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In this paper, we propose the Generative Multi-Adversarial Network (GMAN), a framework that extends GANs to multiple discriminators. In previous work, the successful training of GANs requires modifying the minimax objective to accelerate training early on. In contrast, GMAN can be reliably trained with the original, untampered objective. We explore a number of design perspectives with the discriminator role ranging from formidable adversary to forgiving teacher. Image generation tasks comparing the proposed framework to standard GANs demonstrate GMAN produces higher quality samples in a fraction of the iterations when measured by a pairwise GAM-type metric. + +# 1 INTRODUCTION + +Generative adversarial networks (Goodfellow et al. (2014)) (GANs) are a framework for producing a generative model by way of a two-player minimax game. One player, the generator, attempts to generate realistic data samples by transforming noisy samples, $z$ , drawn from a simple distribution (e.g., $z \sim \mathcal { N } ( 0 , 1 ) )$ using a transformation function $G _ { \theta } ( z )$ with learned weights, $\theta$ . The generator receives feedback as to how realistic its synthetic sample is from another player, the discriminator, which attempts to discern between synthetic data samples produced by the generator and samples drawn from an actual dataset using a function $D _ { \omega } ( x )$ with learned weights, $\omega$ . + +The GAN framework is one of the more recent successes in a line of research on adversarial training in machine learning (Schmidhuber (1992); Bagnell (2005); Ajakan et al. (2014)) where games between learners are carefully crafted so that Nash equilibria coincide with some set of desired optimality criteria. Preliminary work on GANs focused on generating images (e.g., MNIST (LeCun et al. (1998)), CIFAR (Krizhevsky (2009))), however, GANs have proven useful in a variety of application domains including learning censored representations (Edwards & Storkey (2015)), imitating expert policies (Ho & Ermon (2016)), and domain transfer (Yoo et al. (2016)). Work extending GANs to semi-supervised learning (Chen et al. (2016); Mirza & Osindero (2014); Gauthier (2014); Springenberg (2015)), inference (Makhzani et al. (2015); Dumoulin et al. (2016)), feature learning (Donahue et al. (2016)), and improved image generation (Im et al. (2016); Denton et al. (2015); Radford et al. (2015)) have shown promise as well. + +Despite these successes, GANs are reputably difficult to train. While research is still underway to improve training techniques and heuristics (Salimans et al. (2016)), most approaches have focused on understanding and generalizing GANs theoretically with the aim of exploring more tractable formulations (Zhao et al. (2016); Li et al. (2015); Uehara et al. (2016); Nowozin et al. (2016)). + +In this paper, we theoretically and empirically justify generalizing the GAN framework to multiple discriminators. We review GANs and summarize our extension in Section 2. In Sections 3 and 4, we present our $N$ -discriminator extension to the GAN framework (Generative Multi-Adversarial Networks) with several variants which range the role of the discriminator from formidable adversary to forgiving teacher. Section 4.2 explains how this extension makes training with the untampered minimax objective tractable. In Section 5, we define an intuitive metric (GMAM) to quantify GMAN performance and evaluate our framework on a variety of image generation tasks. Section 6 concludes with a summary of our contributions and directions for future research. + +Contributions—To summarize, our main contributions are: i) a multi-discriminator GAN framework, GMAN, that allows training with the original, untampered minimax objective; ii) a generative multi-adversarial metric (GMAM) to perform pairwise evaluation of separately trained frameworks; iii) a particular instance of GMAN, $\mathrm { G M A N ^ { * } }$ , that allows the generator to automatically regulate training and reach higher performance (as measured by GMAM) in a fraction of the training time required for the standard GAN model. + +# 2 GENERATIVE ADVERSARIAL NETWORKS TO GMAN + +The original formulation of a GAN is a minimax game between a generator, $G _ { \theta } ( z ) : z \to x$ , and a discriminator, $D _ { \omega } ( x ) : x [ 0 , 1 ]$ , + +$$ +\underset { G } { \mathrm { m i n } } \underset { D \in \mathcal { D } } { \mathrm { m a x } } V ( D , G ) = \mathbb { E } _ { x \sim p _ { d a t a } ( x ) } \Big [ \log ( D ( x ) ) \Big ] + \mathbb { E } _ { z \sim p _ { z } ( z ) } \Big [ \log ( 1 - D ( G ( z ) ) ) \Big ] , +$$ + +where $p _ { d a t a } ( x )$ is the true data distribution and $p _ { z } ( z )$ is a simple (usually fixed) distribution that is easy to draw samples from (e.g., $\mathcal { N } ( 0 , 1 ) )$ ). We differentiate between the function space of discriminators, $\mathcal { D }$ , and elements of this space, $D$ . Let $p _ { G } ( x )$ be the distribution induced by the generator, $G _ { \theta } ( z )$ . We assume $D , G$ to be deep neural networks as is typically the case. + +In their original work, Goodfellow et al. (2014) proved that given sufficient network capacities and an oracle providing the optimal discriminator, $D ^ { * } = a r g \operatorname* { m a x } _ { \mathcal { D } } V ( D , G )$ , gradient descent on $p _ { G } ( x )$ will recover the desired globally optimal solution, $p _ { G } ( x ) = p _ { d a t a } ( x )$ , so that the generator distribution exactly matches the data distribution. In practice, they replaced the second term, $\log ( 1 -$ $D ( G ( z ) ) )$ , with $- \log ( D ( G ( z ) ) )$ to enhance gradient signals at the start of the game; note this is no longer a zero-sum game. Part of their convergence and optimality proof involves using the oracle, $D ^ { * }$ , to reduce the minimax game to a minimization over $G$ only: + +$$ +\operatorname* { m i n } _ { G } V ( D ^ { * } , G ) = \operatorname* { m i n } _ { G } \Big \{ C ( G ) = - \log ( 4 ) + 2 \cdot J S D ( p _ { d a t a } | | p _ { G } ) \Big \} +$$ + +where $J S D$ denotes Jensen-Shannon divergence. Minimizing $C ( G )$ necessarily minimizes $J S D$ , however, we rarely know $D ^ { * }$ and so we instead minimize $V ( D , G )$ , which is only a lower bound. + +This perspective of minimizing the distance between the distributions, $p _ { d a t a }$ and $p _ { G }$ , motivated Li et al. (2015) to develop a generative model that matches all moments of $p _ { G } ( x )$ with $p _ { d a t a } ( x )$ (at optimality) by minimizing maximum mean discrepancy (MMD). Another approach, EBGAN, (Zhao et al. (2016)) explores a larger class of games (non-zero-sum games) which generalize the generator and discriminator objectives to take real-valued “energies” as input instead of probabilities. Nowozin et al. (2016) and then Uehara et al. (2016) extended the $J S D$ perspective on GANs to more general divergences, specifically $f$ -divergences and then Bregman-divergences respectively. + +In general, these approaches focus on exploring fundamental reformulations of $V ( D , G )$ . Similarly, our work focuses on a fundamental reformulation, however, our aim is to provide a framework that accelerates training of the generator to a more robust state irrespective of the choice of $V$ . + +# 2.1 GMAN: A MULTI-ADVERSARIAL EXTENSION + +We propose introducing multiple discriminators, which brings with it a number of design possibilities. We explore approaches ranging between two extremes: 1) a more discriminating $D$ (better approximating $\operatorname* { m a x } _ { \mathcal { D } } V ( D , G ) )$ and 2) a $D$ better matched to the generator’s capabilities. Mathematically, we reformulate $G$ ’s objective as $\operatorname* { m i n } _ { G }$ max $F ( V ( D _ { 1 } , G ) , \dots , V ( D _ { N } , { \bar { G } } ) )$ for different choices of $F$ (see Figure 1). Each $D _ { i }$ is still expected to independently maximize its own $V ( D _ { i } , G )$ (i.e. no cooperation). We sometimes abbreviate $V ( D _ { i } , G )$ with $V _ { i }$ and $F ( V _ { 1 } , \dots , V _ { N } )$ with $F _ { G } ( V _ { i } )$ . + +# 3 A FORMIDABLE ADVERSARY + +Here, we consider multi-discriminator variants that attempt to better approximate $\operatorname* { m a x } _ { D } V ( D , G )$ , providing a harsher critic to the generator. + +![](images/d7e87896eadef41722f4246a3374c0bd54b131437357aba75558be6c0d6b5810.jpg) +Figure 1: (GMAN) The generator trains using feedback aggregated over multiple discriminators. If $F : = \mathrm { m a x }$ , $G$ trains against the best discriminator. If $F : = { \mathrm { m e a n } }$ , $G$ trains against an ensemble. We explore other alternatives to $F$ in Sections 4.1 & 4.4 that improve on both these options. + +# 3.1 MAXIMIZING V(D,G) + +For a fixed $G$ , maximizing $F _ { G } ( V _ { i } )$ with $F : = \mathrm { m a x }$ and $N$ randomly instantiated copies of our discriminator is functionally equivalent to optimizing $V$ (e.g., stochastic gradient ascent) with random restarts in parallel and then presenting $\mathrm { m a x } _ { i \in \{ 1 , . . . , N \} } V ( D _ { i } , G )$ as the loss to the generator —a very pragmatic approach to the difficulties presented by the non-convexity of $V$ caused by the deep net. Requiring the generator to minimize the max forces $G$ to generate high fidelity samples that must hold up under the scrutiny of all $N$ discriminators, each potentially representing a distinct max. + +In practice, $\operatorname* { m a x } _ { D _ { i } \in \mathcal { D } } V ( D _ { i } , G )$ is not performed to convergence (or global optimality), so the above problem is oversimplified. Furthermore, introducing $N$ discriminators affects the dynamics of the game which affects the trajectories of the discriminators. This prevents us from claiming $\operatorname* { m a x } \{ V _ { 1 } ( t ) , \ldots , V _ { N } ( t ) \} > \operatorname* { m a x } \{ V _ { 1 } ^ { \prime } ( t ) \} \forall t$ even if we initalize $D _ { 1 } ( 0 ) = \mathbf { \bar { \cal { D } } } _ { 1 } ^ { \prime } ( 0 )$ as it is unlikely that $D _ { 1 } ( t ) = D _ { 1 } ^ { \prime } ( t )$ at some time $t$ after the start of the game. + +# 3.2 BOOSTING + +We can also consider taking the max over $N$ discriminators as a form of boosting for the discriminator’s online classification problem (online because $G$ can produce an infinite data stream). The boosted discriminator is given a sample $x _ { t }$ and must predict whether it came from the generator or the dataset. The booster then makes its prediction using the predictions of the $N$ weaker $D _ { i }$ . + +There are a few differences between taking the max (case 1) and online boosting (case 2). In case 1, our booster is limited to selecting a single weak discriminator (i.e. a pure strategy), while in case 2, many boosting algorithms more generally use linear combinations of the discriminators. Moreover, in case 2, a booster must make a prediction before receiving a loss function. In case 1, we assume access to the loss function at prediction time, which allows us to compute the max. + +It is possible to train the weak discriminators using boosting and then ignore the booster’s prediction by instead presenting max $\{ V _ { i } \}$ . We explore both variants in our experiments, using the adaptive algorithm proposed in Beygelzimer et al. (2015). Unfortunately, boosting failed to produce promising results on the image generation tasks. It is possible that boosting produces too strong an adversary for learning which motivates the next section. Boosting results appear in Appendix A.7. + +# 4 A FORGIVING TEACHER + +The previous perspectives focus on improving the discriminator with the goal of presenting a better approximation of $\operatorname* { m a x } _ { \mathcal { D } } V ( D , G )$ to the generator. Our next perspective asks the question, “Is $\operatorname* { m a x } _ { \mathcal { D } } V ( D , G )$ too harsh a critic?” + +# 4.1 Soft-DISCRIMINATOR + +In practice, training against a far superior discriminator can impede the generator’s learning. This is because the generator is unlikely to generate any samples considered “realistic” by the discriminator’s standards, and so the generator will receive uniformly negative feedback. This is problematic because the information contained in the gradient derived from negative feedback only dictates where to drive down $p _ { G } ( x )$ , not specifically where to increase $p _ { G } ( x )$ . Furthermore, driving down $p _ { G } ( x )$ necessarily increases $p _ { G } ( x )$ in other regions of $\mathcal { X }$ (to maintain $\textstyle \int _ { \mathcal { X } } p _ { G } ( x ) = 1 )$ ) which may or may not contain samples from the true dataset (whack-a-mole dilemma). In contrast, a generator is more likely to see positive feedback against a more lenient discriminator, which may better guide a generator towards amassing $p _ { G } ( x )$ in approximately correct regions of $\mathcal { X }$ . + +For this reason, we explore a variety of functions that allow us to soften the max operator. We choose to focus on soft versions of the three classical Pythagorean means parameterized by $\lambda$ where $\lambda = 0$ corresponds to the mean and the max is recovered as $\lambda \to \infty$ : + +$$ +\begin{array} { l } { { \mathrm { { A M } } _ { s o f t } ( V , \lambda ) = \displaystyle \sum _ { i } ^ { N } w _ { i } V _ { i } } } \\ { ~ } \\ { { \mathrm { { G M } } _ { s o f t } ( V , \lambda ) = - \exp \Big ( \displaystyle \sum _ { i } ^ { N } w _ { i } \log ( - V _ { i } ) \Big ) } } \\ { { { } } } \\ { { { \mathrm { H M } } _ { s o f t } ( V , \lambda ) = \Big ( \displaystyle \sum _ { i } ^ { N } w _ { i } V _ { i } ^ { - 1 } \Big ) ^ { - 1 } } } \end{array} +$$ + +where $w _ { i } = e ^ { \lambda V _ { i } } / \Sigma _ { j } e ^ { \lambda V _ { j } }$ with $\lambda \ge 0 , V _ { i } < 0$ . Using a softmax also has the well known advantage of being differentiable (as opposed to subdifferentiable for max). Note that we only require continuity to guarantee that computing the softmax is actually equivalent to computing $V ( \tilde { D } , G )$ where $\tilde { D }$ is some convex combination of $D _ { i }$ (see Appendix A.5). + +# 4.2 USING THE ORIGINAL MINIMAX OBJECTIVE + +To illustrate the effect the softmax has on training, observe that the component of $A M _ { s o f t } ( V , 0 )$ relevant to generator training can be rewritten as + +$$ +\frac { 1 } { N } \sum _ { i } ^ { N } \mathbb { E } _ { x \sim p _ { G } ( x ) } \Big [ \log ( 1 - D _ { i } ( x ) ) \Big ] = \frac { 1 } { N } \mathbb { E } _ { x \sim p _ { G } ( x ) } \Big [ \log ( z ) \Big ] . +$$ + +where $\begin{array} { r } { z = \prod _ { i } ^ { N } ( 1 - D _ { i } ( x ) ) } \end{array}$ . Note that the generator gradient, | ∂ log(z)∂z |, is minimized at z = 1 over $z \in ( 0 , 1 ] ^ { 1 }$ . From this form, it is clear that $z = 1$ if and only if $D _ { i } = 0 \forall i$ , so $G$ only receives a vanishing gradient if all $D _ { i }$ agree that the sample is fake; this is especially unlikely for large $N$ . In other words, $G$ only needs to fool a single $D _ { i }$ to receive constructive feedback. This result allows the generator to successfully minimize the original generator objective, $\log ( 1 - D )$ . This is in contrast to the more popular $- \log ( D )$ introduced to artificially enhance gradients at the start of training. + +At the beginning of training, when $\operatorname* { m a x } _ { D _ { i } } V ( D _ { i } , G )$ is likely too harsh a critic for the generator, we can set $\lambda$ closer to zero to use the mean, increasing the odds of providing constructive feedback to the generator. In addition, the discriminators have the added benefit of functioning as an ensemble, reducing the variance of the feedback presented to the generator, which is especially important when the discriminators are far from optimal and are still learning a reasonable decision boundary. As training progresses and the discriminators improve, we can increase $\lambda$ to become more critical of the generator for more refined training. + +# 4.3 MAINTAINING MULTIPLE HYPOTHESES + +We argue for this ensemble approach on a more fundamental level as well. Here, we draw on the density ratio estimation perspective of GANs (Uehara et al. (2016)). The original GAN proof assumes we have access to $p _ { d a t a } ( x )$ , if only implicitly. In most cases of interest, the discriminator only has access to a finite dataset sampled from $p _ { d a t a } ( x )$ ; therefore, when computing expectations of $\dot { V } ( D , G )$ , we only draw samples from our finite dataset. This is equivalent to training a GAN with $p _ { d a t a } ( x ) = \tilde { p } _ { d a t a }$ which is a distribution consisting of point masses on all the data points in the dataset. For the sake of argument, let’s assume we are training a discriminator and generator, each with infinite capacity. In this case, the global optimum $( p _ { G } ( x ) = \tilde { p } _ { d a t a ( x ) } )$ fails to capture any of the interesting structure from $p _ { d a t a } ( x )$ , the true distribution we are trying to learn. Therefore, it is actually critical that we avoid this global optimum. + +![](images/6825e5bfd09d2c7ad38711198c6dc71b70472969679776d3899bf8813fa65739.jpg) +Figure 2: Consider a dataset consisting of the nine 1-dimensional samples in black. Their corresponding probability mass function is given in light gray. After training GMAN, three discriminators converge to distinct local optima which implicitly define distributions over the data (red, blue, yellow). Each discriminator may specialize in discriminating a region of the data space (placing more diffuse mass in other regions). Averaging over the three discriminators results in the distribution in black, which we expect has higher likelihood under reasonable assumptions on the structure of the true distribution. + +In practice, this degenerate result is avoided by employing learners with limited capacity and corrupting data samples with noise (i.e., dropout), but we might better accomplish this by simultaneously training a variety of limited capacity discriminators. With this approach, we might obtain a diverse set of seemingly tenable hypotheses for the true $p _ { d a t a } ( x )$ . Averaging over these multiple locally optimal discriminators increases the entropy of $\tilde { p } _ { d a t a } ( x )$ by diffusing the probability mass over the data space (see Figure 2 for an example). + +# 4.4 AUTOMATING REGULATION + +The problem of keeping the discriminator and generator in balance has been widely recognized in previous work with GANs. Issues with unstable dynamics, oscillatory behavior, and generator collapse are not uncommon. In addition, the discriminator is often times able to achieve a high degree of classification accuracy (producing a single scalar) before the generator has made sufficient progress on the arguably more difficult generative task (producing a high dimensional sample). Salimans et al. (2016) suggested label smoothing to reduce the vulnerability of the generator to a relatively superior discriminator. Here, we explore an approach that enables the generator to automatically temper the performance of the discriminator when necessary, but still encourages the generator to challenge itself against more accurate adversaries. Specifically, we augment the generator objective: + +$$ +\operatorname* { m i n } _ { G , \lambda > 0 } F _ { G } ( V _ { i } ) - f ( \lambda ) +$$ + +where $f ( \lambda )$ is monotonically increasing in $\lambda$ which appears in the softmax equations, (3)—(5). In experiments, we simply set $f ( \lambda ) = c \lambda$ with $c$ a constant (e.g., 0.001). The generator is incentivized to increase $\lambda$ to reduce its objective at the expense of competing against the best available adversary $D ^ { * }$ (see Appendix A.6). + +# 5 EVALUATION + +Evaluating GANs is still an open problem. In