence, but in later layers become mistaken for their consensus class. “Ambiguous” examples are", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "processed with low confidence and accuracy in the early layers, for both ground truth and consensus", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "classes. In later layers “ambiguous” examples are recognized, with intermediate confidence and", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "accuracy, as members of the consensus class, which matches the ground truth class for a sizeable", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 248, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 248, + 118 + ], + "score": 1.0, + "content": "fraction of “ambiguous” examples.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 613, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "processed with low confidence and accuracy in the early layers, for both ground truth and consensus", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "classes. In later layers “ambiguous” examples are recognized, with intermediate confidence and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "accuracy, as members of the consensus class, which matches the ground truth class for a sizeable", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 248, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 248, + 118 + ], + "score": 1.0, + "content": "fraction of “ambiguous” examples.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 124, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 124, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 506, + 137 + ], + "score": 1.0, + "content": "Improving the prediction accuracy Can the prediction accuracy be improved using our under-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "score": 1.0, + "content": "standing of how each class of difficult examples are processed by deep models? Figure 8 suggest", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 123, + 158 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 123, + 146, + 130, + 156 + ], + "score": 0.36, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "-NN probes in intermediate layers may be more accurate than the full deep model for examples", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 157, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 506, + 169 + ], + "score": 1.0, + "content": "that are “ambiguous without their label” (data points closest to the lower right corner of Figure 7).", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 167, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 368, + 181 + ], + "score": 1.0, + "content": "In order to test this hypothesis, we compare the accuracy of the", + "type": "text" + }, + { + "bbox": [ + 369, + 168, + 375, + 178 + ], + "score": 0.36, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 167, + 506, + 181 + ], + "score": 1.0, + "content": "-NN probe in layer 4 to the full", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "score": 1.0, + "content": "model’s prediction for the 100 examples closest to the lower right corner of Figure 8. We obtain a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 189, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 261, + 203 + ], + "score": 1.0, + "content": "striking improvement in accuracy from", + "type": "text" + }, + { + "bbox": [ + 261, + 189, + 280, + 200 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 189, + 291, + 203 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 291, + 189, + 311, + 200 + ], + "score": 0.88, + "content": "98 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 189, + 505, + 203 + ], + "score": 1.0, + "content": "for these examples. This showcases how insights", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 199, + 379, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 379, + 214 + ], + "score": 1.0, + "content": "from this study can be directly used to improve prediction accuracy.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 106, + 222, + 180, + 235 + ], + "lines": [ + { + "bbox": [ + 104, + 220, + 181, + 238 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 181, + 238 + ], + "score": 1.0, + "content": "5 Discussion", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "Summary We have introduced a notion of example difficulty called the prediction depth, which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "uses the processing of data inside the network to score the difficulty of an example. We have shown", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "how the prediction depth is related to the accuracy and uncertainty of a prediction, the adversarial", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "input margin and the output margin of the learned solution, and that data points that are easier", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "according to the prediction depth are also typically learned earlier in training. We have also shown", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "that the difficulty of an example can be both similar, or very different depending on whether an input", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "appears in the validation split or the training split, and described four extremes of example difficulty.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "For data points that are “ambiguous without their label”, we have demonstrated how returning the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 113, + 340 + ], + "score": 0.35, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "-NN prediction in a middle layer can lead to impressive increases in model accuracy: for CIFAR10", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 335, + 353 + ], + "score": 1.0, + "content": "in ResNet18 we obtained an increase in accuracy from", + "type": "text" + }, + { + "bbox": [ + 336, + 340, + 356, + 351 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 340, + 368, + 353 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 369, + 340, + 389, + 351 + ], + "score": 0.87, + "content": "98 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "for the inputs that are most", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 239, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 239, + 364 + ], + "score": 1.0, + "content": "“ambiguous without their label”.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 370, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "Connecting known phenomena In the literature, the following phenomena are separately reported", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 266, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 266, + 394 + ], + "score": 1.0, + "content": "from different experimental paradigms:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 130, + 395, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 129, + 396, + 453, + 409 + ], + "spans": [ + { + "bbox": [ + 129, + 396, + 453, + 409 + ], + "score": 1.0, + "content": "1. Early layers generalize while later layers memorize (Stephenson et al., 2021).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 128, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 128, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "2. Model layers converge from input layer towards output layer (Raghu et al., 2017; Morcos", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 419, + 195, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 419, + 195, + 432 + ], + "score": 1.0, + "content": "et al., 2018).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 128, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 128, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "3. Deep models learn easy data (Jiang et al., 2021; Toneva et al., 2019) and simple functions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 444, + 322, + 456 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 322, + 456 + ], + "score": 1.0, + "content": "first (Hu et al., 2020; Kalimeris et al., 2019).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 113, + 459, + 382, + 470 + ], + "lines": [ + { + "bbox": [ + 110, + 457, + 383, + 473 + ], + "spans": [ + { + "bbox": [ + 110, + 457, + 383, + 473 + ], + "score": 1.0, + "content": "Following this paper, a coherent and closely related picture emerges:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 130, + 474, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 129, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 129, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "1. 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Data points learned early in training typically have smaller prediction depths than those", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 520, + 312, + 532 + ], + "spans": [ + { + "bbox": [ + 141, + 520, + 312, + 532 + ], + "score": 1.0, + "content": "learned later during training (Section 3.2).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 128, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 128, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "3. On average, deep neural networks exhibit wider input and output margins (common measures", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 543, + 497, + 556 + ], + "spans": [ + { + "bbox": [ + 141, + 543, + 497, + 556 + ], + "score": 1.0, + "content": "of “local simplicity”) in the vicinity of data with smaller prediction depths (Section 3.3).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "Pertinence of example difficulty to topics in machine learning Curriculum Learning attempts", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "to treat hard examples differently from easy examples during training. Robustness to distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "shifts that change the relative frequencies of common and rare subgroups in the test set (which we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "have shown can have different forms of example difficulty) is important for ML Fairness. Methods", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 507, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 507, + 618 + ], + "score": 1.0, + "content": "developed to address heteroscedastic uncertainty typically address example difficulty as a one-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 507, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 507, + 629 + ], + "score": 1.0, + "content": "dimensional quantity. We expand upon the relevance of our work to these three topics in Appendix D.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "Limitations We believe that the results we report stem from a deep model’s representation, which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "is hierarchical by construction. We expect that the same results will therefore apply in larger models,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "larger datasets, and tasks other than image classification, but testing this remains as further work.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "Although we demonstrate that returning the results of a hidden k-NN can yield dramatic increases in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "accuracy for examples that are “ambiguous without their label”, we otherwise do not explore ways", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "to practically apply the insights we present. In particular, we expressly do not claim that all that is", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "required for good accuracy is to reduce the prediction depth: freezing later layers of the network", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 327, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 327, + 723 + ], + "score": 1.0, + "content": "would not be expected to result in good generalization.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 505, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 124, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 124, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 506, + 137 + ], + "score": 1.0, + "content": "Improving the prediction accuracy Can the prediction accuracy be improved using our under-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "score": 1.0, + "content": "standing of how each class of difficult examples are processed by deep models? Figure 8 suggest", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 123, + 158 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 123, + 146, + 130, + 156 + ], + "score": 0.36, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "-NN probes in intermediate layers may be more accurate than the full deep model for examples", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 157, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 506, + 169 + ], + "score": 1.0, + "content": "that are “ambiguous without their label” (data points closest to the lower right corner of Figure 7).", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 167, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 368, + 181 + ], + "score": 1.0, + "content": "In order to test this hypothesis, we compare the accuracy of the", + "type": "text" + }, + { + "bbox": [ + 369, + 168, + 375, + 178 + ], + "score": 0.36, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 167, + 506, + 181 + ], + "score": 1.0, + "content": "-NN probe in layer 4 to the full", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "score": 1.0, + "content": "model’s prediction for the 100 examples closest to the lower right corner of Figure 8. We obtain a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 189, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 261, + 203 + ], + "score": 1.0, + "content": "striking improvement in accuracy from", + "type": "text" + }, + { + "bbox": [ + 261, + 189, + 280, + 200 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 189, + 291, + 203 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 291, + 189, + 311, + 200 + ], + "score": 0.88, + "content": "98 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 189, + 505, + 203 + ], + "score": 1.0, + "content": "for these examples. This showcases how insights", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 199, + 379, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 379, + 214 + ], + "score": 1.0, + "content": "from this study can be directly used to improve prediction accuracy.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 124, + 506, + 214 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 222, + 180, + 235 + ], + "lines": [ + { + "bbox": [ + 104, + 220, + 181, + 238 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 181, + 238 + ], + "score": 1.0, + "content": "5 Discussion", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "Summary We have introduced a notion of example difficulty called the prediction depth, which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "uses the processing of data inside the network to score the difficulty of an example. We have shown", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "how the prediction depth is related to the accuracy and uncertainty of a prediction, the adversarial", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "input margin and the output margin of the learned solution, and that data points that are easier", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "according to the prediction depth are also typically learned earlier in training. We have also shown", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "that the difficulty of an example can be both similar, or very different depending on whether an input", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "appears in the validation split or the training split, and described four extremes of example difficulty.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "For data points that are “ambiguous without their label”, we have demonstrated how returning the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 113, + 340 + ], + "score": 0.35, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "-NN prediction in a middle layer can lead to impressive increases in model accuracy: for CIFAR10", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 335, + 353 + ], + "score": 1.0, + "content": "in ResNet18 we obtained an increase in accuracy from", + "type": "text" + }, + { + "bbox": [ + 336, + 340, + 356, + 351 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 340, + 368, + 353 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 369, + 340, + 389, + 351 + ], + "score": 0.87, + "content": "98 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "for the inputs that are most", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 239, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 239, + 364 + ], + "score": 1.0, + "content": "“ambiguous without their label”.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 242, + 506, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 370, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "Connecting known phenomena In the literature, the following phenomena are separately reported", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 266, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 266, + 394 + ], + "score": 1.0, + "content": "from different experimental paradigms:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 368, + 505, + 394 + ] + }, + { + "type": "list", + "bbox": [ + 130, + 395, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 129, + 396, + 453, + 409 + ], + "spans": [ + { + "bbox": [ + 129, + 396, + 453, + 409 + ], + "score": 1.0, + "content": "1. Early layers generalize while later layers memorize (Stephenson et al., 2021).", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 128, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "2. Model layers converge from input layer towards output layer (Raghu et al., 2017; Morcos", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 419, + 195, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 419, + 195, + 432 + ], + "score": 1.0, + "content": "et al., 2018).", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 128, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "3. Deep models learn easy data (Jiang et al., 2021; Toneva et al., 2019) and simple functions", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 444, + 322, + 456 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 322, + 456 + ], + "score": 1.0, + "content": "first (Hu et al., 2020; Kalimeris et al., 2019).", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + } + ], + "index": 28, + "bbox_fs": [ + 128, + 396, + 505, + 456 + ] + }, + { + "type": "text", + "bbox": [ + 113, + 459, + 382, + 470 + ], + "lines": [ + { + "bbox": [ + 110, + 457, + 383, + 473 + ], + "spans": [ + { + "bbox": [ + 110, + 457, + 383, + 473 + ], + "score": 1.0, + "content": "Following this paper, a coherent and closely related picture emerges:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 110, + 457, + 383, + 473 + ] + }, + { + "type": "list", + "bbox": [ + 130, + 474, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 129, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 129, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "1. Predictions made in early layers are more likely to be consistent than those made in later lay-", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 141, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "ers. Consistent predictions are likely to be correct and the expected accuracy of inconsistent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 496, + 309, + 508 + ], + "spans": [ + { + "bbox": [ + 141, + 496, + 309, + 508 + ], + "score": 1.0, + "content": "predictions is naturally low (Section 3.1).", + "type": "text" + } + ], + "index": 34, + "is_list_end_line": true + }, + { + "bbox": [ + 130, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 130, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "2. 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On average, deep neural networks exhibit wider input and output margins (common measures", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 543, + 497, + 556 + ], + "spans": [ + { + "bbox": [ + 141, + 543, + 497, + 556 + ], + "score": 1.0, + "content": "of “local simplicity”) in the vicinity of data with smaller prediction depths (Section 3.3).", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + } + ], + "index": 35, + "bbox_fs": [ + 128, + 473, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "Pertinence of example difficulty to topics in machine learning Curriculum Learning attempts", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "to treat hard examples differently from easy examples during training. Robustness to distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "shifts that change the relative frequencies of common and rare subgroups in the test set (which we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "have shown can have different forms of example difficulty) is important for ML Fairness. Methods", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 507, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 507, + 618 + ], + "score": 1.0, + "content": "developed to address heteroscedastic uncertainty typically address example difficulty as a one-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 507, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 507, + 629 + ], + "score": 1.0, + "content": "dimensional quantity. We expand upon the relevance of our work to these three topics in Appendix D.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 560, + 507, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "Limitations We believe that the results we report stem from a deep model’s representation, which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "is hierarchical by construction. We expect that the same results will therefore apply in larger models,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "larger datasets, and tasks other than image classification, but testing this remains as further work.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "Although we demonstrate that returning the results of a hidden k-NN can yield dramatic increases in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "accuracy for examples that are “ambiguous without their label”, we otherwise do not explore ways", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "to practically apply the insights we present. In particular, we expressly do not claim that all that is", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "required for good accuracy is to reduce the prediction depth: freezing later layers of the network", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 327, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 327, + 723 + ], + "score": 1.0, + "content": "would not be expected to result in good generalization.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 634, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 279, + 85 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 281, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 281, + 88 + ], + "score": 1.0, + "content": "Funding Transparency Statement", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 91, + 504, + 113 + ], + "lines": [ + { + "bbox": [ + 105, + 89, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 89, + 506, + 105 + ], + "score": 1.0, + "content": "This research was funded by, and undertaken at, Google. All calculations were performed using", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 101, + 245, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 245, + 115 + ], + "score": 1.0, + "content": "Google’s computer infrastructure.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 124, + 197, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 199, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 199, + 140 + ], + "score": 1.0, + "content": "Acknowledgment", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "We would like to thank Hanie Sedghi, Ilya Tolstikhin, Ibrahim Alabdulmohsin, Daniel Keysers and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "Julian Eisenschlos for valuable discussions on the topic and Arthur Baldock for proofreading the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 166, + 155, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 155, + 179 + ], + "score": 1.0, + "content": "manuscript.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 188, + 163, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 165, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 165, + 203 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 104, + 200, + 507, + 724 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "Agarwal, C. and Hooker, S. 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In general, training or fine-tuning a state-of-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 469, + 379 + ], + "score": 1.0, + "content": "the-art deep model on a new domain requires a significant amount of data, which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 377, + 469, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 469, + 391 + ], + "score": 1.0, + "content": "for many applications is simply not available. Transfer of models directly to new", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 470, + 402 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 402 + ], + "score": 1.0, + "content": "domains without adaptation has historically led to poor recognition performance.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 399, + 469, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 469, + 412 + ], + "score": 1.0, + "content": "In this paper, we pose the following question: is a single image dataset, much", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 411, + 469, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 411, + 469, + 423 + ], + "score": 1.0, + "content": "larger than previously explored for adaptation, comprehensive enough to learn", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 422, + 470, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 422, + 470, + 434 + ], + "score": 1.0, + "content": "general deep models that may be effectively applied to new image domains? In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "score": 1.0, + "content": "other words, are deep CNNs trained on large amounts of labeled data as suscep-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 444, + 470, + 456 + ], + "spans": [ + { + "bbox": [ + 142, + 444, + 470, + 456 + ], + "score": 1.0, + "content": "tible to dataset bias as previous methods have been shown to be? We show that a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 454, + 470, + 467 + ], + "spans": [ + { + "bbox": [ + 141, + 454, + 470, + 467 + ], + "score": 1.0, + "content": "generic supervised deep CNN model trained on a large dataset reduces, but does", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 466, + 469, + 478 + ], + "spans": [ + { + "bbox": [ + 141, + 466, + 469, + 478 + ], + "score": 1.0, + "content": "not remove, dataset bias. 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Our experiments show that adaptation of deep", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 498, + 469, + 511 + ], + "spans": [ + { + "bbox": [ + 141, + 498, + 469, + 511 + ], + "score": 1.0, + "content": "models on benchmark visual domain adaptation datasets can provide a significant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 510, + 222, + 520 + ], + "spans": [ + { + "bbox": [ + 141, + 510, + 222, + 520 + ], + "score": 1.0, + "content": "performance boost.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 23.5, + "bbox_fs": [ + 141, + 324, + 470, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 540, + 190, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 192, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 192, + 556 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "score": 1.0, + "content": "Supervised deep convolutional neural networks (CNNs) trained on large-scale classification tasks", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "have been shown to learn impressive mid-level structures and obtain high levels of performance on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 586, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 601 + ], + "score": 1.0, + "content": "contemporary classification challenges [3, 23]. These models generally assume extensive training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "using labeled data, and testing is limited to data from the same domain. In practice, however, the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "images we would like to classify are often produced under different imaging conditions or drawn", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "from a different distribution, leading to a domain shift. Scaling such models to new domains remains", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 631, + 184, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 184, + 646 + ], + "score": 1.0, + "content": "an open challenge.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 566, + 505, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "Deep CNNs require large amounts of training data to learn good mid-level convolutional models", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "and final fully-connected classifier stages. While the continuing expansion of web-based datasets", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "like ImageNet [3] promises to produce labeled data for almost any desired category, such large-scale", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "supervised datasets may not include images of the category across all domains of practical interest.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 505, + 705 + ], + "score": 1.0, + "content": "Earlier deep learning efforts addressed this challenge by learning layers in an unsupervised fashion", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 702, + 506, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 717 + ], + "score": 1.0, + "content": "using unlabeled data to discover salient mid-level structures [6, 8]. While such approaches are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "appealing, they have heretofore been unable to match the level of performance of supervised models,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 490, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 490, + 106 + ], + "score": 1.0, + "content": "and unsupervised training of networks with the same level of depth as [17] remains a challenge.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 648, + 506, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "appealing, they have heretofore been unable to match the level of performance of supervised models,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 490, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 490, + 106 + ], + "score": 1.0, + "content": "and unsupervised training of networks with the same level of depth as [17] remains a challenge.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "score": 1.0, + "content": "Unfortunately, image datasets are inherently biased [21]. Theoretical [2, 4] and practical results from", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "score": 1.0, + "content": "[20, 21] have shown that supervised methods’ test error increases in proportion to the difference", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "between the test and training input distribution. Many visual domain adaptation methods have been", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "put forth to compensate for dataset bias [7, 22, 1, 20, 18, 16, 13, 12, 14, 15], but are limited to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "shallow models. Evaluation for image category classification across visually distinct domains has", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "focused on the Office dataset, which contains 31 image categories and 3 domains [20]. 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However, [9] limited their experiments to small-scale source domains found", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 364, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 364, + 222 + ], + "score": 1.0, + "content": "only in Office, and evaluated on only a subset of relevant layers.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Yet until now, almost none of the previous domain adaptation studies used ImageNet as the source", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "domain, nor utilized the full set of parameters of a deep CNN trained on source data. Recent work by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "Rodner et al. [19] attempted to adapt from ImageNet to the SUN dataset, but did not take advantage", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 232, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 232, + 271 + ], + "score": 1.0, + "content": "of deep convolutional features.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "In this paper, we ask the question: will deep models still suffer from dataset bias when trained with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "all layers of the CNN and a truly large scale source dataset? Here, we provide the first evaluation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "of domain adaptation with deep learned representations in its most natural setting, in which all of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "ImageNet is used as source data for a target category. We use the 1.2 million labeled images available", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "in the 2012 ImageNet 1000-way classification dataset [3] to train the model in [17] and evaluate its", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "generalization to the Office dataset. This constitutes a three orders of magnitude increase in source", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 459, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 459, + 354 + ], + "score": 1.0, + "content": "data compared to the several thousand images available for the largest domain in Office.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "We find that it is easier to adapt from ImageNet than from previous smaller source domains, but", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "that dataset bias remains a major issue. Fine-tuning the parameters on the small amount of labeled", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "target data (we consider one-shot adaptation) turns out to be unsurprisingly problematic. Instead, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "propose a simple yet intuitive adaptation method: train a final domain-adapted classification “layer”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "using various layers of the pre-trained network as features, without any fine-tuning its parameters.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "We provide a comprehensive evaluation of existing methods for classifier adaptation as applied to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "each of the fully connected layers of the network, including the last, task-specific classification", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 436, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 447 + ], + "score": 1.0, + "content": "layer. When adapting from ImageNet to Office, it turns out to be possible to achieve target domain", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "performance on par with source domain performance using only a single labeled example per target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 145, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 145, + 471 + ], + "score": 1.0, + "content": "category.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 504, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 504, + 487 + ], + "score": 1.0, + "content": "We examine both the setting where there are a few labeled examples from the target domain (super-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "vised adaptation) and the setting where there are no labeled target examples (unsupervised adap-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 497, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 508 + ], + "score": 1.0, + "content": "tation). We also describe practical solutions for choosing between the various adaptation methods", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 507, + 381, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 381, + 519 + ], + "score": 1.0, + "content": "based on experimental constraints such as limited computation time.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 390, + 546 + ], + "lines": [ + { + "bbox": [ + 103, + 529, + 392, + 550 + ], + "spans": [ + { + "bbox": [ + 103, + 529, + 392, + 550 + ], + "score": 1.0, + "content": "2 Background: Deep Domain Adaptation Approaches", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "For our task we consider adapting between a large source domain and a target domain with few or", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "or no labeled examples. A typical approach to domain adaptation or transfer learning with deep", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "architectures is to take the representation learned via back-propagation on a large dataset, and then", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 504, + 600 + ], + "score": 1.0, + "content": "transfer the representation to a smaller dataset by fine-tuning, i.e. backpropagation at a lower learn-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "ing rate [11, 23]. However, fine-tuning requires an ample amount of labeled target data and so should", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "not be expected to work well when we consider the very sparse label condition, such as the one-shot", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "learning scenario we evaluate below, where we have just one labeled example per category in the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 166, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 166, + 644 + ], + "score": 1.0, + "content": "target domain.