their original work, Goodfellow et al. (2014) report log likelihood estimates from Gaussian Parzen windows, which they admit, has high variance and is known not to perform well in high dimensions. Theis et al. (2016) recommend avoiding Parzen windows and argue that generative models should be evaluated with respect to their intended application. Salimans et al. (2016) suggest an Inception score, however, it assumes labels exist for the dataset. Recently, Im et al. (2016) introduced the Generative Adversarial Metric (GAM) for making pairwise comparisons between independently trained GAN models. The core idea behind their approach is given two generator, discriminator pairs $( G _ { 1 } , D _ { 1 } )$ and $( G _ { 2 } , D _ { 2 } )$ , we should be able to learn their relative performance by judging each generator under the opponent’s discriminator. + +# 5.1 METRIC + +In GMAN, the opponent may have multiple discriminators, which makes it unclear how to perform the swaps needed for GAM. We introduce a variant of GAM, the generative multi-adversarial metric (GMAM), that is amenable to training with multiple discriminators, + +$$ +\mathrm { G M A M } = \log \Big ( \frac { F _ { G _ { b } } ^ { a } ( V _ { i } ^ { a } ) } { F _ { G _ { a } } ^ { a } ( V _ { i } ^ { a } ) } \Big / \frac { F _ { G _ { a } } ^ { b } ( V _ { i } ^ { b } ) } { F _ { G _ { b } } ^ { b } ( V _ { i } ^ { b } ) } \Big ) . +$$ + +where $a$ and $b$ refer to the two GMAN variants (see Section 3 for notation $F _ { G } ( V _ { i } ) ,$ ). The idea here is similar. If $G _ { 2 }$ performs better than $G _ { 1 }$ with respect to both $D _ { 1 }$ and $D _ { 2 }$ , then $\mathrm { G M A M { > } 0 }$ (remember $V { \le } 0$ always). If $G _ { 1 }$ performs better in both cases, GMAM $< 0$ , otherwise, the result is indeterminate. + +# 5.2 EXPERIMENTS + +We evaluate the aforementioned variations of GMAN on a variety of image generation tasks: MNIST (LeCun et al. (1998)), CIFAR-10 (Krizhevsky (2009)) and CelebA (Liu et al. (2015)). We focus on rates of convergence to steady state along with quality of the steady state generator according to the GMAM metric. To summarize, loosely in order of increasing discriminator leniency, we compare + +• F-boost: A single AdaBoost.OL-boosted discriminator (see Appendix A.7). +• P-boost: $D _ { i }$ is trained according to AdaBoost.OL. A max over the weak learner losses is presented to the generator instead of the boosted prediction (see Appendix A.7). +• GMAN-max: max $\{ V _ { i } \}$ is presented to the generator. +• GAN: Standard GAN with a single discriminator (see Appendix A.2). +• mod-GAN: GAN with modified objective (generator minimizes − $\log ( D ( G ( z ) ) )$ . +• GMAN- $\lambda$ : GMAN with $F$ : $=$ arithmetic softmax with parameter $\lambda$ . +• GMAN∗: The arithmetic softmax is controlled by the generator through $\lambda$ . + +All generator and discriminator models are deep (de)convolutional networks (Radford et al. (2015)), and aside from the boosted variants, all are trained with Adam (Kingma & Ba (2014)) and batch normalization (Ioffe & Szegedy (2015)). Discriminators convert the real-valued outputs of their rksivedis $( \epsilon \dot { + } \frac { 1 - 2 \epsilon } { 1 + e ^ { - z } } )$ ies with squashed-sigmoids to prevent saturating logarithms in the min. See Appendix A.8 for further details. We test GMAN systems with We maintain discriminator diversity by varying dropout and network de $N =$ $\{ 2 , 5 \}$ + +# 5.2.1 MNIST + +Figure 3 reveals that increasing the number of discriminators reduces the number of iterations to steady-state by $2 \mathbf { x }$ on MNIST; increasing $N$ (the size of the discriminator ensemble) also has the added benefit of reducing the variance the minimax objective over runs. Figure 4 displays the variance of the same objective over a sliding time window, reaffirming GMAN’s acceleration to steadystate. Figure 5 corroborates this conclusion with recognizable digits appearing approximately an epoch before the single discriminator run; digits at steady-state appear slightly sharper as well. + +Our GMAM metric (see Table 1) agrees with the relative quality of images in Figure 5 with GMAN∗ achieving the best overall performance. Figure 6 reveals GMAN∗’s attempt to regulate the difficulty + +
ScoreVariantGMAN*GMAN-0GMAN-maxmod-GAN
←Feneg0.127GMAN*-0.020±0.009-0.028 ± 0.019-0.089± 0.036
0.007GMAN-00.020 ±0.009-0.013 ± 0.015-0.018 ± 0.027
-0.034GMAN-max0.028 ± 0.0190.013 ± 0.015-0.011 ± 0.024
-0.122mod-GAN0.089 ± 0.0360.018 ± 0.0270.011 ± 0.024
+ +Table 1: Pairwise GMAM metric means with stdev for select models on MNIST. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each variant’s column. + +![](images/08c9b309155352dc2819d633838b3a4d5befc6b6345a4dc3783dc239c759e699.jpg) +Figure 3: Generator objective, $F$ , averaged over 5 training runs on MNIST. Increasing the number of discriminators accelerates convergence of $F$ to steady state (solid line) and reduces its variance, $\scriptstyle { \dot { \sigma } } ^ { 2 }$ (filled shadow $\pm 1 \sigma )$ . Figure 4 provides alternative evidence of GMAN∗’s accelerated convergence. + +![](images/91baa284ba5198c489e8fe62abdaa1cd4493d944a3ed1d655748b8ce3d35238a.jpg) +Figure 4: Stdev, $\sigma$ , of the generator objective over a sliding window of 500 iterations. Lower values indicate a more steady-state. $\mathrm { G M A N ^ { * } }$ with $N = 5$ achieves steady-state at ${ \approx } 2 \mathbf { x }$ speed of GAN $N = 1$ ). Note Figure 3’s filled shadows reveal stdev of $F$ over runs, while this plot shows stdev over time. + +![](images/927417a6a7e30513a07171408412de620cc44add4f1baf38e0437e716f97524e.jpg) +Figure 5: Comparison of image quality across epochs for $N = \{ 1 , 2 , 5 \}$ using GMAN-0 on MNIST. + +of the game to accelerate learning. Figure 7 displays the GMAM scores comparing fixed $\lambda$ ’s to the variable $\lambda$ controlled by $\mathrm { G M A N ^ { * } }$ . + +
Score入 (N=5)*入=1入=0
1eeieg0.028*=-0.008 ±0.009-0.019 ±0.010 -0.008 ±0.010
0.001入=10.008 ±0.009
-0.025入=00.019 ±0.0100.008 ±0.010
+ +![](images/e8aa4f28694600150894ad026b0802a1d5d5dbe36b1d2be2f0f461a34fa2385d.jpg) +Figure 6: GMAN∗ regulates difficulty of the game by adjusting $\lambda$ . Initially, $G$ reduces $\lambda$ to ease learning and then gradually increases $\lambda$ for a more challenging learning environment. +Figure 7: Pairwise $\frac { \mathbf { G } \mathbf { M } \mathbf { A } \mathbf { M } } { s t d e \nu ( \mathbf { G } \mathbf { M } \mathbf { A } \mathbf { M } ) }$ for GMAN- $\lambda$ and GMAN∗ $( \lambda ^ { * } )$ over 5 runs on MNIST. + +# 5.2.2 CELEBA & CIFAR-10 + +We see similar accelerated convergence behavior for the CelebA dataset in Figure 8. + +![](images/0fec7eacde54564201c48cef48a66b25d97982a4db342eb0cb73c0d2af2e480b.jpg) +Figure 8: Image quality improvement across number of generators at same number of iterations for GMAN-0 on CelebA. + +![](images/4e3d61aa52dc0162b9f5a5f3fa6e3cc1b7828787aa84570d77f19a2ba6bc225e.jpg) +Figure 9 displays images generated by GMAN-0 on CIFAR-10. See Appendix A.3 for more results. +Figure 9: Images generated by GMAN-0 on the CIFAR-10 dataset. + +We also found that GMAN is robust to mode collapse. We believe this is because the generator must appease a diverse set of discriminators in each minibatch. Emitting a single sample will score well for one discriminator at the expense of the rest of the discriminators. Current solutions (e.g., minibatch discrimination) are quadratic in batch size. GMAN, however, is linear in batch size. + +# 6 CONCLUSION + +We introduced multiple discriminators into the GAN framework and explored discriminator roles ranging from a formidable adversary to a forgiving teacher. Allowing the generator to automatically tune its learning schedule $\mathbf { ( G M A N ^ { * } }$ ) outperformed GANs with a single discriminator on MNIST. In general, GMAN variants achieved faster convergence to a higher quality steady state on a variety of tasks as measured by a GAM-type metric (GMAM). In addition, GMAN makes using the original GAN objective possible by increasing the odds of the generator receiving constructive feedback. + +In future work, we will look at more sophisticated mechanisms for letting the generator control the game as well as other ways to ensure diversity among the discriminators. Introducing multiple generators is conceptually an obvious next step, however, we expect difficulties to arise from more complex game dynamics. For this reason, game theory and game design will likely be important. + +# ACKNOWLEDGMENTS + +We acknowledge helpful conversations with Stefan Dernbach, Archan Ray, Luke Vilnis, Ben Turtel, Stephen Giguere, Rajarshi Das, and Subhransu Maji. We also thank NVIDIA for donating a K40 GPU. This material is based upon work supported by the National Science Foundation under Grant Nos. IIS-1564032. 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Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016. + +# A APPENDIX + +# A.1 ACCELERATED CONVERGENCE & REDUCED VARIANCE + +See Figures 10, 11, 12, and 13. + +![](images/b86d64796a6407863f16b73360d23ccb0605cbb461cdf3aa67ee32ce2732ac42.jpg) +Figure 10: Generator objective, $F$ , averaged over 5 training runs on CelebA. Increasing $N$ (# of $D$ ) accelerates convergence of $F$ to steady state (solid line) and reduces its variance, $\sigma ^ { 2 }$ (filled shadow $\pm 1 \sigma$ ). Figure 11 provides alternative evidence of GMAN-0’s accelerated convergence. + +![](images/9622c7b37ca5ddacd7af81de88f321c6e0817503fdc2273800e82b66e56f1305.jpg) +Figure 11: Stdev, $\sigma$ , of the generator objective over a sliding window of 500 iterations. Lower values indicate a more steady-state. GMAN-0 with $N = 5$ achieves steady-state at ${ \approx } 2 \mathbf { x }$ speed of GAN $( N = 1 )$ ). Note Figure $1 0 ^ { \circ } \mathrm { s }$ filled shadows reveal stdev of $F$ over runs, while this plot shows stdev over time. + +![](images/aee15738e9c5166c1e19d8010f35de87c81b2f5d02bae4c11f691b2967a0f3b0.jpg) +Figure 12: Generator objective, $F$ , averaged over 5 training runs on CIFAR-10. Increasing $N$ (# of $D$ ) accelerates convergence of $F$ to steady state (solid line) and reduces its variance, $\textstyle { \bar { \sigma } } ^ { 2 }$ (filled shadow $\pm 1 \sigma$ ). Figure 13 provides alternative evidence of GMAN-0’s accelerated convergence. + +![](images/61640db03eb7afae558dd80985fbefff02d9bc9af9635f45b050ee336a25f67b.jpg) +Figure 13: Stdev, $\sigma$ , of the generator objective over a sliding window of 500 iterations. Lower values indicate a more steady-state. GMAN-0 with $N = 5$ achieves steady-state at ${ \approx } 2 \mathbf { x }$ speed of GAN $N = 1$ ). Note Figure 12’s filled shadows reveal stdev of $F$ over runs, while this plot shows stdev over time. + +# A.2 ADDITIONAL GMAM TABLES + +See Tables 2, 3, 4, 5, 6. Increasing the number of discriminators from 2 to 5 on CIFAR-10 significantly improves scores over the standard GAN both in terms of the GMAM metric and Inception scores. + +# A.3 GENERATED IMAGES + +See Figures 14 and 15. + +
ScoreVariantGMAN*GMAN-1GANGMAN-0GMAN-maxmod-GAN
←Feneg0.184 0.067GMAN*=-0.007-0.040-0.020-0.028-0.089
0.030GMAN-10.0071-0.008-0.008-0.021-0.037
GAN0.0400.008-0.002-0.018-0.058
0.005GMAN-00.0200.0080.002--0.013-0.018
-0.091GMAN-max0.0280.0210.0180.013--0.011
-0.213mod-GAN0.0890.0370.0580.0180.011=
+ +Table 2: Pairwise GMAM metric means for select models on MNIST. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each column. + +
ScoreVariantGMAN-0GMAN-1GMAN*mod-GAN
Beeneg0.172GMAN-01-0.022-0.062-0.088
0.050GMAN-10.0220.006-0.078
-0.055GMAN*0.062-0.006=-0.001
-0.167mod-GAN0.0880.0780.001=
+ +Table 3: Pairwise GMAM metric means for select models on CIFAR-10. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each column. GMAN variants were trained with two discriminators. + +Table 4: Inception score means with standard deviations for select models on CIFAR-10. Higher scores are better. GMAN variants were trained with two discriminators. + +
GMAN-0GMAN-1mod-GANGMAN*
Score5.878± 0.1935.765 ± 0.1685.738± 0.1765.539± 0.099
+ +
ScoreVariantGMAN-0GMAN*GMAN-1mod-GAN
FFeeeg0.180GMAN-0=-0.008-0.041-0.132
0.122GMAN*0.008=-0.038-0.092
0.010GMAN-10.0410.038-0.089
-0.313mod-GAN0.1320.0920.0891
+ +Table 5: Pairwise GMAM metric means for select models on CIFAR-10. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each column. GMAN variants were trained with five discriminators. + +
GMAN-1GMAN-0GMAN*mod-GAN
Score6.001± 0.1945.957 ± 0.1355.955 ± 0.1535.738 ± 0.176
+ +Table 6: Inception score means with standard deviations for select models on CIFAR-10. Higher scores are better. GMAN variants were trained with five discriminators. + +![](images/020e890b6cb8dc91c0185efe78a1ef059894f20210a4366160346a2c3173932f.jpg) +Figure 14: Sample of pictures generated on CelebA cropped dataset. + +![](images/5738407f4436e1f5c6e9f59db0134286b0d0355991c78a133dd9476e96d7f73a.jpg) +Figure 15: Sample of pictures generated by GMAN-0 on CIFAR dataset. + +# A.4 SOMEWHAT RELATED WORK + +A GAN framework with two discriminators appeared in Yoo et al. (2016), however, it is applicable only in a semi-supervised case where a label can be assigned to subsets of the dataset (e.g., $\mathcal { X } \ = \ \{ \mathcal { X } _ { 1 } = \operatorname { D o m a i n } \ 1 , \mathcal { X } _ { 2 } = \operatorname { D o m a i n } \ 2 , \ldots \} .$ ). In contrast, our framework applies to an unsupervised scenario where an obvious partition of the dataset is unknown. Furthermore, extending GMAN to the semi-supervised domain-adaptation scenario would suggest multiple discriminators per domain, therefore our line of research is strictly orthogonal to that of their multi-domain discriminator approach. Also, note that assigning a discriminator to each domain is akin to prescribing a new discriminator to each value of a conditional variable in conditional GANs (Mirza & Osindero (2014)). In this case, we interpret GMAN as introducing multiple conditional discriminators and not a discriminator for each of the possibly exponentially many conditional labels. + +In Section 4.4, we describe an approach to customize adversarial training to better suit the development of the generator. An approach with similar conceptual underpinnings was described in Ravanbakhsh et al. (2016), however, similar to the above, it is only admissible in a semi-supervised scenario whereas our applies to the unsupervised case. + +# A.5 Softmax REPRESENTABILITY + +Let soft $n a x ( V _ { i } ) = \hat { V } \in [ \operatorname* { m i n } _ { V _ { i } } , \operatorname* { m a x } _ { V _ { i } } ]$ . Also let $a = a r g \operatorname* { m i n } _ { i } V _ { i }$ , $b = a r g \operatorname* { m a x } _ { i } V _ { i }$ , and $\gamma ( t ) =$ $V ( ( 1 - t ) D _ { a } + t D _ { b } )$ so that $\nu ( 0 ) = V _ { a }$ and $\nu ( 1 ) = V _ { b }$ . The softmax and minimax objective $V ( D _ { i } , G )$ are both continuous in their inputs, so by the intermediate value theorem, we have that $\exists { \hat { t } } \in [ 0 , 1 ]$ s.t. $\mathcal { V } ( \hat { t } ) = \hat { V }$ , which implies $\exists \hat { D } \in \mathcal { D }$ s.t. $V ( \hat { D } , G ) = \hat { V }$ . This result implies that the softmax (and any other continuous substitute) can be interpreted as returning $V ( \hat { D } , G )$ for some $\hat { D }$ selected by computing an another, unknown function over the space of the discriminators. This result holds even if $\hat { D }$ is not representable by the architecture chosen for $D$ ’s neural network. + +# A.6 UNCONSTRAINED OPTIMIZATION + +To convert GMAN∗ minimax formulation to an unconstrained minimax formulation, we introduce an auxiliary variable, $\Lambda$ , define $\lambda ( \Lambda ) = \log ( 1 + e ^ { \Lambda } )$ , and let the generator minimize over $\Lambda \in \mathbb { R }$ . + +# A.7 BOOSTING WITH AdaBoost.OL + +AdaBoost.OL (Beygelzimer et al. (2015)) does not require knowledge of the weak learner’s slight edge over random guessing $( P ( { \mathrm { c o r r e c t 1 a b e l } } ) = 0 . 5 + \gamma \in ( 0 , 0 . 5 ] )$ , and in fact, allows $\gamma < 0$ . This is crucial because our weak learners are deep nets with unknown, possibly negative, $\gamma$ ’s. + +![](images/4734cd63537b4f2336b3d8c856ffdb1731187e1ee2c98fca4b386b349c0f1212.jpg) +Figure 16: Sample of pictures generated across 4 independent runs on MNIST with F-boost (similar results with P-boost). + +# A.8 EXPERIMENTAL SETUP + +All experiments were conducted using an architecture similar to DCGAN (Radford et al. (2015)). We use convolutional transpose layers (Zeiler et al. (2010)) for $G$ and strided convolutions for $D$ except for the input of $G$ and the last layer of $D$ . We use the single step gradient method as in (Nowozin et al. (2016)), and batch normalization (Ioffe & Szegedy (2015)) was used in each of the generator layers. The different discriminators were trained with varying dropout rates from [0.3, 0.7]. Variations in the discriminators were effected in two ways. We varied the architecture by varying the number of filters in the discriminator layers (reduced by factors of 2, 4 and so on), as well as varying dropout rates. Secondly we also decorrelated the samples that the disriminators were training on by splitting the minibatch across the discriminators. The code was written in Tensorflow (Abadi et al. (2016)) and run on Nvidia GTX 980 GPUs. Code to reproduce experiments and plots is at https://github.com/iDurugkar/GMAN. Specifics for the MNIST architecture and training are: + +• Generator latent variables z ∼ U (−1, 1)100 +• Generator convolution transpose layers: $\left( 4 , 4 , 1 2 8 \right) , \left( 8 , 8 , 6 4 \right) , \left( 1 6 , 1 6 , 3 2 \right) , \left( 3 2 , 3 2 , 1 \right)$ +• Base Discriminator architecture: $\left( 3 2 , 3 2 , 1 \right) , \left( 1 6 , 1 6 , 3 2 \right) , \left( 8 , 8 , 6 4 \right) , \left( 4 , 4 , 1 2 8 \right) .$ +• Variants have either convolution 3 $( 4 , 4 , 1 2 8 )$ removed or all the filter sizes are divided by 2 or 4. That is, $\left( 3 2 , 3 2 , 1 \right) , \left( 1 6 , 1 6 , 1 6 \right) , \left( 8 , 8 , 3 2 \right) , \left( 4 , 4 , 6 4 \right)$ or $\left( 3 2 , 3 2 , 1 \right) , \left( 1 6 , 1 6 , 8 \right) , \left( 8 , 8 , 1 6 \right) , \left( 4 , 4 , 3 2 \right)$ . +• ReLu activations for all the hidden units. Tanh activation at the output units of the generator. Sigmoid at the output of the Discriminator. +• Training was performed with Adam (Kingma & Ba (2014)) $( l r = 2 \times 1 0 ^ { - 4 }$ , $\beta _ { 1 } = 0 . 