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "In fact, in our experiments under this setting, fine-tuning actually reduces performance. Specifically,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 175, + 672 + ], + "score": 1.0, + "content": "on the ImageNet", + "type": "text" + }, + { + "bbox": [ + 176, + 661, + 186, + 670 + ], + "score": 0.72, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "Webcam task reported in Section 4, using the final output layer as a predictor", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 231, + 684 + ], + "score": 1.0, + "content": "in the target domain received", + "type": "text" + }, + { + "bbox": [ + 231, + 671, + 251, + 681 + ], + "score": 0.87, + "content": "6 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "accuracy, while using the final output layer after fine tuning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 682, + 264, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 240, + 694 + ], + "score": 1.0, + "content": "produced a degraded accuracy of", + "type": "text" + }, + { + "bbox": [ + 241, + 682, + 259, + 692 + ], + "score": 0.87, + "content": "61 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 682, + 264, + 694 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "A separate method that was recently proposed for deep adaptation is called Deep Learning for do-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "main adaptation by Interpolating between Domains (DLID) [5]. This method learns multiple unsu-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "pervised deep models directly on the source, target, and combined datasets and uses a representation", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 506, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "score": 1.0, + "content": "Unfortunately, image datasets are inherently biased [21]. Theoretical [2, 4] and practical results from", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "score": 1.0, + "content": "[20, 21] have shown that supervised methods’ test error increases in proportion to the difference", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "between the test and training input distribution. Many visual domain adaptation methods have been", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "put forth to compensate for dataset bias [7, 22, 1, 20, 18, 16, 13, 12, 14, 15], but are limited to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "shallow models. Evaluation for image category classification across visually distinct domains has", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "focused on the Office dataset, which contains 31 image categories and 3 domains [20]. Recently, [9]", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "showed that using the deep mid-level features learned on ImageNet, instead of the more conventional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "bag-of-words features, effectively removed the bias in some of the domain adaptation settings in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "Office dataset [20]. However, [9] limited their experiments to small-scale source domains found", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 364, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 364, + 222 + ], + "score": 1.0, + "content": "only in Office, and evaluated on only a subset of relevant layers.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 111, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Yet until now, almost none of the previous domain adaptation studies used ImageNet as the source", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "domain, nor utilized the full set of parameters of a deep CNN trained on source data. Recent work by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "Rodner et al. [19] attempted to adapt from ImageNet to the SUN dataset, but did not take advantage", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 232, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 232, + 271 + ], + "score": 1.0, + "content": "of deep convolutional features.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 225, + 506, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "In this paper, we ask the question: will deep models still suffer from dataset bias when trained with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "all layers of the CNN and a truly large scale source dataset? Here, we provide the first evaluation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "of domain adaptation with deep learned representations in its most natural setting, in which all of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "ImageNet is used as source data for a target category. We use the 1.2 million labeled images available", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "in the 2012 ImageNet 1000-way classification dataset [3] to train the model in [17] and evaluate its", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "generalization to the Office dataset. This constitutes a three orders of magnitude increase in source", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 459, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 459, + 354 + ], + "score": 1.0, + "content": "data compared to the several thousand images available for the largest domain in Office.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 275, + 506, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "We find that it is easier to adapt from ImageNet than from previous smaller source domains, but", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "that dataset bias remains a major issue. Fine-tuning the parameters on the small amount of labeled", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "target data (we consider one-shot adaptation) turns out to be unsurprisingly problematic. Instead, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "propose a simple yet intuitive adaptation method: train a final domain-adapted classification “layer”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "using various layers of the pre-trained network as features, without any fine-tuning its parameters.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "We provide a comprehensive evaluation of existing methods for classifier adaptation as applied to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "each of the fully connected layers of the network, including the last, task-specific classification", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 436, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 447 + ], + "score": 1.0, + "content": "layer. When adapting from ImageNet to Office, it turns out to be possible to achieve target domain", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "performance on par with source domain performance using only a single labeled example per target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 145, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 145, + 471 + ], + "score": 1.0, + "content": "category.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 358, + 506, + 471 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 504, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 504, + 487 + ], + "score": 1.0, + "content": "We examine both the setting where there are a few labeled examples from the target domain (super-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "vised adaptation) and the setting where there are no labeled target examples (unsupervised adap-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 497, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 508 + ], + "score": 1.0, + "content": "tation). We also describe practical solutions for choosing between the various adaptation methods", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 507, + 381, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 381, + 519 + ], + "score": 1.0, + "content": "based on experimental constraints such as limited computation time.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 473, + 505, + 519 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 390, + 546 + ], + "lines": [ + { + "bbox": [ + 103, + 529, + 392, + 550 + ], + "spans": [ + { + "bbox": [ + 103, + 529, + 392, + 550 + ], + "score": 1.0, + "content": "2 Background: Deep Domain Adaptation Approaches", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "For our task we consider adapting between a large source domain and a target domain with few or", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "or no labeled examples. A typical approach to domain adaptation or transfer learning with deep", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "architectures is to take the representation learned via back-propagation on a large dataset, and then", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 504, + 600 + ], + "score": 1.0, + "content": "transfer the representation to a smaller dataset by fine-tuning, i.e. backpropagation at a lower learn-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "ing rate [11, 23]. However, fine-tuning requires an ample amount of labeled target data and so should", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "not be expected to work well when we consider the very sparse label condition, such as the one-shot", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "learning scenario we evaluate below, where we have just one labeled example per category in the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 166, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 166, + 644 + ], + "score": 1.0, + "content": "target domain.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 554, + 506, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "In fact, in our experiments under this setting, fine-tuning actually reduces performance. Specifically,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 175, + 672 + ], + "score": 1.0, + "content": "on the ImageNet", + "type": "text" + }, + { + "bbox": [ + 176, + 661, + 186, + 670 + ], + "score": 0.72, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "Webcam task reported in Section 4, using the final output layer as a predictor", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 231, + 684 + ], + "score": 1.0, + "content": "in the target domain received", + "type": "text" + }, + { + "bbox": [ + 231, + 671, + 251, + 681 + ], + "score": 0.87, + "content": "6 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "accuracy, while using the final output layer after fine tuning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 682, + 264, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 240, + 694 + ], + "score": 1.0, + "content": "produced a degraded accuracy of", + "type": "text" + }, + { + "bbox": [ + 241, + 682, + 259, + 692 + ], + "score": 0.87, + "content": "61 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 682, + 264, + 694 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 648, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "A separate method that was recently proposed for deep adaptation is called Deep Learning for do-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "main adaptation by Interpolating between Domains (DLID) [5]. This method learns multiple unsu-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "pervised deep models directly on the source, target, and combined datasets and uses a representation", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "which is the concatenation of the outputs of each model as its adaptation approach. While this was", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 474, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 474, + 106 + ], + "score": 1.0, + "content": "shown to be an interesting approach, it is limited by its use of unsupervised deep structures.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 698, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "which is the concatenation of the outputs of each model as its adaptation approach. While this was", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 474, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 474, + 106 + ], + "score": 1.0, + "content": "shown to be an interesting approach, it is limited by its use of unsupervised deep structures.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "score": 1.0, + "content": "In general, unsupervised deep convolutional models have been unable to achieve the performance of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "supervised deep CNNs. However, training a supervised deep model requires sufficient labeled data.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "Our insight is that the extensive labeled data available in the source domain can be exploited using", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 430, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 430, + 156 + ], + "score": 1.0, + "content": "a supervised model without requiring a significant amount of labeled target data.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 174 + ], + "score": 1.0, + "content": "Therefore, we propose using a supervised deep source model with supervised or unsupervised adap-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "score": 1.0, + "content": "tation algorithms that are applied to models learned on the target data directly. This hybrid approach", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "will utilize the strong representation available from the supervised deep model trained on a large", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "score": 1.0, + "content": "source dataset while requiring only enough target labeled data to train a shallow model with far", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "fewer parameters. Specifically, we consider training a convolutional neural network (CNN) on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "source domain and using that network to extract features on the target data that can then be used to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "train an auxiliary shallow learner. For extracting features from the deep source model, we follow", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 361, + 250 + ], + "score": 1.0, + "content": "the setup of Donahue et al. [9], which extracts a visual feature", + "type": "text" + }, + { + "bbox": [ + 361, + 237, + 393, + 247 + ], + "score": 0.39, + "content": "D e C A F", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "from the ImageNet-trained", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 188, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 188, + 260 + ], + "score": 1.0, + "content": "architecture of [17].", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 270, + 420, + 284 + ], + "lines": [ + { + "bbox": [ + 104, + 269, + 420, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 420, + 288 + ], + "score": 1.0, + "content": "3 Adapting Deep CNNs with Few Labeled Target Examples", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 292, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "We propose a general framework for selectively adapting the parameters of a convolutional neural", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "network (CNN) whose representation and classifier weights are trained on a large-scale source do-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "main, such as ImageNet. Our framework adds a final domain-adaptive classification “layer” that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "takes the activations of one of the existing network’s layers as input features. Note that the net-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "work cannot be effectively fine-tuned without access to more labeled target data. This adapted layer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "is a linear classifier that combines source and target training data using an adaptation method. To", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "demonstrate the generality of our framework, we select a representative set of popular linear clas-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "sifier adaptation approaches that we empirically evaluate in Section 4. We separate our discussion", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 359, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 359, + 394 + ], + "score": 1.0, + "content": "into the set of supervised and unsupervised adaptation settings.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 397, + 503, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 369, + 409 + ], + "score": 1.0, + "content": "Below we denote the features extracted over the source domain as", + "type": "text" + }, + { + "bbox": [ + 369, + 398, + 380, + 408 + ], + "score": 0.79, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "and the features extracted over", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 189, + 422 + ], + "score": 1.0, + "content": "the target domain as", + "type": "text" + }, + { + "bbox": [ + 189, + 407, + 200, + 418 + ], + "score": 0.85, + "content": "\\tilde { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 407, + 440, + 422 + ], + "score": 1.0, + "content": ". Similarly, we denote the source domain image classifier as", + "type": "text" + }, + { + "bbox": [ + 440, + 409, + 447, + 418 + ], + "score": 0.76, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 407, + 506, + 422 + ], + "score": 1.0, + "content": "and the target", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 419, + 226, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 214, + 433 + ], + "score": 1.0, + "content": "domain image classifier as", + "type": "text" + }, + { + "bbox": [ + 214, + 419, + 222, + 431 + ], + "score": 0.83, + "content": "\\tilde { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 420, + 226, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 107, + 450, + 239, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 240, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 240, + 464 + ], + "score": 1.0, + "content": "3.1 Unsupervised Adaptation", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 504, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "Many unsupervised adaptation techniques seek to minimize the distance between subspaces that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 483, + 482, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 390, + 497 + ], + "score": 1.0, + "content": "represent the source and target domains. We denote these subspaces as", + "type": "text" + }, + { + "bbox": [ + 390, + 484, + 400, + 495 + ], + "score": 0.77, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 484, + 417, + 497 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 483, + 427, + 495 + ], + "score": 0.84, + "content": "\\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 484, + 482, + 497 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "GFK [12] The Geodesic Flow Kernel (GFK) method [12] is an unsupervised domain adaptation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "approach which seeks embeddings for the source and target points that minimize domain shift.