5 )$ ). +• MNIST was trained for 20 epochs with a minibatch of size 100. +• CelebA and CIFAR were trained over 24000 iterations with a minibatch of size 100. \ No newline at end of file diff --git a/parse/train/Byk-VI9eg/Byk-VI9eg_content_list.json b/parse/train/Byk-VI9eg/Byk-VI9eg_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..7940638b97df6f7058a6533ca55660a7120f8e7c --- /dev/null +++ b/parse/train/Byk-VI9eg/Byk-VI9eg_content_list.json @@ -0,0 +1,1814 @@ +[ + { + "type": "text", + "text": "GENERATIVE MULTI-ADVERSARIAL NETWORKS ", + "text_level": 1, + "bbox": [ + 173, + 98, + 756, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ishan Durugkar∗, Ian Gemp∗, Sridhar Mahadevan ", + "bbox": [ + 184, + 143, + 537, + 160 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "College of Information and Computer Sciences University of Massachusetts, Amherst Amherst, MA 01060, USA {idurugkar,imgemp,mahadeva}@cs.umass.edu ", + "bbox": [ + 184, + 160, + 573, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 252, + 544, + 267 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative adversarial networks (GANs) are a framework for producing a generative model by way of a two-player minimax game. In this paper, we propose the Generative Multi-Adversarial Network (GMAN), a framework that extends GANs to multiple discriminators. In previous work, the successful training of GANs requires modifying the minimax objective to accelerate training early on. In contrast, GMAN can be reliably trained with the original, untampered objective. We explore a number of design perspectives with the discriminator role ranging from formidable adversary to forgiving teacher. Image generation tasks comparing the proposed framework to standard GANs demonstrate GMAN produces higher quality samples in a fraction of the iterations when measured by a pairwise GAM-type metric. ", + "bbox": [ + 233, + 282, + 764, + 435 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 460, + 336, + 476 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative adversarial networks (Goodfellow et al. (2014)) (GANs) are a framework for producing a generative model by way of a two-player minimax game. One player, the generator, attempts to generate realistic data samples by transforming noisy samples, $z$ , drawn from a simple distribution (e.g., $z \\sim \\mathcal { N } ( 0 , 1 ) )$ using a transformation function $G _ { \\theta } ( z )$ with learned weights, $\\theta$ . The generator receives feedback as to how realistic its synthetic sample is from another player, the discriminator, which attempts to discern between synthetic data samples produced by the generator and samples drawn from an actual dataset using a function $D _ { \\omega } ( x )$ with learned weights, $\\omega$ . ", + "bbox": [ + 174, + 491, + 825, + 589 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The GAN framework is one of the more recent successes in a line of research on adversarial training in machine learning (Schmidhuber (1992); Bagnell (2005); Ajakan et al. (2014)) where games between learners are carefully crafted so that Nash equilibria coincide with some set of desired optimality criteria. Preliminary work on GANs focused on generating images (e.g., MNIST (LeCun et al. (1998)), CIFAR (Krizhevsky (2009))), however, GANs have proven useful in a variety of application domains including learning censored representations (Edwards & Storkey (2015)), imitating expert policies (Ho & Ermon (2016)), and domain transfer (Yoo et al. (2016)). Work extending GANs to semi-supervised learning (Chen et al. (2016); Mirza & Osindero (2014); Gauthier (2014); Springenberg (2015)), inference (Makhzani et al. (2015); Dumoulin et al. (2016)), feature learning (Donahue et al. (2016)), and improved image generation (Im et al. (2016); Denton et al. (2015); Radford et al. (2015)) have shown promise as well. ", + "bbox": [ + 174, + 595, + 825, + 748 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite these successes, GANs are reputably difficult to train. While research is still underway to improve training techniques and heuristics (Salimans et al. (2016)), most approaches have focused on understanding and generalizing GANs theoretically with the aim of exploring more tractable formulations (Zhao et al. (2016); Li et al. (2015); Uehara et al. (2016); Nowozin et al. (2016)). ", + "bbox": [ + 174, + 756, + 823, + 811 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we theoretically and empirically justify generalizing the GAN framework to multiple discriminators. We review GANs and summarize our extension in Section 2. In Sections 3 and 4, we present our $N$ -discriminator extension to the GAN framework (Generative Multi-Adversarial Networks) with several variants which range the role of the discriminator from formidable adversary to forgiving teacher. Section 4.2 explains how this extension makes training with the untampered minimax objective tractable. In Section 5, we define an intuitive metric (GMAM) to quantify GMAN performance and evaluate our framework on a variety of image generation tasks. Section 6 concludes with a summary of our contributions and directions for future research. ", + "bbox": [ + 174, + 819, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions—To summarize, our main contributions are: i) a multi-discriminator GAN framework, GMAN, that allows training with the original, untampered minimax objective; ii) a generative multi-adversarial metric (GMAM) to perform pairwise evaluation of separately trained frameworks; iii) a particular instance of GMAN, $\\mathrm { G M A N ^ { * } }$ , that allows the generator to automatically regulate training and reach higher performance (as measured by GMAM) in a fraction of the training time required for the standard GAN model. ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 GENERATIVE ADVERSARIAL NETWORKS TO GMAN ", + "text_level": 1, + "bbox": [ + 174, + 242, + 642, + 258 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The original formulation of a GAN is a minimax game between a generator, $G _ { \\theta } ( z ) : z \\to x$ , and a discriminator, $D _ { \\omega } ( x ) : x [ 0 , 1 ]$ , ", + "bbox": [ + 174, + 273, + 825, + 303 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/eed6e4351ee579310d34dd2f2da8d953c3dd43a6133196dddc651f428b8e9ce9.jpg", + "text": "$$\n\\underset { G } { \\mathrm { m i n } } \\underset { D \\in \\mathcal { D } } { \\mathrm { m a x } } V ( D , G ) = \\mathbb { E } _ { x \\sim p _ { d a t a } ( x ) } \\Big [ \\log ( D ( x ) ) \\Big ] + \\mathbb { E } _ { z \\sim p _ { z } ( z ) } \\Big [ \\log ( 1 - D ( G ( z ) ) ) \\Big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 228, + 306, + 766, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $p _ { d a t a } ( x )$ is the true data distribution and $p _ { z } ( z )$ is a simple (usually fixed) distribution that is easy to draw samples from (e.g., $\\mathcal { N } ( 0 , 1 ) )$ ). We differentiate between the function space of discriminators, $\\mathcal { D }$ , and elements of this space, $D$ . Let $p _ { G } ( x )$ be the distribution induced by the generator, $G _ { \\theta } ( z )$ . We assume $D , G$ to be deep neural networks as is typically the case. ", + "bbox": [ + 174, + 339, + 825, + 396 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In their original work, Goodfellow et al. (2014) proved that given sufficient network capacities and an oracle providing the optimal discriminator, $D ^ { * } = a r g \\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G )$ , gradient descent on $p _ { G } ( x )$ will recover the desired globally optimal solution, $p _ { G } ( x ) = p _ { d a t a } ( x )$ , so that the generator distribution exactly matches the data distribution. In practice, they replaced the second term, $\\log ( 1 -$ $D ( G ( z ) ) )$ , with $- \\log ( D ( G ( z ) ) )$ to enhance gradient signals at the start of the game; note this is no longer a zero-sum game. Part of their convergence and optimality proof involves using the oracle, $D ^ { * }$ , to reduce the minimax game to a minimization over $G$ only: ", + "bbox": [ + 173, + 401, + 825, + 500 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/7c762ae5742ca040ffe9aeea854cd7d4b7147bbeb6682b0c763660eaf023b9e7.jpg", + "text": "$$\n\\operatorname* { m i n } _ { G } V ( D ^ { * } , G ) = \\operatorname* { m i n } _ { G } \\Big \\{ C ( G ) = - \\log ( 4 ) + 2 \\cdot J S D ( p _ { d a t a } | | p _ { G } ) \\Big \\}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 503, + 720, + 531 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $J S D$ denotes Jensen-Shannon divergence. Minimizing $C ( G )$ necessarily minimizes $J S D$ , however, we rarely know $D ^ { * }$ and so we instead minimize $V ( D , G )$ , which is only a lower bound. ", + "bbox": [ + 174, + 536, + 826, + 565 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This perspective of minimizing the distance between the distributions, $p _ { d a t a }$ and $p _ { G }$ , motivated Li et al. (2015) to develop a generative model that matches all moments of $p _ { G } ( x )$ with $p _ { d a t a } ( x )$ (at optimality) by minimizing maximum mean discrepancy (MMD). Another approach, EBGAN, (Zhao et al. (2016)) explores a larger class of games (non-zero-sum games) which generalize the generator and discriminator objectives to take real-valued “energies” as input instead of probabilities. Nowozin et al. (2016) and then Uehara et al. (2016) extended the $J S D$ perspective on GANs to more general divergences, specifically $f$ -divergences and then Bregman-divergences respectively. ", + "bbox": [ + 173, + 570, + 825, + 670 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In general, these approaches focus on exploring fundamental reformulations of $V ( D , G )$ . Similarly, our work focuses on a fundamental reformulation, however, our aim is to provide a framework that accelerates training of the generator to a more robust state irrespective of the choice of $V$ . ", + "bbox": [ + 174, + 676, + 825, + 718 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 GMAN: A MULTI-ADVERSARIAL EXTENSION ", + "text_level": 1, + "bbox": [ + 174, + 734, + 534, + 750 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We propose introducing multiple discriminators, which brings with it a number of design possibilities. We explore approaches ranging between two extremes: 1) a more discriminating $D$ (better approximating $\\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G ) )$ and 2) a $D$ better matched to the generator’s capabilities. Mathematically, we reformulate $G$ ’s objective as $\\operatorname* { m i n } _ { G }$ max $F ( V ( D _ { 1 } , G ) , \\dots , V ( D _ { N } , { \\bar { G } } ) )$ for different choices of $F$ (see Figure 1). Each $D _ { i }$ is still expected to independently maximize its own $V ( D _ { i } , G )$ (i.e. no cooperation). We sometimes abbreviate $V ( D _ { i } , G )$ with $V _ { i }$ and $F ( V _ { 1 } , \\dots , V _ { N } )$ with $F _ { G } ( V _ { i } )$ . ", + "bbox": [ + 174, + 760, + 825, + 844 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 A FORMIDABLE ADVERSARY ", + "text_level": 1, + "bbox": [ + 176, + 864, + 446, + 880 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Here, we consider multi-discriminator variants that attempt to better approximate $\\operatorname* { m a x } _ { D } V ( D , G )$ , providing a harsher critic to the generator. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/d7e87896eadef41722f4246a3374c0bd54b131437357aba75558be6c0d6b5810.jpg", + "image_caption": [ + "Figure 1: (GMAN) The generator trains using feedback aggregated over multiple discriminators. If $F : = \\mathrm { m a x }$ , $G$ trains against the best discriminator. If $F : = { \\mathrm { m e a n } }$ , $G$ trains against an ensemble. We explore other alternatives to $F$ in Sections 4.1 & 4.4 that improve on both these options. " + ], + "image_footnote": [], + "bbox": [ + 392, + 99, + 604, + 226 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 MAXIMIZING V(D,G)", + "text_level": 1, + "bbox": [ + 176, + 304, + 367, + 319 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For a fixed $G$ , maximizing $F _ { G } ( V _ { i } )$ with $F : = \\mathrm { m a x }$ and $N$ randomly instantiated copies of our discriminator is functionally equivalent to optimizing $V$ (e.g., stochastic gradient ascent) with random restarts in parallel and then presenting $\\mathrm { m a x } _ { i \\in \\{ 1 , . . . , N \\} } V ( D _ { i } , G )$ as the loss to the generator —a very pragmatic approach to the difficulties presented by the non-convexity of $V$ caused by the deep net. Requiring the generator to minimize the max forces $G$ to generate high fidelity samples that must hold up under the scrutiny of all $N$ discriminators, each potentially representing a distinct max. ", + "bbox": [ + 174, + 330, + 825, + 415 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In practice, $\\operatorname* { m a x } _ { D _ { i } \\in \\mathcal { D } } V ( D _ { i } , G )$ is not performed to convergence (or global optimality), so the above problem is oversimplified. Furthermore, introducing $N$ discriminators affects the dynamics of the game which affects the trajectories of the discriminators. This prevents us from claiming $\\operatorname* { m a x } \\{ V _ { 1 } ( t ) , \\ldots , V _ { N } ( t ) \\} > \\operatorname* { m a x } \\{ V _ { 1 } ^ { \\prime } ( t ) \\} \\forall t$ even if we initalize $D _ { 1 } ( 0 ) = \\mathbf { \\bar { \\cal { D } } } _ { 1 } ^ { \\prime } ( 0 )$ as it is unlikely that $D _ { 1 } ( t ) = D _ { 1 } ^ { \\prime } ( t )$ at some time $t$ after the start of the game. ", + "bbox": [ + 174, + 421, + 825, + 492 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 BOOSTING ", + "text_level": 1, + "bbox": [ + 174, + 508, + 289, + 522 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can also consider taking the max over $N$ discriminators as a form of boosting for the discriminator’s online classification problem (online because $G$ can produce an infinite data stream). The boosted discriminator is given a sample $x _ { t }$ and must predict whether it came from the generator or the dataset. The booster then makes its prediction using the predictions of the $N$ weaker $D _ { i }$ . ", + "bbox": [ + 174, + 535, + 825, + 590 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "There are a few differences between taking the max (case 1) and online boosting (case 2). In case 1, our booster is limited to selecting a single weak discriminator (i.e. a pure strategy), while in case 2, many boosting algorithms more generally use linear combinations of the discriminators. Moreover, in case 2, a booster must make a prediction before receiving a loss function. In case 1, we assume access to the loss function at prediction time, which allows us to compute the max. ", + "bbox": [ + 174, + 597, + 825, + 667 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "It is possible to train the weak discriminators using boosting and then ignore the booster’s prediction by instead presenting max $\\{ V _ { i } \\}$ . We explore both variants in our experiments, using the adaptive algorithm proposed in Beygelzimer et al. (2015). Unfortunately, boosting failed to produce promising results on the image generation tasks. It is possible that boosting produces too strong an adversary for learning which motivates the next section. Boosting results appear in Appendix A.7. ", + "bbox": [ + 174, + 674, + 825, + 744 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 A FORGIVING TEACHER ", + "text_level": 1, + "bbox": [ + 176, + 765, + 408, + 781 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The previous perspectives focus on improving the discriminator with the goal of presenting a better approximation of $\\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G )$ to the generator. Our next perspective asks the question, “Is $\\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G )$ too harsh a critic?” ", + "bbox": [ + 174, + 796, + 825, + 838 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4.1 Soft-DISCRIMINATOR ", + "text_level": 1, + "bbox": [ + 174, + 856, + 361, + 869 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In practice, training against a far superior discriminator can impede the generator’s learning. This is because the generator is unlikely to generate any samples considered “realistic” by the discriminator’s standards, and so the generator will receive uniformly negative feedback. This is problematic because the information contained in the gradient derived from negative feedback only dictates where to drive down $p _ { G } ( x )$ , not specifically where to increase $p _ { G } ( x )$ . Furthermore, driving down $p _ { G } ( x )$ necessarily increases $p _ { G } ( x )$ in other regions of $\\mathcal { X }$ (to maintain $\\textstyle \\int _ { \\mathcal { X } } p _ { G } ( x ) = 1 )$ ) which may or may not contain samples from the true dataset (whack-a-mole dilemma). In contrast, a generator is more likely to see positive feedback against a more lenient discriminator, which may better guide a generator towards amassing $p _ { G } ( x )$ in approximately correct regions of $\\mathcal { X }$ . ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 102, + 825, + 188 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For this reason, we explore a variety of functions that allow us to soften the max operator. We choose to focus on soft versions of the three classical Pythagorean means parameterized by $\\lambda$ where $\\lambda = 0$ corresponds to the mean and the max is recovered as $\\lambda \\to \\infty$ : ", + "bbox": [ + 174, + 194, + 825, + 236 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/16a94c76c11169741b475d5a2234908f705713b6e80ce458958475e7bb495df9.jpg", + "text": "$$\n\\begin{array} { l } { { \\mathrm { { A M } } _ { s o f t } ( V , \\lambda ) = \\displaystyle \\sum _ { i } ^ { N } w _ { i } V _ { i } } } \\\\ { ~ } \\\\ { { \\mathrm { { G M } } _ { s o f t } ( V , \\lambda ) = - \\exp \\Big ( \\displaystyle \\sum _ { i } ^ { N } w _ { i } \\log ( - V _ { i } ) \\Big ) } } \\\\ { { { } } } \\\\ { { { \\mathrm { H M } } _ { s o f t } ( V , \\lambda ) = \\Big ( \\displaystyle \\sum _ { i } ^ { N } w _ { i } V _ { i } ^ { - 1 } \\Big ) ^ { - 1 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 351, + 242, + 643, + 375 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $w _ { i } = e ^ { \\lambda V _ { i } } / \\Sigma _ { j } e ^ { \\lambda V _ { j } }$ with $\\lambda \\ge 0 , V _ { i } < 0$ . Using a softmax also has the well known advantage of being differentiable (as opposed to subdifferentiable for max). Note that we only require continuity to guarantee that computing the softmax is actually equivalent to computing $V ( \\tilde { D } , G )$ where $\\tilde { D }$ is some convex combination of $D _ { i }$ (see Appendix A.5). ", + "bbox": [ + 174, + 380, + 825, + 439 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 USING THE ORIGINAL MINIMAX OBJECTIVE ", + "text_level": 1, + "bbox": [ + 174, + 455, + 526, + 470 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To illustrate the effect the softmax has on training, observe that the component of $A M _ { s o f t } ( V , 0 )$ relevant to generator training can be rewritten as ", + "bbox": [ + 171, + 481, + 823, + 511 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b58fcfef8b959b666bec5deb8dfc486508913f6e1d9aae8817c2216645d243aa.jpg", + "text": "$$\n\\frac { 1 } { N } \\sum _ { i } ^ { N } \\mathbb { E } _ { x \\sim p _ { G } ( x ) } \\Big [ \\log ( 1 - D _ { i } ( x ) ) \\Big ] = \\frac { 1 } { N } \\mathbb { E } _ { x \\sim p _ { G } ( x ) } \\Big [ \\log ( z ) \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 516, + 700, + 559 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { z = \\prod _ { i } ^ { N } ( 1 - D _ { i } ( x ) ) } \\end{array}$ . Note that the generator gradient, | ∂ log(z)∂z |, is minimized at z = 1 over $z \\in ( 0 , 1 ] ^ { 1 }$ . From this form, it is clear that $z = 1$ if and only if $D _ { i } = 0 \\forall i$ , so $G$ only receives a vanishing gradient if all $D _ { i }$ agree that the sample is fake; this is especially unlikely for large $N$ . In other words, $G$ only needs to fool a single $D _ { i }$ to receive constructive feedback. This result allows the generator to successfully minimize the original generator objective, $\\log ( 1 - D )$ . This is in contrast to the more popular $- \\log ( D )$ introduced to artificially enhance gradients at the start of training. ", + "bbox": [ + 173, + 565, + 825, + 655 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "At the beginning of training, when $\\operatorname* { m a x } _ { D _ { i } } V ( D _ { i } , G )$ is likely too harsh a critic for the generator, we can set $\\lambda$ closer to zero to use the mean, increasing the odds of providing constructive feedback to the generator. In addition, the discriminators have the added benefit of functioning as an ensemble, reducing the variance of the feedback presented to the generator, which is especially important when the discriminators are far from optimal and are still learning a reasonable decision boundary. As training progresses and the discriminators improve, we can increase $\\lambda$ to become more critical of the generator for more refined training. ", + "bbox": [ + 173, + 661, + 825, + 760 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.3 MAINTAINING MULTIPLE HYPOTHESES ", + "text_level": 1, + "bbox": [ + 176, + 775, + 488, + 790 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We argue for this ensemble approach on a more fundamental level as well. Here, we draw on the density ratio estimation perspective of GANs (Uehara et al. (2016)). The original GAN proof assumes we have access to $p _ { d a t a } ( x )$ , if only implicitly. In most cases of interest, the discriminator only has access to a finite dataset sampled from $p _ { d a t a } ( x )$ ; therefore, when computing expectations of $\\dot { V } ( D , G )$ , we only draw samples from our finite dataset. This is equivalent to training a GAN with $p _ { d a t a } ( x ) = \\tilde { p } _ { d a t a }$ which is a distribution consisting of point masses on all the data points in the dataset. For the sake of argument, let’s assume we are training a discriminator and generator, each with infinite capacity. In this case, the global optimum $( p _ { G } ( x ) = \\tilde { p } _ { d a t a ( x ) } )$ fails to capture any of the interesting structure from $p _ { d a t a } ( x )$ , the true distribution we are trying to learn. Therefore, it is actually critical that we avoid this global optimum. ", + "bbox": [ + 173, + 801, + 825, + 900 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 147 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/6825e5bfd09d2c7ad38711198c6dc71b70472969679776d3899bf8813fa65739.jpg", + "image_caption": [ + "Figure 2: Consider a dataset consisting of the nine 1-dimensional samples in black. Their corresponding probability mass function is given in light gray. After training GMAN, three discriminators converge to distinct local optima which implicitly define distributions over the data (red, blue, yellow). Each discriminator may specialize in discriminating a region of the data space (placing more diffuse mass in other regions). Averaging over the three discriminators results in the distribution in black, which we expect has higher likelihood under reasonable assumptions on the structure of the true distribution. " + ], + "image_footnote": [], + "bbox": [ + 379, + 160, + 617, + 292 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In practice, this degenerate result is avoided by employing learners with limited capacity and corrupting data samples with noise (i.e., dropout), but we might better accomplish this by simultaneously training a variety of limited capacity discriminators. With this approach, we might obtain a diverse set of seemingly tenable hypotheses for the true $p _ { d a t a } ( x )$ . Averaging over these multiple locally optimal discriminators increases the entropy of $\\tilde { p } _ { d a t a } ( x )$ by diffusing the probability mass over the data space (see Figure 2 for an example). ", + "bbox": [ + 173, + 415, + 825, + 500 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.4 AUTOMATING REGULATION", + "text_level": 1, + "bbox": [ + 176, + 515, + 406, + 529 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The problem of keeping the discriminator and generator in balance has been widely recognized in previous work with GANs. Issues with unstable dynamics, oscillatory behavior, and generator collapse are not uncommon. In addition, the discriminator is often times able to achieve a high degree of classification accuracy (producing a single scalar) before the generator has made sufficient progress on the arguably more difficult generative task (producing a high dimensional sample). Salimans et al. (2016) suggested label smoothing to reduce the vulnerability of the generator to a relatively superior discriminator. Here, we explore an approach that enables the generator to automatically temper the performance of the discriminator when necessary, but still encourages the generator to challenge itself against more accurate adversaries. Specifically, we augment the generator objective: ", + "bbox": [ + 173, + 540, + 825, + 666 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6242b41e4b33660e55e229c2930ce1c833008d3f1bb3fc6d601eb5551ff1cbe2.jpg", + "text": "$$\n\\operatorname* { m i n } _ { G , \\lambda > 0 } F _ { G } ( V _ { i } ) - f ( \\lambda )\n$$", + "text_format": "latex", + "bbox": [ + 424, + 666, + 573, + 690 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $f ( \\lambda )$ is monotonically increasing in $\\lambda$ which appears in the softmax equations, (3)—(5). In experiments, we simply set $f ( \\lambda ) = c \\lambda$ with $c$ a constant (e.g., 0.001). The generator is incentivized to increase $\\lambda$ to reduce its objective at the expense of competing against the best available adversary $D ^ { * }$ (see Appendix A.6). ", + "bbox": [ + 174, + 693, + 825, + 750 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 768, + 313, + 784 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Evaluating GANs is still an open problem. In their original work, Goodfellow et al. (2014) report log likelihood estimates from Gaussian Parzen windows, which they admit, has high variance and is known not to perform well in high dimensions. Theis et al. (2016) recommend avoiding Parzen windows and argue that generative models should be evaluated with respect to their intended application. Salimans et al. (2016) suggest an Inception score, however, it assumes labels exist for the dataset. Recently, Im et al. (2016) introduced the Generative Adversarial Metric (GAM) for making pairwise comparisons between independently trained GAN models. The core idea behind their approach is given two generator, discriminator pairs $( G _ { 1 } , D _ { 1 } )$ and $( G _ { 2 } , D _ { 2 } )$ , we should be able to learn their relative performance by judging each generator under the opponent’s discriminator. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.1 METRIC ", + "text_level": 1, + "bbox": [ + 173, + 103, + 272, + 117 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In GMAN, the opponent may have multiple discriminators, which makes it unclear how to perform the swaps needed for GAM. We introduce a variant of GAM, the generative multi-adversarial metric (GMAM), that is amenable to training with multiple discriminators, ", + "bbox": [ + 173, + 130, + 825, + 171 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/220c866917c1348388c96aa39b0f96f0bf439729306391c9f59129cf9f0175ae.jpg", + "text": "$$\n\\mathrm { G M A M } = \\log \\Big ( \\frac { F _ { G _ { b } } ^ { a } ( V _ { i } ^ { a } ) } { F _ { G _ { a } } ^ { a } ( V _ { i } ^ { a } ) } \\Big / \\frac { F _ { G _ { a } } ^ { b } ( V _ { i } ^ { b } ) } { F _ { G _ { b } } ^ { b } ( V _ { i } ^ { b } ) } \\Big ) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 179, + 633, + 218 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $a$ and $b$ refer to the two GMAN variants (see Section 3 for notation $F _ { G } ( V _ { i } ) ,$ ). The idea here is similar. If $G _ { 2 }$ performs better than $G _ { 1 }$ with respect to both $D _ { 1 }$ and $D _ { 2 }$ , then $\\mathrm { G M A M { > } 0 }$ (remember $V { \\le } 0$ always). If $G _ { 1 }$ performs better in both cases, GMAM $< 0$ , otherwise, the result is indeterminate. ", + "bbox": [ + 174, + 224, + 825, + 267 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 285, + 316, + 299 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate the aforementioned variations of GMAN on a variety of image generation tasks: MNIST (LeCun et al. (1998)), CIFAR-10 (Krizhevsky (2009)) and CelebA (Liu et al. (2015)). We focus on rates of convergence to steady state along with quality of the steady state generator according to the GMAM metric. To summarize, loosely in order of increasing discriminator leniency, we compare ", + "bbox": [ + 174, + 310, + 825, + 367 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• F-boost: A single AdaBoost.OL-boosted discriminator (see Appendix A.7). \n• P-boost: $D _ { i }$ is trained according to AdaBoost.OL. A max over the weak learner losses is presented to the generator instead of the boosted prediction (see Appendix A.7). \n• GMAN-max: max $\\{ V _ { i } \\}$ is presented to the generator. \n• GAN: Standard GAN with a single discriminator (see Appendix A.2). \n• mod-GAN: GAN with modified objective (generator minimizes − $\\log ( D ( G ( z ) ) )$ . \n• GMAN- $\\lambda$ : GMAN with $F$ : $=$ arithmetic softmax with parameter $\\lambda$ . \n• GMAN∗: The arithmetic softmax is controlled by the generator through $\\lambda$ . ", + "bbox": [ + 215, + 378, + 826, + 525 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "All generator and discriminator models are deep (de)convolutional networks (Radford et al. (2015)), and aside from the boosted variants, all are trained with Adam (Kingma & Ba (2014)) and batch normalization (Ioffe & Szegedy (2015)). Discriminators convert the real-valued outputs of their rksivedis $( \\epsilon \\dot { + } \\frac { 1 - 2 \\epsilon } { 1 + e ^ { - z } } )$ ies with squashed-sigmoids to prevent saturating logarithms in the min. See Appendix A.8 for further details. We test GMAN systems with We maintain discriminator diversity by varying dropout and network de $N =$ $\\{ 2 , 5 \\}$ ", + "bbox": [ + 173, + 536, + 825, + 622 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2.1 MNIST ", + "text_level": 1, + "bbox": [ + 174, + 637, + 284, + 652 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 3 reveals that increasing the number of discriminators reduces the number of iterations to steady-state by $2 \\mathbf { x }$ on MNIST; increasing $N$ (the size of the discriminator ensemble) also has the added benefit of reducing the variance the minimax objective over runs. Figure 4 displays the variance of the same objective over a sliding time window, reaffirming GMAN’s acceleration to steadystate. Figure 5 corroborates this conclusion with recognizable digits appearing approximately an epoch before the single discriminator run; digits at steady-state appear slightly sharper as well. ", + "bbox": [ + 173, + 662, + 825, + 747 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/efc3513f1ae67a13cacc78e4ee04d210ed8a7debf623c8f8e60d42ff1aab2a57.jpg", + "table_caption": [ + "Our GMAM metric (see Table 1) agrees with the relative quality of images in Figure 5 with GMAN∗ achieving the best overall performance. Figure 6 reveals GMAN∗’s attempt to regulate the difficulty " + ], + "table_footnote": [], + "table_body": "
ScoreVariantGMAN*GMAN-0GMAN-maxmod-GAN
←Feneg0.127GMAN*-0.020±0.009-0.028 ± 0.019-0.089± 0.036
0.007GMAN-00.020 ±0.009-0.013 ± 0.015-0.018 ± 0.027
-0.034GMAN-max0.028 ± 0.0190.013 ± 0.015-0.011 ± 0.024
-0.122mod-GAN0.089 ± 0.0360.018 ± 0.0270.011 ± 0.024
", + "bbox": [ + 174, + 794, + 870, + 868 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 1: Pairwise GMAM metric means with stdev for select models on MNIST. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each variant’s column. ", + "bbox": [ + 173, + 877, + 825, + 920 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/08c9b309155352dc2819d633838b3a4d5befc6b6345a4dc3783dc239c759e699.jpg", + "image_caption": [ + "Figure 3: Generator objective, $F$ , averaged over 5 training runs on MNIST. Increasing the number of discriminators accelerates convergence of $F$ to steady state (solid line) and reduces its variance, $\\scriptstyle { \\dot { \\sigma } } ^ { 2 }$ (filled shadow $\\pm 1 \\sigma )$ . Figure 4 provides alternative evidence of GMAN∗’s accelerated convergence. " + ], + "image_footnote": [], + "bbox": [ + 189, + 108, + 460, + 262 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/91baa284ba5198c489e8fe62abdaa1cd4493d944a3ed1d655748b8ce3d35238a.jpg", + "image_caption": [ + "Figure 4: Stdev, $\\sigma$ , of the generator objective over a sliding window of 500 iterations. Lower values indicate a more steady-state. $\\mathrm { G M A N ^ { * } }$ with $N = 5$ achieves steady-state at ${ \\approx } 2 \\mathbf { x }$ speed of GAN $N = 1$ ). Note Figure 3’s filled shadows reveal stdev of $F$ over runs, while this plot shows stdev over time. " + ], + "image_footnote": [], + "bbox": [ + 542, + 117, + 805, + 258 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/927417a6a7e30513a07171408412de620cc44add4f1baf38e0437e716f97524e.jpg", + "image_caption": [ + "Figure 5: Comparison of image quality across epochs for $N = \\{ 1 , 2 , 5 \\}$ using GMAN-0 on MNIST. " + ], + "image_footnote": [], + "bbox": [ + 173, + 400, + 818, + 556 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "of the game to accelerate learning. Figure 7 displays the GMAM scores comparing fixed $\\lambda$ ’s to the variable $\\lambda$ controlled by $\\mathrm { G M A N ^ { * } }$ . ", + "bbox": [ + 173, + 613, + 825, + 643 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d1288fd5c17b2995cb9baccf68afd0bd47d70210dc6f4eab09ad31281d1fc246.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Score入 (N=5)*入=1入=0
1eeieg0.028*=-0.008 ±0.009-0.019 ±0.010 -0.008 ±0.010
0.001入=10.008 ±0.009
-0.025入=00.019 ±0.0100.008 ±0.010
", + "bbox": [ + 490, + 684, + 856, + 804 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e8aa4f28694600150894ad026b0802a1d5d5dbe36b1d2be2f0f461a34fa2385d.jpg", + "image_caption": [ + "Figure 6: GMAN∗ regulates difficulty of the game by adjusting $\\lambda$ . Initially, $G$ reduces $\\lambda$ to ease learning and then gradually increases $\\lambda$ for a more challenging learning environment. ", + "Figure 7: Pairwise $\\frac { \\mathbf { G } \\mathbf { M } \\mathbf { A } \\mathbf { M } } { s t d e \\nu ( \\mathbf { G } \\mathbf { M } \\mathbf { A } \\mathbf { M } ) }$ for GMAN- $\\lambda$ and GMAN∗ $( \\lambda ^ { * } )$ over 5 runs on MNIST. " + ], + "image_footnote": [], + "bbox": [ + 196, + 670, + 467, + 823 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2.2 CELEBA & CIFAR-10 ", + "text_level": 1, + "bbox": [ + 176, + 103, + 385, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We see similar accelerated convergence behavior for the CelebA dataset in Figure 8. ", + "bbox": [ + 173, + 127, + 723, + 142 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/0fec7eacde54564201c48cef48a66b25d97982a4db342eb0cb73c0d2af2e480b.jpg", + "image_caption": [ + "Figure 8: Image quality improvement across number of generators at same number of iterations for GMAN-0 on CelebA. " + ], + "image_footnote": [], + "bbox": [ + 176, + 154, + 820, + 321 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/4e3d61aa52dc0162b9f5a5f3fa6e3cc1b7828787aa84570d77f19a2ba6bc225e.jpg", + "image_caption": [ + "Figure 9 displays images generated by GMAN-0 on CIFAR-10. See Appendix A.3 for more results. ", + "Figure 9: Images generated by GMAN-0 on the CIFAR-10 dataset. " + ], + "image_footnote": [], + "bbox": [ + 176, + 420, + 823, + 513 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also found that GMAN is robust to mode collapse. We believe this is because the generator must appease a diverse set of discriminators in each minibatch. Emitting a single sample will score well for one discriminator at the expense of the rest of the discriminators. Current solutions (e.g., minibatch discrimination) are quadratic in batch size. GMAN, however, is linear in batch size. ", + "bbox": [ + 174, + 561, + 825, + 617 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 637, + 318, + 654 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We introduced multiple discriminators into the GAN framework and explored discriminator roles ranging from a formidable adversary to a forgiving teacher. Allowing the generator to automatically tune its learning schedule $\\mathbf { ( G M A N ^ { * } }$ ) outperformed GANs with a single discriminator on MNIST. In general, GMAN variants achieved faster convergence to a higher quality steady state on a variety of tasks as measured by a GAM-type metric (GMAM). In addition, GMAN makes using the original GAN objective possible by increasing the odds of the generator receiving constructive feedback. ", + "bbox": [ + 174, + 669, + 825, + 752 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In future work, we will look at more sophisticated mechanisms for letting the generator control the game as well as other ways to ensure diversity among the discriminators. Introducing multiple generators is conceptually an obvious next step, however, we expect difficulties to arise from more complex game dynamics. For this reason, game theory and game design will likely be important. ", + "bbox": [ + 176, + 758, + 825, + 815 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 830, + 326, + 843 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We acknowledge helpful conversations with Stefan Dernbach, Archan Ray, Luke Vilnis, Ben Turtel, Stephen Giguere, Rajarshi Das, and Subhransu Maji. We also thank NVIDIA for donating a K40 GPU. This material is based upon work supported by the National Science Foundation under Grant Nos. IIS-1564032. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. ", + "bbox": [ + 174, + 853, + 823, + 922 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "BIBLIOGRAPHY ", + "text_level": 1, + "bbox": [ + 174, + 102, + 305, + 118 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Martın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. ", + "bbox": [ + 173, + 133, + 826, + 176 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Hana Ajakan, Pascal Germain, Hugo Larochelle, Franc¸ois Laviolette, and Mario Marchand. 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In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pp. 2528–2535. IEEE, 2010. ", + "bbox": [ + 173, + 545, + 823, + 589 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016. ", + "bbox": [ + 171, + 598, + 823, + 627 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 299, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.1 ACCELERATED CONVERGENCE & REDUCED VARIANCE ", + "text_level": 1, + "bbox": [ + 174, + 133, + 604, + 148 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "See Figures 10, 11, 12, and 13. ", + "bbox": [ + 174, + 160, + 379, + 174 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/b86d64796a6407863f16b73360d23ccb0605cbb461cdf3aa67ee32ce2732ac42.jpg", + "image_caption": [ + "Figure 10: Generator objective, $F$ , averaged over 5 training runs on CelebA. Increasing $N$ (# of $D$ ) accelerates convergence of $F$ to steady state (solid line) and reduces its variance, $\\sigma ^ { 2 }$ (filled shadow $\\pm 1 \\sigma$ ). Figure 11 provides alternative evidence of GMAN-0’s accelerated convergence. " + ], + "image_footnote": [], + "bbox": [ + 189, + 195, + 462, + 349 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/9622c7b37ca5ddacd7af81de88f321c6e0817503fdc2273800e82b66e56f1305.jpg", + "image_caption": [ + "Figure 11: Stdev, $\\sigma$ , of the generator objective over a sliding window of 500 iterations. Lower values indicate a more steady-state. GMAN-0 with $N = 5$ achieves steady-state at ${ \\approx } 2 \\mathbf { x }$ speed of GAN $( N = 1 )$ ). Note Figure $1 0 ^ { \\circ } \\mathrm { s }$ filled shadows reveal stdev of $F$ over runs, while this plot shows stdev over time. " + ], + "image_footnote": [], + "bbox": [ + 542, + 204, + 802, + 347 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/aee15738e9c5166c1e19d8010f35de87c81b2f5d02bae4c11f691b2967a0f3b0.jpg", + "image_caption": [ + "Figure 12: Generator objective, $F$ , averaged over 5 training runs on CIFAR-10. Increasing $N$ (# of $D$ ) accelerates convergence of $F$ to steady state (solid line) and reduces its variance, $\\textstyle { \\bar { \\sigma } } ^ { 2 }$ (filled shadow $\\pm 1 \\sigma$ ). Figure 13 provides alternative evidence of GMAN-0’s accelerated convergence. " + ], + "image_footnote": [], + "bbox": [ + 187, + 498, + 465, + 654 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/61640db03eb7afae558dd80985fbefff02d9bc9af9635f45b050ee336a25f67b.jpg", + "image_caption": [ + "Figure 13: Stdev, $\\sigma$ , of the generator objective over a sliding window of 500 iterations. Lower values indicate a more steady-state. GMAN-0 with $N = 5$ achieves steady-state at ${ \\approx } 2 \\mathbf { x }$ speed of GAN $N = 1$ ). Note Figure 12’s filled shadows reveal stdev of $F$ over runs, while this plot shows stdev over time. " + ], + "image_footnote": [], + "bbox": [ + 537, + 498, + 807, + 652 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.2 ADDITIONAL GMAM TABLES ", + "text_level": 1, + "bbox": [ + 176, + 796, + 428, + 811 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "See Tables 2, 3, 4, 5, 6. Increasing the number of discriminators from 2 to 5 on CIFAR-10 significantly improves scores over the standard GAN both in terms of the GMAM metric and Inception scores. ", + "bbox": [ + 174, + 823, + 825, + 864 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.3 GENERATED IMAGES ", + "text_level": 1, + "bbox": [ + 176, + 882, + 364, + 897 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "See Figures 14 and 15. ", + "bbox": [ + 174, + 909, + 325, + 924 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/113b8353d25d9de82b656521500b8bf842c3ef919a08c421aca2174716c6d6f9.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ScoreVariantGMAN*GMAN-1GANGMAN-0GMAN-maxmod-GAN