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 205, + 538 + ], + "score": 1.0, + "content": "Inputs to the method are", + "type": "text" + }, + { + "bbox": [ + 205, + 524, + 215, + 535 + ], + "score": 0.77, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 523, + 233, + 538 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 233, + 523, + 242, + 535 + ], + "score": 0.84, + "content": "\\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 523, + 505, + 538 + ], + "score": 1.0, + "content": ", lower-dimensional embeddings of the source and target domains", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 459, + 550 + ], + "score": 1.0, + "content": "(e.g. from principal component analysis). The method constructs the geodesic flow", + "type": "text" + }, + { + "bbox": [ + 459, + 536, + 478, + 548 + ], + "score": 0.92, + "content": "\\phi ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 534, + 506, + 550 + ], + "score": 1.0, + "content": "along", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 258, + 561 + ], + "score": 1.0, + "content": "the manifold of subspaces such that", + "type": "text" + }, + { + "bbox": [ + 259, + 548, + 304, + 560 + ], + "score": 0.92, + "content": "U = \\phi ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 548, + 325, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 325, + 547, + 371, + 560 + ], + "score": 0.93, + "content": "\\tilde { U } = \\phi ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 548, + 484, + 561 + ], + "score": 1.0, + "content": ". Finally, a transformation", + "type": "text" + }, + { + "bbox": [ + 484, + 549, + 493, + 558 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 214, + 575 + ], + "score": 1.0, + "content": "constructed by computing", + "type": "text" + }, + { + "bbox": [ + 214, + 559, + 299, + 575 + ], + "score": 0.94, + "content": "\\begin{array} { r } { G = \\int _ { 0 } ^ { 1 } \\phi ( t ) \\phi ( t ) ^ { \\intercal } d t } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "using a closed-form solution, and classification is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 454, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 307, + 588 + ], + "score": 1.0, + "content": "performed by training an SVM on the source data", + "type": "text" + }, + { + "bbox": [ + 307, + 576, + 318, + 586 + ], + "score": 0.81, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 575, + 431, + 588 + ], + "score": 1.0, + "content": "and transformed target data", + "type": "text" + }, + { + "bbox": [ + 432, + 574, + 450, + 586 + ], + "score": 0.89, + "content": "G \\tilde { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 575, + 454, + 588 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "SA [10] The Subspace Alignment (SA) method [10] also begins with low-dimensional em-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 287, + 623 + ], + "score": 1.0, + "content": "beddings of the source and target domains", + "type": "text" + }, + { + "bbox": [ + 287, + 610, + 296, + 620 + ], + "score": 0.81, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 609, + 317, + 623 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 609, + 326, + 620 + ], + "score": 0.83, + "content": "\\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 609, + 489, + 623 + ], + "score": 1.0, + "content": ", respectively. It seeks to minimize in", + "type": "text" + }, + { + "bbox": [ + 489, + 610, + 501, + 620 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 609, + 505, + 623 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 268, + 635 + ], + "score": 1.0, + "content": "a transformation matrix, the objective", + "type": "text" + }, + { + "bbox": [ + 268, + 621, + 327, + 635 + ], + "score": 0.93, + "content": "\\| U M - \\tilde { U } \\| _ { F } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 621, + 505, + 635 + ], + "score": 1.0, + "content": ". The analytical solution to this objective", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 633, + 117, + 648 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 633, + 173, + 645 + ], + "score": 0.92, + "content": "M ^ { * } = U ^ { \\boldsymbol { \\mathsf { T } } } \\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 633, + 208, + 648 + ], + "score": 1.0, + "content": ". Given", + "type": "text" + }, + { + "bbox": [ + 209, + 635, + 225, + 645 + ], + "score": 0.87, + "content": "M ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 633, + 375, + 648 + ], + "score": 1.0, + "content": ", an SVM is trained on source data", + "type": "text" + }, + { + "bbox": [ + 376, + 635, + 387, + 645 + ], + "score": 0.77, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "and transformed target data", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 646, + 158, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 154, + 658 + ], + "score": 0.88, + "content": "U M ^ { * } \\tilde { U } ^ { \\dagger } \\tilde { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 646, + 158, + 658 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 107, + 676, + 228, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 229, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 229, + 690 + ], + "score": 1.0, + "content": "3.2 Supervised Adaptation", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Late Fusion Perhaps the simplest supervised adaptation method is to independently train a source", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "and target classifier and combine the scores of the two to create a final scoring function. We call", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "score": 1.0, + "content": "this approach Late Fusion. It has been explored by many for a simple adaptation approach. Let us", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "score": 1.0, + "content": "In general, unsupervised deep convolutional models have been unable to achieve the performance of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "supervised deep CNNs. However, training a supervised deep model requires sufficient labeled data.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "Our insight is that the extensive labeled data available in the source domain can be exploited using", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 430, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 430, + 156 + ], + "score": 1.0, + "content": "a supervised model without requiring a significant amount of labeled target data.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 111, + 505, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 174 + ], + "score": 1.0, + "content": "Therefore, we propose using a supervised deep source model with supervised or unsupervised adap-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "score": 1.0, + "content": "tation algorithms that are applied to models learned on the target data directly. This hybrid approach", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "will utilize the strong representation available from the supervised deep model trained on a large", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "score": 1.0, + "content": "source dataset while requiring only enough target labeled data to train a shallow model with far", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "fewer parameters. Specifically, we consider training a convolutional neural network (CNN) on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "source domain and using that network to extract features on the target data that can then be used to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "train an auxiliary shallow learner. For extracting features from the deep source model, we follow", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 361, + 250 + ], + "score": 1.0, + "content": "the setup of Donahue et al. [9], which extracts a visual feature", + "type": "text" + }, + { + "bbox": [ + 361, + 237, + 393, + 247 + ], + "score": 0.39, + "content": "D e C A F", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "from the ImageNet-trained", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 188, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 188, + 260 + ], + "score": 1.0, + "content": "architecture of [17].", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 159, + 506, + 260 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 270, + 420, + 284 + ], + "lines": [ + { + "bbox": [ + 104, + 269, + 420, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 420, + 288 + ], + "score": 1.0, + "content": "3 Adapting Deep CNNs with Few Labeled Target Examples", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 292, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "We propose a general framework for selectively adapting the parameters of a convolutional neural", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "network (CNN) whose representation and classifier weights are trained on a large-scale source do-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "main, such as ImageNet. Our framework adds a final domain-adaptive classification “layer” that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "takes the activations of one of the existing network’s layers as input features. Note that the net-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "work cannot be effectively fine-tuned without access to more labeled target data. This adapted layer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "is a linear classifier that combines source and target training data using an adaptation method. To", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "demonstrate the generality of our framework, we select a representative set of popular linear clas-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "sifier adaptation approaches that we empirically evaluate in Section 4. We separate our discussion", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 359, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 359, + 394 + ], + "score": 1.0, + "content": "into the set of supervised and unsupervised adaptation settings.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 293, + 506, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 397, + 503, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 369, + 409 + ], + "score": 1.0, + "content": "Below we denote the features extracted over the source domain as", + "type": "text" + }, + { + "bbox": [ + 369, + 398, + 380, + 408 + ], + "score": 0.79, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "and the features extracted over", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 189, + 422 + ], + "score": 1.0, + "content": "the target domain as", + "type": "text" + }, + { + "bbox": [ + 189, + 407, + 200, + 418 + ], + "score": 0.85, + "content": "\\tilde { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 407, + 440, + 422 + ], + "score": 1.0, + "content": ". Similarly, we denote the source domain image classifier as", + "type": "text" + }, + { + "bbox": [ + 440, + 409, + 447, + 418 + ], + "score": 0.76, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 407, + 506, + 422 + ], + "score": 1.0, + "content": "and the target", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 419, + 226, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 214, + 433 + ], + "score": 1.0, + "content": "domain image classifier as", + "type": "text" + }, + { + "bbox": [ + 214, + 419, + 222, + 431 + ], + "score": 0.83, + "content": "\\tilde { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 420, + 226, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 397, + 506, + 433 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 450, + 239, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 240, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 240, + 464 + ], + "score": 1.0, + "content": "3.1 Unsupervised Adaptation", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 504, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "Many unsupervised adaptation techniques seek to minimize the distance between subspaces that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 483, + 482, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 390, + 497 + ], + "score": 1.0, + "content": "represent the source and target domains. We denote these subspaces as", + "type": "text" + }, + { + "bbox": [ + 390, + 484, + 400, + 495 + ], + "score": 0.77, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 484, + 417, + 497 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 483, + 427, + 495 + ], + "score": 0.84, + "content": "\\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 484, + 482, + 497 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 471, + 505, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "GFK [12] The Geodesic Flow Kernel (GFK) method [12] is an unsupervised domain adaptation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "approach which seeks embeddings for the source and target points that minimize domain shift.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 205, + 538 + ], + "score": 1.0, + "content": "Inputs to the method are", + "type": "text" + }, + { + "bbox": [ + 205, + 524, + 215, + 535 + ], + "score": 0.77, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 523, + 233, + 538 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 233, + 523, + 242, + 535 + ], + "score": 0.84, + "content": "\\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 523, + 505, + 538 + ], + "score": 1.0, + "content": ", lower-dimensional embeddings of the source and target domains", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 459, + 550 + ], + "score": 1.0, + "content": "(e.g. from principal component analysis). The method constructs the geodesic flow", + "type": "text" + }, + { + "bbox": [ + 459, + 536, + 478, + 548 + ], + "score": 0.92, + "content": "\\phi ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 534, + 506, + 550 + ], + "score": 1.0, + "content": "along", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 258, + 561 + ], + "score": 1.0, + "content": "the manifold of subspaces such that", + "type": "text" + }, + { + "bbox": [ + 259, + 548, + 304, + 560 + ], + "score": 0.92, + "content": "U = \\phi ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 548, + 325, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 325, + 547, + 371, + 560 + ], + "score": 0.93, + "content": "\\tilde { U } = \\phi ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 548, + 484, + 561 + ], + "score": 1.0, + "content": ". Finally, a transformation", + "type": "text" + }, + { + "bbox": [ + 484, + 549, + 493, + 558 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 214, + 575 + ], + "score": 1.0, + "content": "constructed by computing", + "type": "text" + }, + { + "bbox": [ + 214, + 559, + 299, + 575 + ], + "score": 0.94, + "content": "\\begin{array} { r } { G = \\int _ { 0 } ^ { 1 } \\phi ( t ) \\phi ( t ) ^ { \\intercal } d t } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "using a closed-form solution, and classification is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 454, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 307, + 588 + ], + "score": 1.0, + "content": "performed by training an SVM on the source data", + "type": "text" + }, + { + "bbox": [ + 307, + 576, + 318, + 586 + ], + "score": 0.81, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 575, + 431, + 588 + ], + "score": 1.0, + "content": "and transformed target data", + "type": "text" + }, + { + "bbox": [ + 432, + 574, + 450, + 586 + ], + "score": 0.89, + "content": "G \\tilde { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 575, + 454, + 588 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 502, + 506, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "SA [10] The Subspace Alignment (SA) method [10] also begins with low-dimensional em-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 287, + 623 + ], + "score": 1.0, + "content": "beddings of the source and target domains", + "type": "text" + }, + { + "bbox": [ + 287, + 610, + 296, + 620 + ], + "score": 0.81, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 609, + 317, + 623 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 609, + 326, + 620 + ], + "score": 0.83, + "content": "\\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 609, + 489, + 623 + ], + "score": 1.0, + "content": ", respectively. It seeks to minimize in", + "type": "text" + }, + { + "bbox": [ + 489, + 610, + 501, + 620 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 609, + 505, + 623 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 268, + 635 + ], + "score": 1.0, + "content": "a transformation matrix, the objective", + "type": "text" + }, + { + "bbox": [ + 268, + 621, + 327, + 635 + ], + "score": 0.93, + "content": "\\| U M - \\tilde { U } \\| _ { F } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 621, + 505, + 635 + ], + "score": 1.0, + "content": ". The analytical solution to this objective", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 633, + 117, + 648 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 633, + 173, + 645 + ], + "score": 0.92, + "content": "M ^ { * } = U ^ { \\boldsymbol { \\mathsf { T } } } \\tilde { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 633, + 208, + 648 + ], + "score": 1.0, + "content": ". Given", + "type": "text" + }, + { + "bbox": [ + 209, + 635, + 225, + 645 + ], + "score": 0.87, + "content": "M ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 633, + 375, + 648 + ], + "score": 1.0, + "content": ", an SVM is trained on source data", + "type": "text" + }, + { + "bbox": [ + 376, + 635, + 387, + 645 + ], + "score": 0.77, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "and transformed target data", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 646, + 158, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 154, + 658 + ], + "score": 0.88, + "content": "U M ^ { * } \\tilde { U } ^ { \\dagger } \\tilde { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 646, + 158, + 658 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 597, + 506, + 658 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 676, + 228, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 229, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 229, + 690 + ], + "score": 1.0, + "content": "3.2 Supervised Adaptation", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Late Fusion Perhaps the simplest supervised adaptation method is to independently train a source", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "and target classifier and combine the scores of the two to create a final scoring function. We call", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "score": 1.0, + "content": "this approach Late Fusion. It has been explored by many for a simple adaptation approach. Let us", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 291, + 95 + ], + "score": 1.0, + "content": "denote the score from the source classifier as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 291, + 84, + 302, + 93 + ], + "score": 0.85, + "content": "v _ { s }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 302, + 82, + 473, + 95 + ], + "score": 1.0, + "content": "and the score from the target classifier as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 473, + 84, + 483, + 93 + ], + "score": 0.84, + "content": "v _ { t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 483, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". For", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 493, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 493, + 106 + ], + "score": 1.0, + "content": "our experiments we explore two methods of combining these scores, which are described below:", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 698, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 291, + 95 + ], + "score": 1.0, + "content": "denote the score from the source classifier as", + "type": "text" + }, + { + "bbox": [ + 291, + 84, + 302, + 93 + ], + "score": 0.85, + "content": "v _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 82, + 473, + 95 + ], + "score": 1.0, + "content": "and the score from the target classifier as", + "type": "text" + }, + { + "bbox": [ + 473, + 84, + 483, + 93 + ], + "score": 0.84, + "content": "v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 493, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 493, + 106 + ], + "score": 1.0, + "content": "our experiments we explore two methods of combining these scores, which are described below:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 131, + 114, + 505, + 184 + ], + "lines": [ + { + "bbox": [ + 132, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 132, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "• Max: Produce the scores of both the source and target classifier and simply choose the max", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 140, + 125, + 460, + 139 + ], + "spans": [ + { + "bbox": [ + 140, + 125, + 372, + 139 + ], + "score": 1.0, + "content": "of the two as the final score for each example. Therefore,", + "type": "text" + }, + { + "bbox": [ + 373, + 126, + 456, + 138 + ], + "score": 0.92, + "content": "v _ { \\mathrm { a d a p t } } = \\operatorname* { m a x } ( v _ { s } , v _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 125, + 460, + 139 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 131, + 139, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 131, + 139, + 505, + 153 + ], + "score": 1.0, + "content": "• Linear Interpolation: Set the score for a particular example to equal the convex combi-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 140, + 150, + 507, + 165 + ], + "spans": [ + { + "bbox": [ + 140, + 150, + 339, + 165 + ], + "score": 1.0, + "content": "nation of the source and target classifier scores,", + "type": "text" + }, + { + "bbox": [ + 339, + 151, + 446, + 163 + ], + "score": 0.93, + "content": "v _ { \\mathrm { a d a p t } } = ( 1 - \\alpha ) v _ { s } + \\alpha v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 150, + 507, + 165 + ], + "score": 1.0, + "content": ". This method", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 163, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 141, + 163, + 279, + 174 + ], + "score": 1.0, + "content": "requires setting a hyperparameter,", + "type": "text" + }, + { + "bbox": [ + 279, + 164, + 286, + 172 + ], + "score": 0.69, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 163, + 505, + 174 + ], + "score": 1.0, + "content": ", which determines the weights of the source and target", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 172, + 186, + 185 + ], + "spans": [ + { + "bbox": [ + 142, + 172, + 186, + 185 + ], + "score": 1.0, + "content": "classifiers.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 194, + 505, + 238 + ], + "lines": [ + { + "bbox": [ + 104, + 193, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 193, + 505, + 207 + ], + "score": 1.0, + "content": "Late Fusion has two major advantages: it is easy to implement, and the source classifier it uses may", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "be precomputed to make adaptation very fast. In the case of the linear interpolation combination", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 228 + ], + "score": 1.0, + "content": "rule, however, this method can potentially suffer from having a sensitive hyperparameter. We show", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 266, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 266, + 239 + ], + "score": 1.0, + "content": "a hyperparameter analysis in Section 4.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "Daume III [7] ´ This simple feature replication method was proposed for domain adaptation by [7].", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "The method augments feature vectors with a source component, a target component, and a shared", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 254, + 278 + ], + "score": 1.0, + "content": "component. Each source data point", + "type": "text" + }, + { + "bbox": [ + 254, + 268, + 262, + 276 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 266, + 331, + 278 + ], + "score": 1.0, + "content": "is augmented to", + "type": "text" + }, + { + "bbox": [ + 331, + 266, + 393, + 278 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { x ^ { \\prime } } } = ( \\mathbf { x } ; x ; \\mathbf { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 266, + 505, + 278 + ], + "score": 1.0, + "content": ", and each target data point", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 114, + 287 + ], + "score": 0.79, + "content": "\\tilde { \\pmb { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 277, + 183, + 289 + ], + "score": 1.0, + "content": "is augmented to", + "type": "text" + }, + { + "bbox": [ + 183, + 277, + 243, + 289 + ], + "score": 0.93, + "content": "\\tilde { \\pmb { x } } ^ { \\prime } = ( \\tilde { \\pmb { x } } ; \\mathbf { 0 } ; \\tilde { \\pmb { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 277, + 505, + 289 + ], + "score": 1.0, + "content": ". 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We show", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 266, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 266, + 239 + ], + "score": 1.0, + "content": "a hyperparameter analysis in Section 4.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 193, + 505, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "Daume III [7] ´ This simple feature replication method was proposed for domain adaptation by [7].", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "The method augments feature vectors with a source component, a target component, and a shared", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 254, + 278 + ], + "score": 1.0, + "content": "component. 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In our experiments using Amazon as a source domain, we follow", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "the standard training protocol for this dataset of using 20 source examples per category [20, 12], for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 691, + 195, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 195, + 705 + ], + "score": 1.0, + "content": "a total of 320 images.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 625, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "ImageNet [3] is the largest available dataset of image category labels. We use 1000 categories’", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "worth of data (1.2M images) to train the network, and use the 16 categories that overlap with Office", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 294, + 117 + ], + "score": 1.0, + "content": "(approximately 1200 examples per category or", + "type": "text" + }, + { + "bbox": [ + 294, + 105, + 321, + 115 + ], + "score": 0.84, + "content": "{ \\approx } 2 0 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "images total) as labeled source classifier data.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 131, + 268, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 270, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 270, + 145 + ], + "score": 1.0, + "content": "4.2 Experimental Setup & Baselines", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 151, + 504, + 185 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 506, + 165 + ], + "score": 1.0, + "content": "For our experiments, we use the fully trained deep CNN model described in Section 2, extracting", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "score": 1.0, + "content": "feature representations from three different layers of the CNN. We then train a source classifier using", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 414, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 414, + 186 + ], + "score": 1.0, + "content": "these features on one of two source domains, and adapt to the target domain.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 191, + 504, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 189, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 205 + ], + "score": 1.0, + "content": "The source domains we consider are either the Amazon domain, or the corresponding 16-category", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 202, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 506, + 214 + ], + "score": 1.0, + "content": "ImageNet subset where each category has many more examples. We focus on the Webcam domain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "as our target (test) domain, as Amazon-to-Webcam was shown to be the only challenging shift in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "score": 1.0, + "content": "[9] (the DSLR domain is much more similar to Webcam and did not require adaptation when using", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "deep mid-level features). This combination exemplifies the shift from online web images to real-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "world images taken in typical office/home environments. Note that, regardless of the source domain", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "chosen to learn the classifier, ImageNet data from all 1000 categories was used to train the network.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 273, + 503, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "score": 1.0, + "content": "In addition, for the supervised adaptation setting we assume access to only a single example per", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 283, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 283, + 297 + ], + "score": 1.0, + "content": "category from the target domain (Webcam).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "Each method is then evaluated across 20 random train/test splits, and we report averages and standard", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "errors for each setting. For each random train/test split we choose one example for training and 10", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "other examples for testing (so there is a balanced test set across categories). Therefore, each test", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "split has 160 examples. The unsupervised adaptation methods operate in a transductive setting, so", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 346, + 350, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 350, + 358 + ], + "score": 1.0, + "content": "the target subspaces are learned from the unlabeled test data.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 504, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "Non-adaptive Baselines In addition to the adaptation methods outlined in Section 3, we also", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 382, + 316, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 316, + 393 + ], + "score": 1.0, + "content": "evaluate using the following non-adaptive baselines.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 132, + 404, + 504, + 516 + ], + "lines": [ + { + "bbox": [ + 132, + 404, + 445, + 417 + ], + "spans": [ + { + "bbox": [ + 132, + 404, + 445, + 417 + ], + "score": 1.0, + "content": "• SVM (source only): A support vector machine trained only on source data.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 132, + 420, + 440, + 432 + ], + "spans": [ + { + "bbox": [ + 132, + 420, + 440, + 432 + ], + "score": 1.0, + "content": "• SVM (target only): A support vector machine trained only on target data.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 132, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 132, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "• SVM (source and target): A support vector machine trained on both source and target", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "data. To account for the large discrepancy between the number of training data points in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "the source and target domains, we weighted the data points such that the constraints from", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "the source and target domains effectively contribute equally to the optimization problem.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 141, + 480, + 369, + 493 + ], + "score": 1.0, + "content": "Specifically, each source data point receives a weight of", + "type": "text" + }, + { + "bbox": [ + 370, + 481, + 395, + 495 + ], + "score": 0.93, + "content": "\\frac { n _ { t } } { n _ { s } + n _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ", and each target data point", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 491, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 141, + 491, + 227, + 509 + ], + "score": 1.0, + "content": "receives a weight of", + "type": "text" + }, + { + "bbox": [ + 227, + 493, + 253, + 507 + ], + "score": 0.93, + "content": "\\frac { n _ { s } } { n _ { s } + n _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 491, + 256, + 509 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 257, + 493, + 285, + 506 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 286, + 495, + 311, + 505 + ], + "score": 0.9, + "content": "n _ { s } , n _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "denote the number of data points in the source", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 505, + 239, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 505, + 239, + 519 + ], + "score": 1.0, + "content": "and target, respectively.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 504, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Many of the adaptation methods we evaluate have hyperparameters that must be cross-validated for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 540, + 429, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 429, + 552 + ], + "score": 1.0, + "content": "use in practice, so we set the parameters of the adaptation techniques as follows.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 416, + 569 + ], + "score": 1.0, + "content": "First, the C value used for C-SVM in the classifier for all methods is set to", + "type": "text" + }, + { + "bbox": [ + 416, + 556, + 446, + 567 + ], + "score": 0.89, + "content": "C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 555, + 505, + 569 + ], + "score": 1.0, + "content": ". Without any", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "validation data we are not able to tune this parameter properly, so we choose to leave it as the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "default value. Since all methods we report require setting of this parameter, we feel that the relative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "comparisons between methods is sound even if the absolute numbers could be improved with a new", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "setting for C. For Daume III and MMDT, which look at the source and target data simultaneously, ´", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "we use the same weighting scheme as we did for the source and target SVM. Late Fusion with the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "linear interpolation combination rule is reported across hyperparameter settings in Figure 1(a) to help", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "understand how performance varies as we trade off emphasis between the learned classifiers from", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the source and target domains. Again, we do not have the validation data to tune this parameter so", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 458, + 667 + ], + "score": 1.0, + "content": "we report in the tables the performance averaged across parameter settings. The plot vs", + "type": "text" + }, + { + "bbox": [ + 458, + 657, + 466, + 665 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "indicates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "that there is usually a best parameter setting that could be learned with more available data. For", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 178, + 689 + ], + "score": 1.0, + "content": "PMT, we choose", + "type": "text" + }, + { + "bbox": [ + 178, + 677, + 223, + 687 + ], + "score": 0.9, + "content": "\\Gamma = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 676, + 505, + 689 + ], + "score": 1.0, + "content": ", which corresponds to allowing a large amount of transfer from the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "source classifier to the target classifier. We do this because the source-only classifier is stronger than", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "the target-only classifier (with ImageNet source). For the unsupervised methods GFK and SA, again", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "we evaluated a variety of subspace dimensionalities and Figure 1(b) shows that the overall method", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 394, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 394, + 733 + ], + "score": 1.0, + "content": "performance does not vary significantly with the dimensionality choice.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 41.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "ImageNet [3] is the largest available dataset of image category labels. We use 1000 categories’", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "worth of data (1.2M images) to train the network, and use the 16 categories that overlap with Office", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 294, + 117 + ], + "score": 1.0, + "content": "(approximately 1200 examples per category or", + "type": "text" + }, + { + "bbox": [ + 294, + 105, + 321, + 115 + ], + "score": 0.84, + "content": "{ \\approx } 2 0 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "images total) as labeled source classifier data.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 82, + 505, + 117 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 131, + 268, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 270, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 270, + 145 + ], + "score": 1.0, + "content": "4.2 Experimental Setup & Baselines", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 151, + 504, + 185 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 506, + 165 + ], + "score": 1.0, + "content": "For our experiments, we use the fully trained deep CNN model described in Section 2, extracting", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "score": 1.0, + "content": "feature representations from three different layers of the CNN. We then train a source classifier using", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 414, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 414, + 186 + ], + "score": 1.0, + "content": "these features on one of two source domains, and adapt to the target domain.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 151, + 506, + 186 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 191, + 504, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 189, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 205 + ], + "score": 1.0, + "content": "The source domains we consider are either the Amazon domain, or the corresponding 16-category", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 202, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 506, + 214 + ], + "score": 1.0, + "content": "ImageNet subset where each category has many more examples. We focus on the Webcam domain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "as our target (test) domain, as Amazon-to-Webcam was shown to be the only challenging shift in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "score": 1.0, + "content": "[9] (the DSLR domain is much more similar to Webcam and did not require adaptation when using", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "deep mid-level features). This combination exemplifies the shift from online web images to real-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "world images taken in typical office/home environments. Note that, regardless of the source domain", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "chosen to learn the classifier, ImageNet data from all 1000 categories was used to train the network.