←Feneg0.184 0.067GMAN*=-0.007-0.040-0.020-0.028-0.089
0.030GMAN-10.0071-0.008-0.008-0.021-0.037
GAN0.0400.008-0.002-0.018-0.058
0.005GMAN-00.0200.0080.002--0.013-0.018
-0.091GMAN-max0.0280.0210.0180.013--0.011
-0.213mod-GAN0.0890.0370.0580.0180.011=
", + "bbox": [ + 176, + 102, + 882, + 204 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Table 2: Pairwise GMAM metric means for select models on MNIST. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each column. ", + "bbox": [ + 174, + 213, + 826, + 256 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/b6b6cac20d18c634651c0fe12ccca03103aedf52143210bedd61f6ce8a725dc3.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ScoreVariantGMAN-0GMAN-1GMAN*mod-GAN
Beeneg0.172GMAN-01-0.022-0.062-0.088
0.050GMAN-10.0220.006-0.078
-0.055GMAN*0.062-0.006=-0.001
-0.167mod-GAN0.0880.0780.001=
", + "bbox": [ + 238, + 267, + 761, + 342 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Table 3: Pairwise GMAM metric means for select models on CIFAR-10. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each column. GMAN variants were trained with two discriminators. ", + "bbox": [ + 176, + 351, + 823, + 393 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/26d44dadf530c323264bda28ab841a25ce66b4e06c9dd3ac648e694830a0a29c.jpg", + "table_caption": [ + "Table 4: Inception score means with standard deviations for select models on CIFAR-10. Higher scores are better. GMAN variants were trained with two discriminators. " + ], + "table_footnote": [], + "table_body": "
GMAN-0GMAN-1mod-GANGMAN*
Score5.878± 0.1935.765 ± 0.1685.738± 0.1765.539± 0.099
", + "bbox": [ + 238, + 405, + 758, + 438 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/68ee9c589262f253c9549a8ad23f33896f1af4b1cd69d130b4684785fece9095.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ScoreVariantGMAN-0GMAN*GMAN-1mod-GAN
FFeeeg0.180GMAN-0=-0.008-0.041-0.132
0.122GMAN*0.008=-0.038-0.092
0.010GMAN-10.0410.038-0.089
-0.313mod-GAN0.1320.0920.0891
", + "bbox": [ + 238, + 488, + 761, + 563 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Table 5: Pairwise GMAM metric means for select models on CIFAR-10. For each column, a positive GMAM indicates better performance relative to the row opponent; negative implies worse. Scores are obtained by summing each column. GMAN variants were trained with five discriminators. ", + "bbox": [ + 176, + 571, + 823, + 614 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/d50f648a2f076beadaef6cd28c8faeede816c6089ec186aedb8fb9755f70340b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
GMAN-1GMAN-0GMAN*mod-GAN
Score6.001± 0.1945.957 ± 0.1355.955 ± 0.1535.738 ± 0.176
", + "bbox": [ + 243, + 626, + 758, + 659 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Table 6: Inception score means with standard deviations for select models on CIFAR-10. Higher scores are better. GMAN variants were trained with five discriminators. ", + "bbox": [ + 173, + 667, + 823, + 696 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/020e890b6cb8dc91c0185efe78a1ef059894f20210a4366160346a2c3173932f.jpg", + "image_caption": [ + "Figure 14: Sample of pictures generated on CelebA cropped dataset. " + ], + "image_footnote": [], + "bbox": [ + 176, + 710, + 820, + 893 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/5738407f4436e1f5c6e9f59db0134286b0d0355991c78a133dd9476e96d7f73a.jpg", + "image_caption": [ + "Figure 15: Sample of pictures generated by GMAN-0 on CIFAR dataset. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 816, + 443 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.4 SOMEWHAT RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 512, + 419, + 526 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A GAN framework with two discriminators appeared in Yoo et al. (2016), however, it is applicable only in a semi-supervised case where a label can be assigned to subsets of the dataset (e.g., $\\mathcal { X } \\ = \\ \\{ \\mathcal { X } _ { 1 } = \\operatorname { D o m a i n } \\ 1 , \\mathcal { X } _ { 2 } = \\operatorname { D o m a i n } \\ 2 , \\ldots \\} .$ ). In contrast, our framework applies to an unsupervised scenario where an obvious partition of the dataset is unknown. Furthermore, extending GMAN to the semi-supervised domain-adaptation scenario would suggest multiple discriminators per domain, therefore our line of research is strictly orthogonal to that of their multi-domain discriminator approach. Also, note that assigning a discriminator to each domain is akin to prescribing a new discriminator to each value of a conditional variable in conditional GANs (Mirza & Osindero (2014)). In this case, we interpret GMAN as introducing multiple conditional discriminators and not a discriminator for each of the possibly exponentially many conditional labels. ", + "bbox": [ + 173, + 545, + 825, + 684 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Section 4.4, we describe an approach to customize adversarial training to better suit the development of the generator. An approach with similar conceptual underpinnings was described in Ravanbakhsh et al. (2016), however, similar to the above, it is only admissible in a semi-supervised scenario whereas our applies to the unsupervised case. ", + "bbox": [ + 174, + 690, + 825, + 747 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.5 Softmax REPRESENTABILITY ", + "text_level": 1, + "bbox": [ + 176, + 781, + 413, + 796 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let soft $n a x ( V _ { i } ) = \\hat { V } \\in [ \\operatorname* { m i n } _ { V _ { i } } , \\operatorname* { m a x } _ { V _ { i } } ]$ . Also let $a = a r g \\operatorname* { m i n } _ { i } V _ { i }$ , $b = a r g \\operatorname* { m a x } _ { i } V _ { i }$ , and $\\gamma ( t ) =$ $V ( ( 1 - t ) D _ { a } + t D _ { b } )$ so that $\\nu ( 0 ) = V _ { a }$ and $\\nu ( 1 ) = V _ { b }$ . The softmax and minimax objective $V ( D _ { i } , G )$ are both continuous in their inputs, so by the intermediate value theorem, we have that $\\exists { \\hat { t } } \\in [ 0 , 1 ]$ s.t. $\\mathcal { V } ( \\hat { t } ) = \\hat { V }$ , which implies $\\exists \\hat { D } \\in \\mathcal { D }$ s.t. $V ( \\hat { D } , G ) = \\hat { V }$ . This result implies that the softmax (and any other continuous substitute) can be interpreted as returning $V ( \\hat { D } , G )$ for some $\\hat { D }$ selected by computing an another, unknown function over the space of the discriminators. This result holds even if $\\hat { D }$ is not representable by the architecture chosen for $D$ ’s neural network. ", + "bbox": [ + 173, + 815, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.6 UNCONSTRAINED OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 449, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "To convert GMAN∗ minimax formulation to an unconstrained minimax formulation, we introduce an auxiliary variable, $\\Lambda$ , define $\\lambda ( \\Lambda ) = \\log ( 1 + e ^ { \\Lambda } )$ , and let the generator minimize over $\\Lambda \\in \\mathbb { R }$ . ", + "bbox": [ + 173, + 128, + 825, + 159 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.7 BOOSTING WITH AdaBoost.OL ", + "text_level": 1, + "bbox": [ + 176, + 174, + 426, + 189 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "AdaBoost.OL (Beygelzimer et al. (2015)) does not require knowledge of the weak learner’s slight edge over random guessing $( P ( { \\mathrm { c o r r e c t 1 a b e l } } ) = 0 . 5 + \\gamma \\in ( 0 , 0 . 5 ] )$ , and in fact, allows $\\gamma < 0$ . This is crucial because our weak learners are deep nets with unknown, possibly negative, $\\gamma$ ’s. ", + "bbox": [ + 174, + 200, + 826, + 243 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/4734cd63537b4f2336b3d8c856ffdb1731187e1ee2c98fca4b386b349c0f1212.jpg", + "image_caption": [ + "Figure 16: Sample of pictures generated across 4 independent runs on MNIST with F-boost (similar results with P-boost). " + ], + "image_footnote": [], + "bbox": [ + 263, + 257, + 735, + 440 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.8 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 505, + 379, + 518 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "All experiments were conducted using an architecture similar to DCGAN (Radford et al. (2015)). We use convolutional transpose layers (Zeiler et al. (2010)) for $G$ and strided convolutions for $D$ except for the input of $G$ and the last layer of $D$ . We use the single step gradient method as in (Nowozin et al. (2016)), and batch normalization (Ioffe & Szegedy (2015)) was used in each of the generator layers. The different discriminators were trained with varying dropout rates from [0.3, 0.7]. Variations in the discriminators were effected in two ways. We varied the architecture by varying the number of filters in the discriminator layers (reduced by factors of 2, 4 and so on), as well as varying dropout rates. Secondly we also decorrelated the samples that the disriminators were training on by splitting the minibatch across the discriminators. The code was written in Tensorflow (Abadi et al. (2016)) and run on Nvidia GTX 980 GPUs. Code to reproduce experiments and plots is at https://github.com/iDurugkar/GMAN. Specifics for the MNIST architecture and training are: ", + "bbox": [ + 173, + 531, + 825, + 684 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Generator latent variables z ∼ U (−1, 1)100 \n• Generator convolution transpose layers: $\\left( 4 , 4 , 1 2 8 \\right) , \\left( 8 , 8 , 6 4 \\right) , \\left( 1 6 , 1 6 , 3 2 \\right) , \\left( 3 2 , 3 2 , 1 \\right)$ \n• Base Discriminator architecture: $\\left( 3 2 , 3 2 , 1 \\right) , \\left( 1 6 , 1 6 , 3 2 \\right) , \\left( 8 , 8 , 6 4 \\right) , \\left( 4 , 4 , 1 2 8 \\right) .$ \n• Variants have either convolution 3 $( 4 , 4 , 1 2 8 )$ removed or all the filter sizes are divided by 2 or 4. That is, $\\left( 3 2 , 3 2 , 1 \\right) , \\left( 1 6 , 1 6 , 1 6 \\right) , \\left( 8 , 8 , 3 2 \\right) , \\left( 4 , 4 , 6 4 \\right)$ or $\\left( 3 2 , 3 2 , 1 \\right) , \\left( 1 6 , 1 6 , 8 \\right) , \\left( 8 , 8 , 1 6 \\right) , \\left( 4 , 4 , 3 2 \\right)$ . \n• ReLu activations for all the hidden units. Tanh activation at the output units of the generator. Sigmoid at the output of the Discriminator. \n• Training was performed with Adam (Kingma & Ba (2014)) $( l r = 2 \\times 1 0 ^ { - 4 }$ , $\\beta _ { 1 } = 0 . 5 )$ ). \n• MNIST was trained for 20 epochs with a minibatch of size 100. \n• CelebA and CIFAR were trained over 24000 iterations with a minibatch of size 100. 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Section 6 concludes", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 391, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 391, + 105 + ], + "score": 1.0, + "content": "with a summary of our contributions and directions for future research.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Contributions—To summarize, our main contributions are: i) a multi-discriminator GAN frame-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "work, GMAN, that allows training with the original, untampered minimax objective; ii) a generative", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "multi-adversarial metric (GMAM) to perform pairwise evaluation of separately trained frameworks;", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 255, + 156 + ], + "score": 1.0, + "content": "iii) a particular instance of GMAN,", + "type": "text" + }, + { + "bbox": [ + 256, + 144, + 291, + 154 + ], + "score": 0.33, + "content": "\\mathrm { G M A N ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 143, + 505, + 156 + ], + "score": 1.0, + "content": ", that allows the generator to automatically regulate", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "training and reach higher performance (as measured by GMAM) in a fraction of the training time", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 261, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 261, + 177 + ], + "score": 1.0, + "content": "required for the standard GAN model.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 192, + 393, + 205 + ], + "lines": [ + { + "bbox": [ + 104, + 190, + 394, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 394, + 207 + ], + "score": 1.0, + "content": "2 GENERATIVE ADVERSARIAL NETWORKS TO GMAN", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 415, + 229 + ], + "score": 1.0, + "content": "The original formulation of a GAN is a minimax game between a generator,", + "type": "text" + }, + { + "bbox": [ + 415, + 217, + 477, + 229 + ], + "score": 0.93, + "content": "G _ { \\theta } ( z ) : z \\to x", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 216, + 505, + 229 + ], + "score": 1.0, + "content": ", and a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 227, + 246, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 164, + 241 + ], + "score": 1.0, + "content": "discriminator,", + "type": "text" + }, + { + "bbox": [ + 165, + 228, + 242, + 240 + ], + "score": 0.93, + "content": "D _ { \\omega } ( x ) : x [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 227, + 246, + 241 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 243, + 469, + 264 + ], + "lines": [ + { + "bbox": [ + 140, + 243, + 469, + 264 + ], + "spans": [ + { + "bbox": [ + 140, + 243, + 469, + 264 + ], + "score": 0.9, + "content": "\\underset { G } { \\mathrm { m i n } } \\underset { D \\in \\mathcal { D } } { \\mathrm { m a x } } V ( D , G ) = \\mathbb { E } _ { x \\sim p _ { d a t a } ( x ) } \\Big [ \\log ( D ( x ) ) \\Big ] + \\mathbb { E } _ { z \\sim p _ { z } ( z ) } \\Big [ \\log ( 1 - 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Let", + "type": "text" + }, + { + "bbox": [ + 296, + 291, + 322, + 303 + ], + "score": 0.92, + "content": "p _ { G } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 292, + 505, + 303 + ], + "score": 1.0, + "content": "be the distribution induced by the generator,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 302, + 412, + 315 + ], + "spans": [ + { + "bbox": [ + 107, + 302, + 132, + 314 + ], + "score": 0.91, + "content": "G _ { \\theta } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 302, + 184, + 315 + ], + "score": 1.0, + "content": ". We assume", + "type": "text" + }, + { + "bbox": [ + 185, + 302, + 207, + 314 + ], + "score": 0.9, + "content": "D , G", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 302, + 412, + 315 + ], + "score": 1.0, + "content": "to be deep neural networks as is typically the case.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "In their original work, Goodfellow et al. 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We explore approaches ranging between two extremes: 1) a more discriminating", + "type": "text" + }, + { + "bbox": [ + 465, + 614, + 474, + 624 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "(better", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 168, + 638 + ], + "score": 1.0, + "content": "approximating", + "type": "text" + }, + { + "bbox": [ + 168, + 625, + 235, + 637 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 624, + 275, + 638 + ], + "score": 1.0, + "content": "and 2) a", + "type": "text" + }, + { + "bbox": [ + 276, + 626, + 285, + 635 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 624, + 506, + 638 + ], + "score": 1.0, + "content": "better matched to the generator’s capabilities. 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Each", + "type": "text" + }, + { + "bbox": [ + 243, + 647, + 255, + 658 + ], + "score": 0.91, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 645, + 463, + 660 + ], + "score": 1.0, + "content": "is still expected to independently maximize its own", + "type": "text" + }, + { + "bbox": [ + 463, + 647, + 504, + 659 + ], + "score": 0.91, + "content": "V ( D _ { i } , G )", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 297, + 670 + ], + "score": 1.0, + "content": "(i.e. no cooperation). 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Let", + "type": "text" + }, + { + "bbox": [ + 296, + 291, + 322, + 303 + ], + "score": 0.92, + "content": "p _ { G } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 292, + 505, + 303 + ], + "score": 1.0, + "content": "be the distribution induced by the generator,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 302, + 412, + 315 + ], + "spans": [ + { + "bbox": [ + 107, + 302, + 132, + 314 + ], + "score": 0.91, + "content": "G _ { \\theta } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 302, + 184, + 315 + ], + "score": 1.0, + "content": ". We assume", + "type": "text" + }, + { + "bbox": [ + 185, + 302, + 207, + 314 + ], + "score": 0.9, + "content": "D , G", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 302, + 412, + 315 + ], + "score": 1.0, + "content": "to be deep neural networks as is typically the case.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 268, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "In their original work, Goodfellow et al. 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In practice, they replaced the second term,", + "type": "text" + }, + { + "bbox": [ + 472, + 352, + 505, + 364 + ], + "score": 0.86, + "content": "\\log ( 1 -", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 363, + 148, + 375 + ], + "score": 0.89, + "content": "D ( G ( z ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 362, + 172, + 376 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 172, + 363, + 240, + 375 + ], + "score": 0.87, + "content": "- \\log ( D ( G ( z ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "to enhance gradient signals at the start of the game; note this is no", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "longer a zero-sum game. 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Minimizing", + "type": "text" + }, + { + "bbox": [ + 361, + 425, + 385, + 437 + ], + "score": 0.91, + "content": "C ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 425, + 478, + 438 + ], + "score": 1.0, + "content": "necessarily minimizes", + "type": "text" + }, + { + "bbox": [ + 479, + 425, + 501, + 436 + ], + "score": 0.78, + "content": "J S D", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 425, + 505, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 436, + 497, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 209, + 449 + ], + "score": 1.0, + "content": "however, we rarely know", + "type": "text" + }, + { + "bbox": [ + 209, + 437, + 223, + 447 + ], + "score": 0.88, + "content": "D ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 436, + 338, + 449 + ], + "score": 1.0, + "content": "and so we instead minimize", + "type": "text" + }, + { + "bbox": [ + 338, + 437, + 375, + 448 + ], + "score": 0.92, + "content": "V ( D , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 436, + 497, + 449 + ], + "score": 1.0, + "content": ", which is only a lower bound.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 425, + 505, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 401, + 467 + ], + "score": 1.0, + "content": "This perspective of minimizing the distance between the distributions,", + "type": "text" + }, + { + "bbox": [ + 401, + 455, + 424, + 465 + ], + "score": 0.83, + "content": "p _ { d a t a }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 452, + 444, + 467 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 455, + 457, + 465 + ], + "score": 0.8, + "content": "p _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 452, + 506, + 467 + ], + "score": 1.0, + "content": ", motivated", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 406, + 477 + ], + "score": 1.0, + "content": "Li et al. (2015) to develop a generative model that matches all moments of", + "type": "text" + }, + { + "bbox": [ + 407, + 465, + 433, + 476 + ], + "score": 0.9, + "content": "p _ { G } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 462, + 455, + 477 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 455, + 465, + 490, + 476 + ], + "score": 0.91, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "(at", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "optimality) by minimizing maximum mean discrepancy (MMD). Another approach, EBGAN, (Zhao", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "et al. (2016)) explores a larger class of games (non-zero-sum games) which generalize the generator", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "and discriminator objectives to take real-valued “energies” as input instead of probabilities. Nowozin", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 328, + 520 + ], + "score": 1.0, + "content": "et al. (2016) and then Uehara et al. (2016) extended the", + "type": "text" + }, + { + "bbox": [ + 329, + 508, + 351, + 519 + ], + "score": 0.75, + "content": "J S D", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "perspective on GANs to more general", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 442, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 206, + 532 + ], + "score": 1.0, + "content": "divergences, specifically", + "type": "text" + }, + { + "bbox": [ + 207, + 519, + 213, + 531 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 518, + 442, + 532 + ], + "score": 1.0, + "content": "-divergences and then Bregman-divergences respectively.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 452, + 506, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 421, + 549 + ], + "score": 1.0, + "content": "In general, these approaches focus on exploring fundamental reformulations of", + "type": "text" + }, + { + "bbox": [ + 422, + 536, + 459, + 548 + ], + "score": 0.93, + "content": "V ( D , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 535, + 505, + 549 + ], + "score": 1.0, + "content": ". 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Each", + "type": "text" + }, + { + "bbox": [ + 243, + 647, + 255, + 658 + ], + "score": 0.91, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 645, + 463, + 660 + ], + "score": 1.0, + "content": "is still expected to independently maximize its own", + "type": "text" + }, + { + "bbox": [ + 463, + 647, + 504, + 659 + ], + "score": 0.91, + "content": "V ( D _ { i } , G )", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 297, + 670 + ], + "score": 1.0, + "content": "(i.e. no cooperation). 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Boosting results appear in Appendix A.7.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 250, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 251, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 251, + 620 + ], + "score": 1.0, + "content": "4 A FORGIVING TEACHER", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "The previous perspectives focus on improving the discriminator with the goal of presenting a better", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 641, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 181, + 656 + ], + "score": 1.0, + "content": "approximation of", + "type": "text" + }, + { + "bbox": [ + 181, + 642, + 246, + 654 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 641, + 505, + 656 + ], + "score": 1.0, + "content": "to the generator. 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This", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is because the generator is unlikely to generate any samples considered “realistic” by the discrimi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "nator’s standards, and so the generator will receive uniformly negative feedback. 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This prevents us from claiming", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 279, + 379 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } \\{ V _ { 1 } ( t ) , \\ldots , V _ { N } ( t ) \\} > \\operatorname* { m a x } \\{ V _ { 1 } ^ { \\prime } ( t ) \\} \\forall t", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 366, + 358, + 380 + ], + "score": 1.0, + "content": "even if we initalize", + "type": "text" + }, + { + "bbox": [ + 359, + 367, + 424, + 379 + ], + "score": 0.93, + "content": "D _ { 1 } ( 0 ) = \\mathbf { \\bar { \\cal { D } } } _ { 1 } ^ { \\prime } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "as it is unlikely that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 377, + 336, + 390 + ], + "spans": [ + { + "bbox": [ + 107, + 379, + 169, + 390 + ], + "score": 0.92, + "content": "D _ { 1 } ( t ) = D _ { 1 } ^ { \\prime } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 377, + 223, + 390 + ], + "score": 1.0, + "content": "at some time", + "type": "text" + }, + { + "bbox": [ + 224, + 380, + 229, + 388 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 377, + 336, + 390 + ], + "score": 1.0, + "content": "after the start of the game.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 334, + 505, + 390 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 403, + 177, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 178, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 178, + 416 + ], + "score": 1.0, + "content": "3.2 BOOSTING", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 279, + 435 + ], + "score": 1.0, + "content": "We can also consider taking the max over", + "type": "text" + }, + { + "bbox": [ + 279, + 424, + 289, + 434 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 424, + 505, + 435 + ], + "score": 1.0, + "content": "discriminators as a form of boosting for the discrim-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 324, + 446 + ], + "score": 1.0, + "content": "inator’s online classification problem (online because", + "type": "text" + }, + { + "bbox": [ + 324, + 435, + 334, + 445 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "can produce an infinite data stream). The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 268, + 459 + ], + "score": 1.0, + "content": "boosted discriminator is given a sample", + "type": "text" + }, + { + "bbox": [ + 268, + 447, + 279, + 457 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "and must predict whether it came from the generator or", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 477, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 417, + 469 + ], + "score": 1.0, + "content": "the dataset. The booster then makes its prediction using the predictions of the", + "type": "text" + }, + { + "bbox": [ + 417, + 457, + 428, + 466 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 456, + 460, + 469 + ], + "score": 1.0, + "content": "weaker", + "type": "text" + }, + { + "bbox": [ + 460, + 457, + 473, + 468 + ], + "score": 0.9, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 456, + 477, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 424, + 505, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "There are a few differences between taking the max (case 1) and online boosting (case 2). In case 1,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "our booster is limited to selecting a single weak discriminator (i.e. a pure strategy), while in case 2,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "many boosting algorithms more generally use linear combinations of the discriminators. Moreover,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "in case 2, a booster must make a prediction before receiving a loss function. In case 1, we assume", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 518, + 440, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 440, + 530 + ], + "score": 1.0, + "content": "access to the loss function at prediction time, which allows us to compute the max.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 474, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "It is possible to train the weak discriminators using boosting and then ignore the booster’s prediction", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 213, + 558 + ], + "score": 1.0, + "content": "by instead presenting max", + "type": "text" + }, + { + "bbox": [ + 213, + 545, + 232, + 558 + ], + "score": 0.82, + "content": "\\{ V _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 545, + 505, + 558 + ], + "score": 1.0, + "content": ". We explore both variants in our experiments, using the adaptive al-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 555, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 570 + ], + "score": 1.0, + "content": "gorithm proposed in Beygelzimer et al. (2015). Unfortunately, boosting failed to produce promising", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "score": 1.0, + "content": "results on the image generation tasks. It is possible that boosting produces too strong an adversary", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 578, + 459, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 459, + 590 + ], + "score": 1.0, + "content": "for learning which motivates the next section. Boosting results appear in Appendix A.7.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 534, + 506, + 590 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 250, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 251, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 251, + 620 + ], + "score": 1.0, + "content": "4 A FORGIVING TEACHER", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "The previous perspectives focus on improving the discriminator with the goal of presenting a better", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 641, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 181, + 656 + ], + "score": 1.0, + "content": "approximation of", + "type": "text" + }, + { + "bbox": [ + 181, + 642, + 246, + 654 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 641, + 505, + 656 + ], + "score": 1.0, + "content": "to the generator. Our next perspective asks the question, “Is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 653, + 251, + 665 + ], + "spans": [ + { + "bbox": [ + 107, + 653, + 171, + 665 + ], + "score": 0.92, + "content": "\\operatorname* { m a x } _ { \\mathcal { D } } V ( D , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 653, + 251, + 664 + ], + "score": 1.0, + "content": "too harsh a critic?”", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 630, + 505, + 665 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 678, + 221, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 222, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 222, + 691 + ], + "score": 1.0, + "content": "4.1 Soft-DISCRIMINATOR", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "In practice, training against a far superior discriminator can impede the generator’s learning. This", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is because the generator is unlikely to generate any samples considered “realistic” by the discrimi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "nator’s standards, and so the generator will receive uniformly negative feedback. 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Using a softmax also has the well known advantage of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "being differentiable (as opposed to subdifferentiable for max). 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Note that the generator gradient,", + "type": "text" + }, + { + "bbox": [ + 356, + 446, + 507, + 466 + ], + "score": 1.0, + "content": "| ∂ log(z)∂z |, is minimized at z = 1 over", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 107, + 463, + 152, + 476 + ], + "score": 0.92, + "content": "z \\in ( 0 , 1 ] ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 463, + 285, + 477 + ], + "score": 1.0, + "content": ". From this form, it is clear that", + "type": "text" + }, + { + "bbox": [ + 285, + 464, + 312, + 474 + ], + "score": 0.9, + "content": "z = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 463, + 370, + 477 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 371, + 464, + 414, + 475 + ], + "score": 0.9, + "content": "D _ { i } = 0 \\forall i", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 463, + 431, + 477 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 431, + 464, + 440, + 474 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "only receives a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 204, + 488 + ], + "score": 1.0, + "content": "vanishing gradient if all", + "type": "text" + }, + { + "bbox": [ + 205, + 475, + 217, + 486 + ], + "score": 0.88, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 474, + 479, + 488 + ], + "score": 1.0, + "content": "agree that the sample is fake; this is especially unlikely for large", + "type": "text" + }, + { + "bbox": [ + 479, + 475, + 489, + 484 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 474, + 505, + 488 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 158, + 498 + ], + "score": 1.0, + "content": "other words,", + "type": "text" + }, + { + "bbox": [ + 158, + 486, + 167, + 496 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 485, + 272, + 498 + ], + "score": 1.0, + "content": "only needs to fool a single", + "type": "text" + }, + { + "bbox": [ + 273, + 486, + 285, + 497 + ], + "score": 0.89, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "to receive constructive feedback. This result allows the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 378, + 509 + ], + "score": 1.0, + "content": "generator to successfully minimize the original generator objective,", + "type": "text" + }, + { + "bbox": [ + 379, + 497, + 426, + 509 + ], + "score": 0.91, + "content": "\\log ( 1 - D )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 496, + 506, + 509 + ], + "score": 1.0, + "content": ". This is in contrast", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 492, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 187, + 520 + ], + "score": 1.0, + "content": "to the more popular", + "type": "text" + }, + { + "bbox": [ + 188, + 507, + 227, + 519 + ], + "score": 0.9, + "content": "- \\log ( D )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 507, + 492, + 520 + ], + "score": 1.0, + "content": "introduced to artificially enhance gradients at the start of training.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 245, + 538 + ], + "score": 1.0, + "content": "At the beginning of training, when", + "type": "text" + }, + { + "bbox": [ + 246, + 524, + 316, + 537 + ], + "score": 0.92, + "content": "\\operatorname* { m a x } _ { D _ { i } } V ( D _ { i } , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "is likely too harsh a critic for the generator, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 137, + 547 + ], + "score": 1.0, + "content": "can set", + "type": "text" + }, + { + "bbox": [ + 137, + 536, + 144, + 545 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "closer to zero to use the mean, increasing the odds of providing constructive feedback to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 504, + 558 + ], + "score": 1.0, + "content": "the generator. In addition, the discriminators have the added benefit of functioning as an ensemble,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "reducing the variance of the feedback presented to the generator, which is especially important", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "when the discriminators are far from optimal and are still learning a reasonable decision boundary.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 397, + 591 + ], + "score": 1.0, + "content": "As training progresses and the discriminators improve, we can increase", + "type": "text" + }, + { + "bbox": [ + 398, + 580, + 405, + 589 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "to become more critical", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 589, + 275, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 275, + 603 + ], + "score": 1.0, + "content": "of the generator for more refined training.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 614, + 299, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 300, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 300, + 627 + ], + "score": 1.0, + "content": "4.3 MAINTAINING MULTIPLE HYPOTHESES", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "We argue for this ensemble approach on a more fundamental level as well. Here, we draw on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 647, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 658 + ], + "score": 1.0, + "content": "the density ratio estimation perspective of GANs (Uehara et al. (2016)). The original GAN proof", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 104, + 657, + 217, + 671 + ], + "score": 1.0, + "content": "assumes we have access to", + "type": "text" + }, + { + "bbox": [ + 217, + 658, + 253, + 669 + ], + "score": 0.92, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 657, + 506, + 671 + ], + "score": 1.0, + "content": ", if only implicitly. In most cases of interest, the discriminator", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 301, + 682 + ], + "score": 1.0, + "content": "only has access to a finite dataset sampled from", + "type": "text" + }, + { + "bbox": [ + 301, + 669, + 337, + 680 + ], + "score": 0.92, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "; therefore, when computing expectations", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 118, + 691 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 679, + 155, + 691 + ], + "score": 0.92, + "content": "\\dot { V } ( D , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 679, + 505, + 691 + ], + "score": 1.0, + "content": ", we only draw samples from our finite dataset. This is equivalent to training a GAN", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 127, + 703 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 691, + 197, + 703 + ], + "score": 0.92, + "content": "p _ { d a t a } ( x ) = \\tilde { p } _ { d a t a }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "which is a distribution consisting of point masses on all the data points in the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 714 + ], + "score": 1.0, + "content": "dataset. For the sake of argument, let’s assume we are training a discriminator and generator, each", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "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": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 166 + ], + "score": 1.0, + "content": "For this reason, we explore a variety of functions that allow us to soften the max operator. We", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 470, + 178 + ], + "score": 1.0, + "content": "choose to focus on soft versions of the three classical Pythagorean means parameterized by", + "type": "text" + }, + { + "bbox": [ + 470, + 166, + 477, + 175 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 176, + 384, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 176, + 132, + 186 + ], + "score": 0.91, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 176, + 347, + 189 + ], + "score": 1.0, + "content": "corresponds to the mean and the max is recovered as", + "type": "text" + }, + { + "bbox": [ + 347, + 177, + 379, + 186 + ], + "score": 0.9, + "content": "\\lambda \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 176, + 384, + 189 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 153, + 505, + 189 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 192, + 394, + 297 + ], + "lines": [ + { + "bbox": [ + 215, + 192, + 394, + 297 + ], + "spans": [ + { + "bbox": [ + 215, + 192, + 394, + 297 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\mathrm { { A M } } _ { s o f t } ( V , \\lambda ) = \\displaystyle \\sum _ { i } ^ { N } w _ { i } V _ { i } } } \\\\ { ~ } \\\\ { { \\mathrm { { G M } } _ { s o f t } ( V , \\lambda ) = - \\exp \\Big ( \\displaystyle \\sum _ { i } ^ { N } w _ { i } \\log ( - V _ { i } ) \\Big ) } } \\\\ { { { } } } \\\\ { { { \\mathrm { H M } } _ { s o f t } ( V , \\lambda ) = \\Big ( \\displaystyle \\sum _ { i } ^ { N } w _ { i } V _ { i } ^ { - 1 } \\Big ) ^ { - 1 } } } \\end{array}", + "type": "interline_equation", + "image_path": "16a94c76c11169741b475d5a2234908f705713b6e80ce458958475e7bb495df9.