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 189, + 506, + 269 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 273, + 503, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "score": 1.0, + "content": "In addition, for the supervised adaptation setting we assume access to only a single example per", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 283, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 283, + 297 + ], + "score": 1.0, + "content": "category from the target domain (Webcam).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 272, + 505, + 297 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "Each method is then evaluated across 20 random train/test splits, and we report averages and standard", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "errors for each setting. For each random train/test split we choose one example for training and 10", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "other examples for testing (so there is a balanced test set across categories). Therefore, each test", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "split has 160 examples. The unsupervised adaptation methods operate in a transductive setting, so", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 346, + 350, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 350, + 358 + ], + "score": 1.0, + "content": "the target subspaces are learned from the unlabeled test data.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 302, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 504, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "Non-adaptive Baselines In addition to the adaptation methods outlined in Section 3, we also", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 382, + 316, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 316, + 393 + ], + "score": 1.0, + "content": "evaluate using the following non-adaptive baselines.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 369, + 505, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 404, + 504, + 516 + ], + "lines": [ + { + "bbox": [ + 132, + 404, + 445, + 417 + ], + "spans": [ + { + "bbox": [ + 132, + 404, + 445, + 417 + ], + "score": 1.0, + "content": "• SVM (source only): A support vector machine trained only on source data.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 132, + 420, + 440, + 432 + ], + "spans": [ + { + "bbox": [ + 132, + 420, + 440, + 432 + ], + "score": 1.0, + "content": "• SVM (target only): A support vector machine trained only on target data.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 132, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 132, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "• SVM (source and target): A support vector machine trained on both source and target", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "data. To account for the large discrepancy between the number of training data points in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "the source and target domains, we weighted the data points such that the constraints from", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "the source and target domains effectively contribute equally to the optimization problem.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 141, + 480, + 369, + 493 + ], + "score": 1.0, + "content": "Specifically, each source data point receives a weight of", + "type": "text" + }, + { + "bbox": [ + 370, + 481, + 395, + 495 + ], + "score": 0.93, + "content": "\\frac { n _ { t } } { n _ { s } + n _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ", and each target data point", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 491, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 141, + 491, + 227, + 509 + ], + "score": 1.0, + "content": "receives a weight of", + "type": "text" + }, + { + "bbox": [ + 227, + 493, + 253, + 507 + ], + "score": 0.93, + "content": "\\frac { n _ { s } } { n _ { s } + n _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 491, + 256, + 509 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 257, + 493, + 285, + 506 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 286, + 495, + 311, + 505 + ], + "score": 0.9, + "content": "n _ { s } , n _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "denote the number of data points in the source", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 505, + 239, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 505, + 239, + 519 + ], + "score": 1.0, + "content": "and target, respectively.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 132, + 404, + 505, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 504, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Many of the adaptation methods we evaluate have hyperparameters that must be cross-validated for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 540, + 429, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 429, + 552 + ], + "score": 1.0, + "content": "use in practice, so we set the parameters of the adaptation techniques as follows.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 528, + 505, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 416, + 569 + ], + "score": 1.0, + "content": "First, the C value used for C-SVM in the classifier for all methods is set to", + "type": "text" + }, + { + "bbox": [ + 416, + 556, + 446, + 567 + ], + "score": 0.89, + "content": "C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 555, + 505, + 569 + ], + "score": 1.0, + "content": ". Without any", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "validation data we are not able to tune this parameter properly, so we choose to leave it as the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "default value. Since all methods we report require setting of this parameter, we feel that the relative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "comparisons between methods is sound even if the absolute numbers could be improved with a new", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "setting for C. For Daume III and MMDT, which look at the source and target data simultaneously, ´", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "we use the same weighting scheme as we did for the source and target SVM. Late Fusion with the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "linear interpolation combination rule is reported across hyperparameter settings in Figure 1(a) to help", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "understand how performance varies as we trade off emphasis between the learned classifiers from", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the source and target domains. Again, we do not have the validation data to tune this parameter so", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 458, + 667 + ], + "score": 1.0, + "content": "we report in the tables the performance averaged across parameter settings. The plot vs", + "type": "text" + }, + { + "bbox": [ + 458, + 657, + 466, + 665 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "indicates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "that there is usually a best parameter setting that could be learned with more available data. For", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 178, + 689 + ], + "score": 1.0, + "content": "PMT, we choose", + "type": "text" + }, + { + "bbox": [ + 178, + 677, + 223, + 687 + ], + "score": 0.9, + "content": "\\Gamma = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 676, + 505, + 689 + ], + "score": 1.0, + "content": ", which corresponds to allowing a large amount of transfer from the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "source classifier to the target classifier. We do this because the source-only classifier is stronger than", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "the target-only classifier (with ImageNet source). For the unsupervised methods GFK and SA, again", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "we evaluated a variety of subspace dimensionalities and Figure 1(b) shows that the overall method", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 394, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 394, + 733 + ], + "score": 1.0, + "content": "performance does not vary significantly with the dimensionality choice.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 555, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 80, + 483, + 240 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 128, + 80, + 483, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 80, + 483, + 240 + ], + "spans": [ + { + "bbox": [ + 128, + 80, + 483, + 240 + ], + "score": 0.986, + "html": "
Adaptation MethodTraining DataDeCAF6DeCAF7
SVM (source only)Amazon50.28 ± 1.854.08 ± 1.7
SVM (target only)Webcam62.28 ± 1.864.97 ± 1.8
GFK[12]Amazon53.13 ±1.153.39 ± 1.1
SA [10]Amazon51.74 ± 1.253.86 ± 1.0
SVM (source and target)Amazon+Webcam62.91 ±1.865.82 ± 1.4
Late Fusion (Max)Amazon+Webcam65.35 ± 1.758.42 ± 1.1
Late Fusion (Lin. Int. Avg)Amazon+Webcam63.23 ± 1.464.29 ± 1.3
Daumé III [7]Amazon+Webcam68.89 ± 1.972.09 ± 1.4
PMT[1]Amazon+Webcam64.84 ± 1.565.63 ± 1.8
MMDT[15]Amazon+Webcam65.47 ± 1.868.10 ± 1.5
Late Fusion (Lin. Int. Oracle)Amazon+Webcam71.1 ± 1.772.82 ±1.4
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We show here multiclass accuracy on the target", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 258, + 504, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 504, + 272 + ], + "score": 1.0, + "content": "domain test set for both supervised and unsupervised adaptation experiments across the two fully", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 284 + ], + "score": 1.0, + "content": "connected layer features (similar to [9], but with one labeled target example). The best performing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "unsupervised adaptation algorithms are shown in blue and the best performing supervised adaptation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 291, + 222, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 222, + 304 + ], + "score": 1.0, + "content": "algorithms are shown in red.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 325, + 254, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 255, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 255, + 338 + ], + "score": 1.0, + "content": "4.3 Effect of Source Domain Size", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 503, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "Previous studies considered source domains from the Office dataset. In this section, we ask what", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 358, + 376, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 376, + 369 + ], + "score": 1.0, + "content": "happens when an orders-of-magnitute larger source dataset is used.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 504, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "For completeness we begin by evaluating Amazon as a source domain. Preliminary results on this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "setting are reported in [9], here we extend the comparison here by presenting the results with more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "adaptation algorithms and more complete evaluation of hyperparameter settings. Table 1 presents", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "multiclass accuracies for each algorithm using either layer 6 or 7 from the deep network, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 418, + 371, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 371, + 432 + ], + "score": 1.0, + "content": "corresponds to the output from each of the fully connected layers.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 314, + 447 + ], + "score": 1.0, + "content": "An SVM trained using only Amazon data achieves", + "type": "text" + }, + { + "bbox": [ + 314, + 435, + 342, + 446 + ], + "score": 0.87, + "content": "7 8 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "in-domain accuracy (tested on the same", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 446, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 206, + 459 + ], + "score": 1.0, + "content": "domain) when using the", + "type": "text" + }, + { + "bbox": [ + 207, + 447, + 243, + 458 + ], + "score": 0.89, + "content": "{ \\mathrm { D e C A F } } _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 446, + 292, + 459 + ], + "score": 1.0, + "content": "feature and", + "type": "text" + }, + { + "bbox": [ + 293, + 447, + 320, + 457 + ], + "score": 0.87, + "content": "8 0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 446, + 466, + 459 + ], + "score": 1.0, + "content": "in-domain accuracy when using the", + "type": "text" + }, + { + "bbox": [ + 467, + 447, + 504, + 458 + ], + "score": 0.87, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "feature. These numbers are significantly higher than the performance of the same classifier on", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Webcam test data, indicating that even with the DeCAF features, there is a still a domain shift", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 479, + 284, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 284, + 491 + ], + "score": 1.0, + "content": "between the Amazon and Webcam datasets.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Next, we consider an unsupervised adaptation setting where no labeled examples are available from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "the target dataset. In this scenario, we apply two state-of-the-art unsupervised adaptation methods,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "GFK [12] and SA [10]. Both of these methods make use of a subspace dimensionality hyperpa-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 528, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 542 + ], + "score": 1.0, + "content": "rameter. We show the results using a 100-dimensional subspace and leave the discussion of setting", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "this parameter until Section 4.6. For this shift the adaptation algorithms increase performance when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 550, + 480, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 480, + 564 + ], + "score": 1.0, + "content": "using the layer 6 feature, but offer no additional improvement when using the layer 7 feature.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "We finally assume that a single example per category is available in the target domain. As the bottom", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "rows of Table 1 show, supervised adaptation algorithms are able to provide significant improvement", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "regardless of the feature space chosen, even in the one-shot scenario. 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Adaptation MethodTraining DataDeCAF6DeCAF7
SVM (source only)Amazon50.28 ± 1.854.08 ± 1.7
SVM (target only)Webcam62.28 ± 1.864.97 ± 1.8
GFK[12]Amazon53.13 ±1.153.39 ± 1.1
SA [10]Amazon51.74 ± 1.253.86 ± 1.0
SVM (source and target)Amazon+Webcam62.91 ±1.865.82 ± 1.4
Late Fusion (Max)Amazon+Webcam65.35 ± 1.758.42 ± 1.1
Late Fusion (Lin. Int. Avg)Amazon+Webcam63.23 ± 1.464.29 ± 1.3
Daumé III [7]Amazon+Webcam68.89 ± 1.972.09 ± 1.4
PMT[1]Amazon+Webcam64.84 ± 1.565.63 ± 1.8
MMDT[15]Amazon+Webcam65.47 ± 1.868.10 ± 1.5
Late Fusion (Lin. Int. Oracle)Amazon+Webcam71.1 ± 1.772.82 ±1.4
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We show here multiclass accuracy on the target", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 258, + 504, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 504, + 272 + ], + "score": 1.0, + "content": "domain test set for both supervised and unsupervised adaptation experiments across the two fully", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 284 + ], + "score": 1.0, + "content": "connected layer features (similar to [9], but with one labeled target example). The best performing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "unsupervised adaptation algorithms are shown in blue and the best performing supervised adaptation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 291, + 222, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 222, + 304 + ], + "score": 1.0, + "content": "algorithms are shown in red.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 248, + 506, + 304 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 325, + 254, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 255, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 255, + 338 + ], + "score": 1.0, + "content": "4.3 Effect of Source Domain Size", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 503, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "Previous studies considered source domains from the Office dataset. In this section, we ask what", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 358, + 376, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 376, + 369 + ], + "score": 1.0, + "content": "happens when an orders-of-magnitute larger source dataset is used.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 346, + 505, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 504, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "For completeness we begin by evaluating Amazon as a source domain. Preliminary results on this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "setting are reported in [9], here we extend the comparison here by presenting the results with more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "adaptation algorithms and more complete evaluation of hyperparameter settings. Table 1 presents", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "multiclass accuracies for each algorithm using either layer 6 or 7 from the deep network, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 418, + 371, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 371, + 432 + ], + "score": 1.0, + "content": "corresponds to the output from each of the fully connected layers.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 375, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 314, + 447 + ], + "score": 1.0, + "content": "An SVM trained using only Amazon data achieves", + "type": "text" + }, + { + "bbox": [ + 314, + 435, + 342, + 446 + ], + "score": 0.87, + "content": "7 8 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "in-domain accuracy (tested on the same", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 446, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 206, + 459 + ], + "score": 1.0, + "content": "domain) when using the", + "type": "text" + }, + { + "bbox": [ + 207, + 447, + 243, + 458 + ], + "score": 0.89, + "content": "{ \\mathrm { D e C A F } } _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 446, + 292, + 459 + ], + "score": 1.0, + "content": "feature and", + "type": "text" + }, + { + "bbox": [ + 293, + 447, + 320, + 457 + ], + "score": 0.87, + "content": "8 0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 446, + 466, + 459 + ], + "score": 1.0, + "content": "in-domain accuracy when using the", + "type": "text" + }, + { + "bbox": [ + 467, + 447, + 504, + 458 + ], + "score": 0.87, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "feature. These numbers are significantly higher than the performance of the same classifier on", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Webcam test data, indicating that even with the DeCAF features, there is a still a domain shift", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 479, + 284, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 284, + 491 + ], + "score": 1.0, + "content": "between the Amazon and Webcam datasets.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 435, + 505, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Next, we consider an unsupervised adaptation setting where no labeled examples are available from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "the target dataset. 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Compare this to the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 266, + 550, + 286, + 560 + ], + "score": 0.87, + "content": "54 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 286, + 549, + 304, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 304, + 550, + 324, + 560 + ], + "score": 0.87, + "content": "59 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 325, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "for Webcam evaluation and a dataset bias is", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 560, + 187, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 187, + 572 + ], + "score": 1.0, + "content": "still clearly evident.", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 699, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 130, + 80, + 482, + 240 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 130, + 80, + 482, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 130, + 80, + 482, + 240 + ], + "spans": [ + { + "bbox": [ + 130, + 80, + 482, + 240 + ], + "score": 0.985, + "html": "