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 192, + 394, + 205.125 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 215, + 205.125, + 394, + 218.25 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 215, + 218.25, + 394, + 231.375 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 215, + 231.375, + 394, + 244.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 215, + 244.5, + 394, + 257.625 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 215, + 257.625, + 394, + 270.75 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 215, + 270.75, + 394, + 283.875 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 215, + 283.875, + 394, + 297.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 133, + 317 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 301, + 209, + 315 + ], + "score": 0.93, + "content": "w _ { i } = e ^ { \\lambda V _ { i } } / \\Sigma _ { j } e ^ { \\lambda V _ { j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 300, + 231, + 317 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 231, + 302, + 288, + 314 + ], + "score": 0.91, + "content": "\\lambda \\ge 0 , V _ { i } < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 300, + 507, + 317 + ], + "score": 1.0, + "content": ". Using a softmax also has the well known advantage of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "being differentiable (as opposed to subdifferentiable for max). Note that we only require continuity", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 417, + 338 + ], + "score": 1.0, + "content": "to guarantee that computing the softmax is actually equivalent to computing", + "type": "text" + }, + { + "bbox": [ + 417, + 324, + 455, + 338 + ], + "score": 0.94, + "content": "V ( \\tilde { D } , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 326, + 484, + 338 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 484, + 324, + 494, + 335 + ], + "score": 0.87, + "content": "\\tilde { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 321, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 224, + 349 + ], + "score": 1.0, + "content": "some convex combination of", + "type": "text" + }, + { + "bbox": [ + 224, + 337, + 236, + 348 + ], + "score": 0.88, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 337, + 321, + 349 + ], + "score": 1.0, + "content": "(see Appendix A.5).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 300, + 507, + 349 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 361, + 322, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 322, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 322, + 374 + ], + "score": 1.0, + "content": "4.2 USING THE ORIGINAL MINIMAX OBJECTIVE", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 105, + 381, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 504, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 446, + 395 + ], + "score": 1.0, + "content": "To illustrate the effect the softmax has on training, observe that the component of", + "type": "text" + }, + { + "bbox": [ + 446, + 382, + 504, + 394 + ], + "score": 0.93, + "content": "A M _ { s o f t } ( V , 0 )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 393, + 302, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 302, + 405 + ], + "score": 1.0, + "content": "relevant to generator training can be rewritten as", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 381, + 504, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 409, + 429, + 443 + ], + "lines": [ + { + "bbox": [ + 182, + 409, + 429, + 443 + ], + "spans": [ + { + "bbox": [ + 182, + 409, + 429, + 443 + ], + "score": 0.94, + "content": "\\frac { 1 } { N } \\sum _ { i } ^ { N } \\mathbb { E } _ { x \\sim p _ { G } ( x ) } \\Big [ \\log ( 1 - D _ { i } ( x ) ) \\Big ] = \\frac { 1 } { N } \\mathbb { E } _ { x \\sim p _ { G } ( x ) } \\Big [ \\log ( z ) \\Big ] .", + "type": "interline_equation", + "image_path": "b58fcfef8b959b666bec5deb8dfc486508913f6e1d9aae8817c2216645d243aa.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 182, + 409, + 429, + 420.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 182, + 420.3333333333333, + 429, + 431.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 182, + 431.66666666666663, + 429, + 442.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 507, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 133, + 465 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 448, + 220, + 463 + ], + "score": 0.92, + "content": "\\begin{array} { r } { z = \\prod _ { i } ^ { N } ( 1 - D _ { i } ( x ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 447, + 356, + 465 + ], + "score": 1.0, + "content": ". Note that the generator gradient,", + "type": "text" + }, + { + "bbox": [ + 356, + 446, + 507, + 466 + ], + "score": 1.0, + "content": "| ∂ log(z)∂z |, is minimized at z = 1 over", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 107, + 463, + 152, + 476 + ], + "score": 0.92, + "content": "z \\in ( 0 , 1 ] ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 463, + 285, + 477 + ], + "score": 1.0, + "content": ". From this form, it is clear that", + "type": "text" + }, + { + "bbox": [ + 285, + 464, + 312, + 474 + ], + "score": 0.9, + "content": "z = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 463, + 370, + 477 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 371, + 464, + 414, + 475 + ], + "score": 0.9, + "content": "D _ { i } = 0 \\forall i", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 463, + 431, + 477 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 431, + 464, + 440, + 474 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "only receives a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 204, + 488 + ], + "score": 1.0, + "content": "vanishing gradient if all", + "type": "text" + }, + { + "bbox": [ + 205, + 475, + 217, + 486 + ], + "score": 0.88, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 474, + 479, + 488 + ], + "score": 1.0, + "content": "agree that the sample is fake; this is especially unlikely for large", + "type": "text" + }, + { + "bbox": [ + 479, + 475, + 489, + 484 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 474, + 505, + 488 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 158, + 498 + ], + "score": 1.0, + "content": "other words,", + "type": "text" + }, + { + "bbox": [ + 158, + 486, + 167, + 496 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 485, + 272, + 498 + ], + "score": 1.0, + "content": "only needs to fool a single", + "type": "text" + }, + { + "bbox": [ + 273, + 486, + 285, + 497 + ], + "score": 0.89, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "to receive constructive feedback. This result allows the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 378, + 509 + ], + "score": 1.0, + "content": "generator to successfully minimize the original generator objective,", + "type": "text" + }, + { + "bbox": [ + 379, + 497, + 426, + 509 + ], + "score": 0.91, + "content": "\\log ( 1 - D )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 496, + 506, + 509 + ], + "score": 1.0, + "content": ". This is in contrast", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 492, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 187, + 520 + ], + "score": 1.0, + "content": "to the more popular", + "type": "text" + }, + { + "bbox": [ + 188, + 507, + 227, + 519 + ], + "score": 0.9, + "content": "- \\log ( D )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 507, + 492, + 520 + ], + "score": 1.0, + "content": "introduced to artificially enhance gradients at the start of training.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 446, + 507, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 245, + 538 + ], + "score": 1.0, + "content": "At the beginning of training, when", + "type": "text" + }, + { + "bbox": [ + 246, + 524, + 316, + 537 + ], + "score": 0.92, + "content": "\\operatorname* { m a x } _ { D _ { i } } V ( D _ { i } , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "is likely too harsh a critic for the generator, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 137, + 547 + ], + "score": 1.0, + "content": "can set", + "type": "text" + }, + { + "bbox": [ + 137, + 536, + 144, + 545 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "closer to zero to use the mean, increasing the odds of providing constructive feedback to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 504, + 558 + ], + "score": 1.0, + "content": "the generator. In addition, the discriminators have the added benefit of functioning as an ensemble,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "reducing the variance of the feedback presented to the generator, which is especially important", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "when the discriminators are far from optimal and are still learning a reasonable decision boundary.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 397, + 591 + ], + "score": 1.0, + "content": "As training progresses and the discriminators improve, we can increase", + "type": "text" + }, + { + "bbox": [ + 398, + 580, + 405, + 589 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "to become more critical", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 589, + 275, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 275, + 603 + ], + "score": 1.0, + "content": "of the generator for more refined training.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 524, + 506, + 603 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 614, + 299, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 300, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 300, + 627 + ], + "score": 1.0, + "content": "4.3 MAINTAINING MULTIPLE HYPOTHESES", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "We argue for this ensemble approach on a more fundamental level as well. Here, we draw on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 647, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 658 + ], + "score": 1.0, + "content": "the density ratio estimation perspective of GANs (Uehara et al. (2016)). The original GAN proof", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 104, + 657, + 217, + 671 + ], + "score": 1.0, + "content": "assumes we have access to", + "type": "text" + }, + { + "bbox": [ + 217, + 658, + 253, + 669 + ], + "score": 0.92, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 657, + 506, + 671 + ], + "score": 1.0, + "content": ", if only implicitly. In most cases of interest, the discriminator", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 301, + 682 + ], + "score": 1.0, + "content": "only has access to a finite dataset sampled from", + "type": "text" + }, + { + "bbox": [ + 301, + 669, + 337, + 680 + ], + "score": 0.92, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "; therefore, when computing expectations", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 118, + 691 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 679, + 155, + 691 + ], + "score": 0.92, + "content": "\\dot { V } ( D , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 679, + 505, + 691 + ], + "score": 1.0, + "content": ", we only draw samples from our finite dataset. This is equivalent to training a GAN", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 127, + 703 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 691, + 197, + 703 + ], + "score": 0.92, + "content": "p _ { d a t a } ( x ) = \\tilde { p } _ { d a t a }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "which is a distribution consisting of point masses on all the data points in the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 714 + ], + "score": 1.0, + "content": "dataset. For the sake of argument, let’s assume we are training a discriminator and generator, each", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 333, + 96 + ], + "score": 1.0, + "content": "with infinite capacity. In this case, the global optimum", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 334, + 82, + 411, + 95 + ], + "score": 0.91, + "content": "( p _ { G } ( x ) = \\tilde { p } _ { d a t a ( x ) } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 412, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "fails to capture any of", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 226, + 108 + ], + "score": 1.0, + "content": "the interesting structure from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 226, + 94, + 262, + 106 + ], + "score": 0.93, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 262, + 94, + 506, + 108 + ], + "score": 1.0, + "content": ", the true distribution we are trying to learn. Therefore, it is", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 312, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 312, + 119 + ], + "score": 1.0, + "content": "actually critical that we avoid this global optimum.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 44, + "bbox_fs": [ + 104, + 635, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 333, + 96 + ], + "score": 1.0, + "content": "with infinite capacity. In this case, the global optimum", + "type": "text" + }, + { + "bbox": [ + 334, + 82, + 411, + 95 + ], + "score": 0.91, + "content": "( p _ { G } ( x ) = \\tilde { p } _ { d a t a ( x ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "fails to capture any of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 226, + 108 + ], + "score": 1.0, + "content": "the interesting structure from", + "type": "text" + }, + { + "bbox": [ + 226, + 94, + 262, + 106 + ], + "score": 0.93, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 94, + 506, + 108 + ], + "score": 1.0, + "content": ", the true distribution we are trying to learn. Therefore, it is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 312, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 312, + 119 + ], + "score": 1.0, + "content": "actually critical that we avoid this global optimum.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "image", + "bbox": [ + 232, + 127, + 378, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 232, + 127, + 378, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 232, + 127, + 378, + 232 + ], + "spans": [ + { + "bbox": [ + 232, + 127, + 378, + 232 + ], + "score": 0.948, + "type": "image", + "image_path": "6825e5bfd09d2c7ad38711198c6dc71b70472969679776d3899bf8813fa65739.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 127, + 378, + 179.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 232, + 179.5, + 378, + 232.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 240, + 505, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "Figure 2: Consider a dataset consisting of the nine 1-dimensional samples in black. Their corre-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "sponding probability mass function is given in light gray. After training GMAN, three discrimina-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "tors converge to distinct local optima which implicitly define distributions over the data (red, blue,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 505, + 288 + ], + "score": 1.0, + "content": "yellow). Each discriminator may specialize in discriminating a region of the data space (placing", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "more diffuse mass in other regions). Averaging over the three discriminators results in the distribu-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "tion in black, which we expect has higher likelihood under reasonable assumptions on the structure", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 306, + 201, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 201, + 319 + ], + "score": 1.0, + "content": "of the true distribution.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + } + ], + "index": 5.75 + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "score": 1.0, + "content": "In practice, this degenerate result is avoided by employing learners with limited capacity and corrupt-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "ing data samples with noise (i.e., dropout), but we might better accomplish this by simultaneously", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "training a variety of limited capacity discriminators. With this approach, we might obtain a diverse", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 306, + 375 + ], + "score": 1.0, + "content": "set of seemingly tenable hypotheses for the true", + "type": "text" + }, + { + "bbox": [ + 306, + 362, + 342, + 374 + ], + "score": 0.91, + "content": "p _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 362, + 505, + 375 + ], + "score": 1.0, + "content": ". Averaging over these multiple locally", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 297, + 387 + ], + "score": 1.0, + "content": "optimal discriminators increases the entropy of", + "type": "text" + }, + { + "bbox": [ + 298, + 374, + 333, + 385 + ], + "score": 0.92, + "content": "\\tilde { p } _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 372, + 506, + 387 + ], + "score": 1.0, + "content": "by diffusing the probability mass over the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 384, + 273, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 273, + 397 + ], + "score": 1.0, + "content": "data space (see Figure 2 for an example).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 249, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 251, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 251, + 421 + ], + "score": 1.0, + "content": "4.4 AUTOMATING REGULATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "The problem of keeping the discriminator and generator in balance has been widely recognized in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "previous work with GANs. Issues with unstable dynamics, oscillatory behavior, and generator col-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "lapse are not uncommon. In addition, the discriminator is often times able to achieve a high degree of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "classification accuracy (producing a single scalar) before the generator has made sufficient progress", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "on the arguably more difficult generative task (producing a high dimensional sample). Salimans", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "et al. (2016) suggested label smoothing to reduce the vulnerability of the generator to a relatively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "score": 1.0, + "content": "superior discriminator. Here, we explore an approach that enables the generator to automatically", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "score": 1.0, + "content": "temper the performance of the discriminator when necessary, but still encourages the generator to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 516, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 504, + 529 + ], + "score": 1.0, + "content": "challenge itself against more accurate adversaries. 