Adaptation MethodTraining DataDeCAF6DeCAF7
SVM (source only)ImageNet53.51 ± 1.159.15 ± 1.1
SVM (target only)Webcam62.28 ± 1.864.97 ± 1.8
GFK [12]ImageNet65.16 ± 1.167.97 ± 1.4
SA [10]ImageNet59.30 ± 1.466.08 ± 1.4
SVM (source and target)ImageNet+Webcam56.68 ± 1.266.93 ± 1.3
Late Fusion (Max)ImageNet+Webcam59.59 ± 1.368.86 ± 1.2
Late Fusion (Lin. Int. Avg)ImageNet+Webcam60.64 ± 1.366.45 ± 1.1
Daumé III [7]ImageNet+Webcam59.21 ± 1.771.39 ± 1.5
PMT[1]ImageNet+Webcam66.30 ± 2.169.81 ± 1.8
MMDT[15]ImageNet+Webcam59.21 ± 1.367.75 ± 1.4
Late Fusion (Lin. Int. Oracle)ImageNet+Webcam71.65 ± 2.076.76 ± 1.3
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Adaptation MethodTraining DataSource=ImageNetSource=Amazon
SVM (source only)Source66.23 ± 0.853.23 ± 1.6
SVM (target only)Webcam63.13 ± 1.963.13 ± 1.9
GFK[12]Source68.73 ±1.154.56 ± 1.2
SA[10]Source66.08 ± 1.155.98 ± 1.0
SVM (source and target)Source+Webcam75.13 ± 1.163.20 ± 1.7
Late Fusion (Max)Source+Webcam71.77 ± 1.462.25 ± 0.8
Late Fusion (LinInt Avg)Source+Webcam70.56 ± 1.264.56 ± 1.3
Daumé III [7]Source+Webcam77.15 ± 1.170.51 ± 1.7
PMT[1]Source+Webcam70.28 ± 1.866.77 ± 2.1
MMDT[15]Source+Webcam73.96 ± 1.266.23 ± 1.4
Late Fusion (Lin. Int. Oracle)Source+Webcam76.61 ± 1.571.49 ± 1.3
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Again, we compare multiclass accu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "racy of various unsupervised and supervised adaptation methods. 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Adaptation MethodTraining DataDeCAF6DeCAF7
SVM (source only)ImageNet53.51 ± 1.159.15 ± 1.1
SVM (target only)Webcam62.28 ± 1.864.97 ± 1.8
GFK [12]ImageNet65.16 ± 1.167.97 ± 1.4
SA [10]ImageNet59.30 ± 1.466.08 ± 1.4
SVM (source and target)ImageNet+Webcam56.68 ± 1.266.93 ± 1.3
Late Fusion (Max)ImageNet+Webcam59.59 ± 1.368.86 ± 1.2
Late Fusion (Lin. Int. Avg)ImageNet+Webcam60.64 ± 1.366.45 ± 1.1
Daumé III [7]ImageNet+Webcam59.21 ± 1.771.39 ± 1.5
PMT[1]ImageNet+Webcam66.30 ± 2.169.81 ± 1.8
MMDT[15]ImageNet+Webcam59.21 ± 1.367.75 ± 1.4
Late Fusion (Lin. Int. Oracle)ImageNet+Webcam71.65 ± 2.076.76 ± 1.3
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Adaptation MethodTraining DataSource=ImageNetSource=Amazon
SVM (source only)Source66.23 ± 0.853.23 ± 1.6
SVM (target only)Webcam63.13 ± 1.963.13 ± 1.9
GFK[12]Source68.73 ±1.154.56 ± 1.2
SA[10]Source66.08 ± 1.155.98 ± 1.0
SVM (source and target)Source+Webcam75.13 ± 1.163.20 ± 1.7
Late Fusion (Max)Source+Webcam71.77 ± 1.462.25 ± 0.8
Late Fusion (LinInt Avg)Source+Webcam70.56 ± 1.264.56 ± 1.3
Daumé III [7]Source+Webcam77.15 ± 1.170.51 ± 1.7
PMT[1]Source+Webcam70.28 ± 1.866.77 ± 2.1
MMDT[15]Source+Webcam73.96 ± 1.266.23 ± 1.4
Late Fusion (Lin. Int. Oracle)Source+Webcam76.61 ± 1.571.49 ± 1.3
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Again, we compare multiclass accu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "racy of various unsupervised and supervised adaptation methods. The best performing unsupervised", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "adaptation algorithm is shown in blue and the best performing supervised adaptation algorithms are", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 517, + 162, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 162, + 528 + ], + "score": 1.0, + "content": "shown in red.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + } + ], + "index": 10.0 + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 504, + 572 + ], + "lines": [], + "index": 15.5, + "bbox_fs": [ + 106, + 549, + 506, + 572 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 108, + 577, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Note that when using ImageNet as a source domain, overall performance of all algorithms improves.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "In addition, unsupervised adaptation approaches are more effective than for the smaller source do-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 600, + 178, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 178, + 612 + ], + "score": 1.0, + "content": "main experiment.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 577, + 505, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 626, + 354, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 356, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 356, + 640 + ], + "score": 1.0, + "content": "4.5 Adapting a Pre-trained Classifier to a New Label Set", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 144, + 660 + ], + "score": 0.84, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "differs from the other DeCAF features in that it constitutes the 1000 activations corre-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "sponding to the 1000 labels in the ImageNet classification task. In the CNN proposed by [17], these", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "activations are fed into a softmax unit to compute the label probabilities. We instead experiment", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 680, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 168, + 696 + ], + "score": 1.0, + "content": "with using the", + "type": "text" + }, + { + "bbox": [ + 168, + 682, + 205, + 693 + ], + "score": 0.9, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 680, + 506, + 696 + ], + "score": 1.0, + "content": "activations directly as a feature representation, which is akin to training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 693, + 374, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 374, + 705 + ], + "score": 1.0, + "content": "another classifier using the output of the 1000-way CNN classifier.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 649, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Table 3 shows results for various adaptation techniques using both ImageNet and Amazon as source", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 347, + 732 + ], + "score": 1.0, + "content": "domains. We use the same setup as before, but instead use", + "type": "text" + }, + { + "bbox": [ + 348, + 721, + 385, + 732 + ], + "score": 0.86, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "as the feature representation.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 296, + 95 + ], + "score": 1.0, + "content": "The ImageNet results are uniformly better with", + "type": "text" + }, + { + "bbox": [ + 297, + 83, + 333, + 94 + ], + "score": 0.86, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 82, + 374, + 95 + ], + "score": 1.0, + "content": "than with", + "type": "text" + }, + { + "bbox": [ + 374, + 83, + 411, + 94 + ], + "score": 0.88, + "content": "{ \\mathrm { D e C A F } } _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 82, + 423, + 95 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 423, + 83, + 459, + 94 + ], + "score": 0.89, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", likely due", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 165, + 106 + ], + "score": 1.0, + "content": "to the fact that", + "type": "text" + }, + { + "bbox": [ + 165, + 94, + 202, + 105 + ], + "score": 0.89, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "was explicitly trained on ImageNet data to effectively discriminate between", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "ImageNet categories. Because it can more effectively classify images from the source domain, it is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 367, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 367, + 128 + ], + "score": 1.0, + "content": "able to better adapt from the source domain to the target domain.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "However, we see a negligible difference in performance for Amazon, with performance actually de-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 202, + 155 + ], + "score": 1.0, + "content": "creasing with respect to", + "type": "text" + }, + { + "bbox": [ + 202, + 144, + 239, + 155 + ], + "score": 0.88, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "for certain adaptation methods. We believe this is because the final", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 399, + 167 + ], + "score": 1.0, + "content": "activation vector is too specific to the 1000-way ImageNet task, and that", + "type": "text" + }, + { + "bbox": [ + 400, + 154, + 437, + 165 + ], + "score": 0.88, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "provides a more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "general representation that is better suited to the Amazon domain. This, in turn, results in improved", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "adaptation. In general, however, the difference between the various DeCAF representations with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 335, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 335, + 199 + ], + "score": 1.0, + "content": "Amazon as a source are small enough to be insignificant.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 107, + 217, + 292, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 293, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 293, + 231 + ], + "score": 1.0, + "content": "4.6 Analysis and Practical Considerations", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "Our adaptation experiments show that, despite its large size, even ImageNet is not large enough", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "to cover all domains, and that traditional domain adaptation methods go a long way in increasing", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "performance and mitigating the effects of this shift. Depending on the characteristics of the problem", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 379, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 379, + 285 + ], + "score": 1.0, + "content": "at hand, our results suggest different methods may be most suitable.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "If no labels exist in the target domain, then there are unsupervised adaptation algorithms that are easy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 315 + ], + "score": 1.0, + "content": "to use and fast to compute at adaptation time, yet still achieve increased performance over source-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "only methods. For this scenario, we experimented with two subspace alignment based methods that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "both require setting a parameter that indicates the dimensionality of the input subspaces. Figure 1(b)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "shows the effect that changing the subspace dimensionality has on the overall method performance.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "In general, we noticed that these methods were not particularly sensitive to this parameter so long", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "as the dimensionality remains larger than the number of categories in our label set. Below this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "threshold, the subspace is less likely to capture all important discriminative information needed for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 163, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 163, + 390 + ], + "score": 1.0, + "content": "classification.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "In the case where we have a large source dataset and a limited number of labeled target examples,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "it may be preferable to compute source classifier parameters in advance, then examine only the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "source parameters and the target data at adaptation time. Examples of these kinds of methods are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "Late Fusion and PMT. These methods are unaffected by the number of data points in the source", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "domain at adaptation time, and can thus be applied quickly. In our experiments, we found that a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "properly tuned Late Fusion classifier with linear interpolation was the fastest and most effective", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "approach. Figure 1(a) shows the performance of linear interpolation Late Fusion as we vary the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 172, + 485 + ], + "score": 1.0, + "content": "hyperparameter", + "type": "text" + }, + { + "bbox": [ + 172, + 474, + 180, + 482 + ], + "score": 0.7, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 471, + 331, + 485 + ], + "score": 1.0, + "content": ". Although the method is sensitive to", + "type": "text" + }, + { + "bbox": [ + 331, + 474, + 339, + 482 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 471, + 506, + 485 + ], + "score": 1.0, + "content": ", we found that for both source domains,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 220, + 496 + ], + "score": 1.0, + "content": "the basic strategy of setting", + "type": "text" + }, + { + "bbox": [ + 221, + 485, + 228, + 493 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "around 0.8 provides a close approximation to optimal performance.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "This setting can be interpreted as trusting the target classifier more than the source, but not so much", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "as to completely discount the information available from the source classifier. In each table we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "report both the performance of linear interpolation both averaged across hyper parameter settings", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 526, + 147, + 538 + ], + "score": 0.92, + "content": "\\bar { \\alpha \\in [ 0 , 1 ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 525, + 480, + 540 + ], + "score": 1.0, + "content": "as well as the performance of linear interpolation with the best possible setting of", + "type": "text" + }, + { + "bbox": [ + 480, + 528, + 488, + 536 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "per", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 325, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 325, + 550 + ], + "score": 1.0, + "content": "experiment – this is denoted as “Oracle” performance.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "If there are no computational constraints and there are very few labels in the target domain, the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "best-performing method seems to be the “frustratingly easy” approach originally proposed by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 576, + 330, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 330, + 589 + ], + "score": 1.0, + "content": "Daume III [7] and applied again for deep models in [5]. ´", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 593, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "Finally, we found that feature representation can have a significant impact on adaptation perfor-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 405, + 616 + ], + "score": 1.0, + "content": "mance. Our results show that ImageNet as source performs best with the", + "type": "text" + }, + { + "bbox": [ + 405, + 604, + 442, + 615 + ], + "score": 0.89, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 605, + 505, + 616 + ], + "score": 1.0, + "content": "representation,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 315, + 627 + ], + "score": 1.0, + "content": "whereas Amazon as source performs best with the", + "type": "text" + }, + { + "bbox": [ + 315, + 615, + 352, + 627 + ], + "score": 0.89, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 615, + 506, + 627 + ], + "score": 1.0, + "content": "representation. This, combined with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "score": 1.0, + "content": "our intuition, seems to indicate that for adaptation from source domains other than ImageNet, an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 259, + 650 + ], + "score": 1.0, + "content": "intermediate representation other than", + "type": "text" + }, + { + "bbox": [ + 260, + 637, + 297, + 649 + ], + "score": 0.89, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "is more powerful for adaptation, whereas ImageNet", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 405, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 405, + 660 + ], + "score": 1.0, + "content": "classification works best with the full representation that was trained on it.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + }, + { + "type": "title", + "bbox": [ + 107, + 681, + 182, + 694 + ], + "lines": [ + { + "bbox": [ + 104, + 679, + 185, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 185, + 697 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In this paper, we presented the first evaluation of domain adaptation from a large-scale source dataset", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "with deep features. We demonstrated that, although using ImageNet as a source domain generalizes", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 296, + 95 + ], + "score": 1.0, + "content": "The ImageNet results are uniformly better with", + "type": "text" + }, + { + "bbox": [ + 297, + 83, + 333, + 94 + ], + "score": 0.86, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 82, + 374, + 95 + ], + "score": 1.0, + "content": "than with", + "type": "text" + }, + { + "bbox": [ + 374, + 83, + 411, + 94 + ], + "score": 0.88, + "content": "{ \\mathrm { D e C A F } } _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 82, + 423, + 95 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 423, + 83, + 459, + 94 + ], + "score": 0.89, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", likely due", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 165, + 106 + ], + "score": 1.0, + "content": "to the fact that", + "type": "text" + }, + { + "bbox": [ + 165, + 94, + 202, + 105 + ], + "score": 0.89, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "was explicitly trained on ImageNet data to effectively discriminate between", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "ImageNet categories. Because it can more effectively classify images from the source domain, it is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 367, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 367, + 128 + ], + "score": 1.0, + "content": "able to better adapt from the source domain to the target domain.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "However, we see a negligible difference in performance for Amazon, with performance actually de-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 202, + 155 + ], + "score": 1.0, + "content": "creasing with respect to", + "type": "text" + }, + { + "bbox": [ + 202, + 144, + 239, + 155 + ], + "score": 0.88, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "for certain adaptation methods. We believe this is because the final", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 399, + 167 + ], + "score": 1.0, + "content": "activation vector is too specific to the 1000-way ImageNet task, and that", + "type": "text" + }, + { + "bbox": [ + 400, + 154, + 437, + 165 + ], + "score": 0.88, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "provides a more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "general representation that is better suited to the Amazon domain. This, in turn, results in improved", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "adaptation. In general, however, the difference between the various DeCAF representations with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 335, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 335, + 199 + ], + "score": 1.0, + "content": "Amazon as a source are small enough to be insignificant.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 132, + 506, + 199 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 217, + 292, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 293, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 293, + 231 + ], + "score": 1.0, + "content": "4.6 Analysis and Practical Considerations", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "Our adaptation experiments show that, despite its large size, even ImageNet is not large enough", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "to cover all domains, and that traditional domain adaptation methods go a long way in increasing", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "performance and mitigating the effects of this shift. Depending on the characteristics of the problem", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 379, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 379, + 285 + ], + "score": 1.0, + "content": "at hand, our results suggest different methods may be most suitable.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 240, + 505, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "If no labels exist in the target domain, then there are unsupervised adaptation algorithms that are easy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 315 + ], + "score": 1.0, + "content": "to use and fast to compute at adaptation time, yet still achieve increased performance over source-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "only methods. For this scenario, we experimented with two subspace alignment based methods that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "both require setting a parameter that indicates the dimensionality of the input subspaces. Figure 1(b)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "shows the effect that changing the subspace dimensionality has on the overall method performance.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "In general, we noticed that these methods were not particularly sensitive to this parameter so long", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "as the dimensionality remains larger than the number of categories in our label set. Below this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "threshold, the subspace is less likely to capture all important discriminative information needed for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 163, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 163, + 390 + ], + "score": 1.0, + "content": "classification.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 289, + 506, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "In the case where we have a large source dataset and a limited number of labeled target examples,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "it may be preferable to compute source classifier parameters in advance, then examine only the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "source parameters and the target data at adaptation time. Examples of these kinds of methods are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "Late Fusion and PMT. These methods are unaffected by the number of data points in the source", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "domain at adaptation time, and can thus be applied quickly. In our experiments, we found that a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "properly tuned Late Fusion classifier with linear interpolation was the fastest and most effective", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "approach. Figure 1(a) shows the performance of linear interpolation Late Fusion as we vary the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 172, + 485 + ], + "score": 1.0, + "content": "hyperparameter", + "type": "text" + }, + { + "bbox": [ + 172, + 474, + 180, + 482 + ], + "score": 0.7, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 471, + 331, + 485 + ], + "score": 1.0, + "content": ". Although the method is sensitive to", + "type": "text" + }, + { + "bbox": [ + 331, + 474, + 339, + 482 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 471, + 506, + 485 + ], + "score": 1.0, + "content": ", we found that for both source domains,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 220, + 496 + ], + "score": 1.0, + "content": "the basic strategy of setting", + "type": "text" + }, + { + "bbox": [ + 221, + 485, + 228, + 493 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "around 0.8 provides a close approximation to optimal performance.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "This setting can be interpreted as trusting the target classifier more than the source, but not so much", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "as to completely discount the information available from the source classifier. In each table we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "report both the performance of linear interpolation both averaged across hyper parameter settings", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 526, + 147, + 538 + ], + "score": 0.92, + "content": "\\bar { \\alpha \\in [ 0 , 1 ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 525, + 480, + 540 + ], + "score": 1.0, + "content": "as well as the performance of linear interpolation with the best possible setting of", + "type": "text" + }, + { + "bbox": [ + 480, + 528, + 488, + 536 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "per", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 325, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 325, + 550 + ], + "score": 1.0, + "content": "experiment – this is denoted as “Oracle” performance.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 394, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "If there are no computational constraints and there are very few labels in the target domain, the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "best-performing method seems to be the “frustratingly easy” approach originally proposed by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 576, + 330, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 330, + 589 + ], + "score": 1.0, + "content": "Daume III [7] and applied again for deep models in [5]. ´", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 554, + 505, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 593, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "Finally, we found that feature representation can have a significant impact on adaptation perfor-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 405, + 616 + ], + "score": 1.0, + "content": "mance. Our results show that ImageNet as source performs best with the", + "type": "text" + }, + { + "bbox": [ + 405, + 604, + 442, + 615 + ], + "score": 0.89, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 605, + 505, + 616 + ], + "score": 1.0, + "content": "representation,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 315, + 627 + ], + "score": 1.0, + "content": "whereas Amazon as source performs best with the", + "type": "text" + }, + { + "bbox": [ + 315, + 615, + 352, + 627 + ], + "score": 0.89, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 615, + 506, + 627 + ], + "score": 1.0, + "content": "representation. This, combined with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 638 + ], + "score": 1.0, + "content": "our intuition, seems to indicate that for adaptation from source domains other than ImageNet, an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 259, + 650 + ], + "score": 1.0, + "content": "intermediate representation other than", + "type": "text" + }, + { + "bbox": [ + 260, + 637, + 297, + 649 + ], + "score": 0.89, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "is more powerful for adaptation, whereas ImageNet", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 405, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 405, + 660 + ], + "score": 1.0, + "content": "classification works best with the full representation that was trained on it.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 592, + 506, + 660 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 681, + 182, + 694 + ], + "lines": [ + { + "bbox": [ + 104, + 679, + 185, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 185, + 697 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In this paper, we presented the first evaluation of domain adaptation from a large-scale source dataset", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "with deep features. We demonstrated that, although using ImageNet as a source domain generalizes", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "better than other smaller source domains, there is still a domain shift when adapting to other visual", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 336, + 146, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 146, + 349 + ], + "score": 1.0, + "content": "domains.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 120, + 87, + 491, + 258 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 87, + 491, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 87, + 491, + 258 + ], + "spans": [ + { + "bbox": [ + 120, + 87, + 491, + 258 + ], + "score": 0.97, + "type": "image", + "image_path": "3bc68fe9e1a528a694c293fb0f4369506b121f899b4f01134593f6e12b3afc0f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 120, + 87, + 491, + 144.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 144.0, + 491, + 201.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 201.0, + 491, + 258.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 271, + 505, + 305 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "Figure 1: Evaluation of hyperparameters for domain adaptation methods. (a) Analysis of the com-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 207, + 295 + ], + "score": 1.0, + "content": "bination hyperparameter", + "type": "text" + }, + { + "bbox": [ + 208, + 285, + 216, + 293 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "for Late Fusion with linear interpolation. (b) Analysis of the subspace", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 294, + 341, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 341, + 306 + ], + "score": 1.0, + "content": "dimensionality for the unsupervised adaptation algorithms", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 108, + 326, + 503, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "better than other smaller source domains, there is still a domain shift when adapting to other visual", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 336, + 146, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 146, + 349 + ], + "score": 1.0, + "content": "domains.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 504, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "Our experimental results show that deep adaptation methods can go a long way in mitigating the ef-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "fects of this domain shift. Based on our results, we also provided a set of practical recommendations", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "score": 1.0, + "content": "for choosing a feature representation and adaptation method accounting for constraints on runtime", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 386, + 163, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 163, + 400 + ], + "score": 1.0, + "content": "and accuracy.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "There are a number of interesting directions to take given our results. First we notice that though", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 415, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 143, + 426 + ], + "score": 0.89, + "content": "\\mathrm { D e C A F _ { 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 415, + 424, + 426 + ], + "score": 1.0, + "content": "is the strongest feature to use for learning a classifier on ImageNet data,", + "type": "text" + }, + { + "bbox": [ + 425, + 415, + 462, + 426 + ], + "score": 0.85, + "content": "{ \\mathrm { D e C A F } } _ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 415, + 504, + 426 + ], + "score": 1.0, + "content": "is actually", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "a better feature to use with the Amazon source domain and the Webcam target domain. This could", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "lead to a hybrid approach where one uses different feature representations for the various domains", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "and produces a combined adapted model. Another interesting direction that should be explored is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "to integrate the adaption algorithms into the deep models explicitly and even allow for feedback", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "between the two stages. Current deep models although allow information flow between the final", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "classifier and the representation learning architecture. We feel that the next step is to have a separate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "task specific adaptable layer that does not simply learn a new final layer, but instead learns a separate,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 502, + 473, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 473, + 515 + ], + "score": 1.0, + "content": "but equivalent final layer, that is regularized by the final layer learned on the source dataset.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "This future work is a natural extension of the result we have shown in this paper: that pre-trained", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "deep representations with large source domains can be effectively adapted to new target domains", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "using only shallow, linear adaptation methods, and that in cases where the target data is limited, this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 552, + 303, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 303, + 564 + ], + "score": 1.0, + "content": "approach is the best way to mitigate dataset bias.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 579, + 163, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 165, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 165, + 594 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 110, + 600, + 506, + 731 + ], + "lines": [ + { + "bbox": [ + 110, + 600, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 110, + 600, + 505, + 611 + ], + "score": 1.0, + "content": "[1] Y. 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(a) Analysis of the com-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 207, + 295 + ], + "score": 1.0, + "content": "bination hyperparameter", + "type": "text" + }, + { + "bbox": [ + 208, + 285, + 216, + 293 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "for Late Fusion with linear interpolation. 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Based on our results, we also provided a set of practical recommendations", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 389 + ], + "score": 1.0, + "content": "for choosing a feature representation and adaptation method accounting for constraints on runtime", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 386, + 163, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 163, + 400 + ], + "score": 1.0, + "content": "and accuracy.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 353, + 506, + 400 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "There are a number of interesting directions to take given our results. 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This could", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "lead to a hybrid approach where one uses different feature representations for the various domains", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "and produces a combined adapted model. 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We feel that the next step is to have a separate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "task specific adaptable layer that does not simply learn a new final layer, but instead learns a separate,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 502, + 473, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 473, + 515 + ], + "score": 1.0, + "content": "but equivalent final layer, that is regularized by the final layer learned on the source dataset.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 403, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "This future work is a natural extension of the result we have shown in this paper: that pre-trained", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "deep representations with large source domains can be effectively adapted to new target domains", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "using only shallow, linear adaptation methods, and that in cases where the target data is limited, this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 552, + 303, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 303, + 564 + ], + "score": 1.0, + "content": "approach is the best way to mitigate dataset bias.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 519, + 505, + 564 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 579, + 163, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 165, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 165, + 594 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "list", + "bbox": [ + 110, + 600, + 506, + 731 + ], + "lines": [ + { + "bbox": [ + 110, + 600, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 110, + 600, + 505, + 611 + ], + "score": 1.0, + "content": "[1] Y. 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