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In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 218, + 572 + ], + "score": 1.0, + "content": "experiments, we simply set", + "type": "text" + }, + { + "bbox": [ + 218, + 560, + 262, + 572 + ], + "score": 0.93, + "content": "f ( \\lambda ) = c \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 560, + 283, + 572 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 283, + 563, + 289, + 570 + ], + "score": 0.74, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "a constant (e.g., 0.001). 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(2016) recommend avoiding Parzen", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "windows and argue that generative models should be evaluated with respect to their intended appli-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "cation. Salimans et al. (2016) suggest an Inception score, however, it assumes labels exist for the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 688, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 504, + 700 + ], + "score": 1.0, + "content": "dataset. Recently, Im et al. 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Their corre-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "sponding probability mass function is given in light gray. After training GMAN, three discrimina-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "tors converge to distinct local optima which implicitly define distributions over the data (red, blue,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 505, + 288 + ], + "score": 1.0, + "content": "yellow). Each discriminator may specialize in discriminating a region of the data space (placing", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "more diffuse mass in other regions). Averaging over the three discriminators results in the distribu-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "tion in black, which we expect has higher likelihood under reasonable assumptions on the structure", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 306, + 201, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 201, + 319 + ], + "score": 1.0, + "content": "of the true distribution.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + } + ], + "index": 5.75 + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "score": 1.0, + "content": "In practice, this degenerate result is avoided by employing learners with limited capacity and corrupt-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "ing data samples with noise (i.e., dropout), but we might better accomplish this by simultaneously", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "training a variety of limited capacity discriminators. 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Averaging over these multiple locally", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 297, + 387 + ], + "score": 1.0, + "content": "optimal discriminators increases the entropy of", + "type": "text" + }, + { + "bbox": [ + 298, + 374, + 333, + 385 + ], + "score": 0.92, + "content": "\\tilde { p } _ { d a t a } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 372, + 506, + 387 + ], + "score": 1.0, + "content": "by diffusing the probability mass over the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 384, + 273, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 273, + 397 + ], + "score": 1.0, + "content": "data space (see Figure 2 for an example).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 329, + 506, + 397 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 249, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 251, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 251, + 421 + ], + "score": 1.0, + "content": "4.4 AUTOMATING REGULATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "The problem of keeping the discriminator and generator in balance has been widely recognized in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "previous work with GANs. Issues with unstable dynamics, oscillatory behavior, and generator col-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "lapse are not uncommon. In addition, the discriminator is often times able to achieve a high degree of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "classification accuracy (producing a single scalar) before the generator has made sufficient progress", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "on the arguably more difficult generative task (producing a high dimensional sample). Salimans", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "et al. (2016) suggested label smoothing to reduce the vulnerability of the generator to a relatively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "score": 1.0, + "content": "superior discriminator. Here, we explore an approach that enables the generator to automatically", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "score": 1.0, + "content": "temper the performance of the discriminator when necessary, but still encourages the generator to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 516, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 504, + 529 + ], + "score": 1.0, + "content": "challenge itself against more accurate adversaries. 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In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 218, + 572 + ], + "score": 1.0, + "content": "experiments, we simply set", + "type": "text" + }, + { + "bbox": [ + 218, + 560, + 262, + 572 + ], + "score": 0.93, + "content": "f ( \\lambda ) = c \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 560, + 283, + 572 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 283, + 563, + 289, + 570 + ], + "score": 0.74, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "a constant (e.g., 0.001). The generator is incentivized", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 151, + 584 + ], + "score": 1.0, + "content": "to increase", + "type": "text" + }, + { + "bbox": [ + 152, + 572, + 159, + 581 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "to reduce its objective at the expense of competing against the best available adversary", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 581, + 205, + 596 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 120, + 592 + ], + "score": 0.85, + "content": "D ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 581, + 205, + 596 + ], + "score": 1.0, + "content": "(see Appendix A.6).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 549, + 506, + 596 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 609, + 192, + 621 + ], + "lines": [ + { + "bbox": [ + 104, + 607, + 195, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 607, + 195, + 624 + ], + "score": 1.0, + "content": "5 EVALUATION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Evaluating GANs is still an open problem. In their original work, Goodfellow et al. (2014) report", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "log likelihood estimates from Gaussian Parzen windows, which they admit, has high variance and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "is known not to perform well in high dimensions. Theis et al. 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(2016) introduced the Generative Adversarial Metric (GAM) for mak-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "ing pairwise comparisons between independently trained GAN models. 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The idea here is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 148, + 202 + ], + "score": 1.0, + "content": "similar. 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If", + "type": "text" + }, + { + "bbox": [ + 174, + 201, + 187, + 212 + ], + "score": 0.89, + "content": "G _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 201, + 338, + 213 + ], + "score": 1.0, + "content": "performs better in both cases, GMAM", + "type": "text" + }, + { + "bbox": [ + 339, + 201, + 352, + 211 + ], + "score": 0.72, + "content": "< 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 201, + 505, + 213 + ], + "score": 1.0, + "content": ", otherwise, the result is indeterminate.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 194, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 194, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 194, + 239 + ], + "score": 1.0, + "content": "5.2 EXPERIMENTS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "We evaluate the aforementioned variations of GMAN on a variety of image generation tasks: MNIST", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "(LeCun et al. (1998)), CIFAR-10 (Krizhevsky (2009)) and CelebA (Liu et al. (2015)). We focus on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "rates of convergence to steady state along with quality of the steady state generator according to the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 279, + 497, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 497, + 292 + ], + "score": 1.0, + "content": "GMAM metric. To summarize, loosely in order of increasing discriminator leniency, we compare", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 132, + 300, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 132, + 301, + 443, + 313 + ], + "spans": [ + { + "bbox": [ + 132, + 301, + 443, + 313 + ], + "score": 1.0, + "content": "• F-boost: A single AdaBoost.OL-boosted discriminator (see Appendix A.7).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 132, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 132, + 316, + 179, + 329 + ], + "score": 1.0, + "content": "• P-boost:", + "type": "text" + }, + { + "bbox": [ + 180, + 316, + 192, + 327 + ], + "score": 0.88, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "is trained according to AdaBoost.OL. 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Figure 4 displays the vari-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 559, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 504, + 570 + ], + "score": 1.0, + "content": "ance of the same objective over a sliding time window, reaffirming GMAN’s acceleration to steady-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "state. Figure 5 corroborates this conclusion with recognizable digits appearing approximately an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 580, + 486, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 486, + 592 + ], + "score": 1.0, + "content": "epoch before the single discriminator run; digits at steady-state appear slightly sharper as well.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "table", + "bbox": [ + 107, + 629, + 533, + 688 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 596, + 504, + 620 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 596, + 504, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 504, + 609 + ], + "score": 1.0, + "content": "Our GMAM metric (see Table 1) agrees with the relative quality of images in Figure 5 with GMAN∗", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 606, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 622 + ], + "score": 1.0, + "content": "achieving the best overall performance. Figure 6 reveals GMAN∗’s attempt to regulate the difficulty", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 629, + 533, + 688 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 629, + 533, + 688 + ], + "spans": [ + { + "bbox": [ + 107, + 629, + 533, + 688 + ], + "score": 0.977, + "html": "
ScoreVariantGMAN*GMAN-0GMAN-maxmod-GAN
←Feneg0.127GMAN*-0.020±0.009-0.028 ± 0.019-0.089± 0.036
0.007GMAN-00.020 ±0.009-0.013 ± 0.015-0.018 ± 0.027
-0.034GMAN-max0.028 ± 0.0190.013 ± 0.015-0.011 ± 0.024
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The idea here is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 148, + 202 + ], + "score": 1.0, + "content": "similar. 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If", + "type": "text" + }, + { + "bbox": [ + 174, + 201, + 187, + 212 + ], + "score": 0.89, + "content": "G _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 201, + 338, + 213 + ], + "score": 1.0, + "content": "performs better in both cases, GMAM", + "type": "text" + }, + { + "bbox": [ + 339, + 201, + 352, + 211 + ], + "score": 0.72, + "content": "< 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 201, + 505, + 213 + ], + "score": 1.0, + "content": ", otherwise, the result is indeterminate.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 178, + 505, + 213 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 194, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 194, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 194, + 239 + ], + "score": 1.0, + "content": "5.2 EXPERIMENTS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "We evaluate the aforementioned variations of GMAN on a variety of image generation tasks: MNIST", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "(LeCun et al. (1998)), CIFAR-10 (Krizhevsky (2009)) and CelebA (Liu et al. (2015)). We focus on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "rates of convergence to steady state along with quality of the steady state generator according to the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 279, + 497, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 497, + 292 + ], + "score": 1.0, + "content": "GMAM metric. To summarize, loosely in order of increasing discriminator leniency, we compare", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 247, + 505, + 292 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 300, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 132, + 301, + 443, + 313 + ], + "spans": [ + { + "bbox": [ + 132, + 301, + 443, + 313 + ], + "score": 1.0, + "content": "• F-boost: A single AdaBoost.OL-boosted discriminator (see Appendix A.7).", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 132, + 316, + 179, + 329 + ], + "score": 1.0, + "content": "• P-boost:", + "type": "text" + }, + { + "bbox": [ + 180, + 316, + 192, + 327 + ], + "score": 0.88, + "content": "D _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "is trained according to AdaBoost.OL. 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Figure 4 displays the vari-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 559, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 504, + 570 + ], + "score": 1.0, + "content": "ance of the same objective over a sliding time window, reaffirming GMAN’s acceleration to steady-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "state. Figure 5 corroborates this conclusion with recognizable digits appearing approximately an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 580, + 486, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 486, + 592 + ], + "score": 1.0, + "content": "epoch before the single discriminator run; digits at steady-state appear slightly sharper as well.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 525, + 505, + 592 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 629, + 533, + 688 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 596, + 504, + 620 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 596, + 504, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 504, + 609 + ], + "score": 1.0, + "content": "Our GMAM metric (see Table 1) agrees with the relative quality of images in Figure 5 with GMAN∗", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 606, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 622 + ], + "score": 1.0, + "content": "achieving the best overall performance. Figure 6 reveals GMAN∗’s attempt to regulate the difficulty", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 629, + 533, + 688 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 629, + 533, + 688 + ], + "spans": [ + { + "bbox": [ + 107, + 629, + 533, + 688 + ], + "score": 0.977, + "html": "
ScoreVariantGMAN*GMAN-0GMAN-maxmod-GAN
←Feneg0.127GMAN*-0.020±0.009-0.028 ± 0.019-0.089± 0.036
0.007GMAN-00.020 ±0.009-0.013 ± 0.015-0.018 ± 0.027
-0.034GMAN-max0.028 ± 0.0190.013 ± 0.015-0.011 ± 0.024
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In addition, GMAN makes using the original", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 585, + 492, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 492, + 597 + ], + "score": 1.0, + "content": "GAN objective possible by increasing the odds of the generator receiving constructive feedback.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 530, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 601, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "In future work, we will look at more sophisticated mechanisms for letting the generator control", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "the game as well as other ways to ensure diversity among the discriminators. Introducing multiple", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "generators is conceptually an obvious next step, however, we expect difficulties to arise from more", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 635, + 496, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 496, + 647 + ], + "score": 1.0, + "content": "complex game dynamics. 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We also thank NVIDIA for donating a K40", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "GPU. This material is based upon work supported by the National Science Foundation under Grant", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Nos. IIS-1564032. 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ScoreVariantGMAN*GMAN-1GANGMAN-0GMAN-maxmod-GAN
←Feneg0.184 0.067GMAN*=-0.007-0.040-0.020-0.028-0.089
0.030GMAN-10.0071-0.008-0.008-0.021-0.037
GAN0.0400.008-0.002-0.018-0.058
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ScoreVariantGMAN-0GMAN-1GMAN*mod-GAN
Beeneg0.172GMAN-01-0.022-0.062-0.088
0.050GMAN-10.0220.006-0.078
-0.055GMAN*0.062-0.006=-0.001
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GMAN-0GMAN-1mod-GANGMAN*
Score5.878± 0.1935.765 ± 0.1685.738± 0.1765.539± 0.099
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ScoreVariantGMAN-0GMAN*GMAN-1mod-GAN
FFeeeg0.180GMAN-0=-0.008-0.041-0.132
0.122GMAN*0.008=-0.038-0.092
0.010GMAN-10.0410.038-0.089
-0.313mod-GAN0.1320.0920.0891
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GMAN-1GMAN-0GMAN*mod-GAN
Score6.001± 0.1945.957 ± 0.1355.955 ± 0.1535.738 ± 0.176
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ScoreVariantGMAN*GMAN-1GANGMAN-0GMAN-maxmod-GAN
←Feneg0.184 0.067GMAN*=-0.007-0.040-0.020-0.028-0.089
0.030GMAN-10.0071-0.008-0.008-0.021-0.037
GAN0.0400.008-0.002-0.018-0.058
0.005GMAN-00.0200.0080.002--0.013-0.018
-0.091GMAN-max0.0280.0210.0180.013--0.011
-0.213mod-GAN0.0890.0370.0580.0180.011=
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ScoreVariantGMAN-0GMAN-1GMAN*mod-GAN
Beeneg0.172GMAN-01-0.022-0.062-0.088
0.050GMAN-10.0220.006-0.078
-0.055GMAN*0.062-0.006=-0.001
-0.167mod-GAN0.0880.0780.001=
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GMAN-0GMAN-1mod-GANGMAN*
Score5.878± 0.1935.765 ± 0.1685.738± 0.1765.539± 0.099
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ScoreVariantGMAN-0GMAN*GMAN-1mod-GAN
FFeeeg0.180GMAN-0=-0.008-0.041-0.132
0.122GMAN*0.008=-0.038-0.092
0.010GMAN-10.0410.038-0.089
-0.313mod-GAN0.1320.0920.0891
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GMAN-1GMAN-0GMAN*mod-GAN
Score6.001± 0.1945.957 ± 0.1355.955 ± 0.1535.738 ± 0.176
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In this case, we interpret GMAN as introducing multiple conditional discriminators and not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 530, + 423, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 530, + 423, + 543 + ], + "score": 1.0, + "content": "a discriminator for each of the possibly exponentially many conditional labels.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 431, + 506, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "In Section 4.4, we describe an approach to customize adversarial training to better suit the devel-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "opment of the generator. An approach with similar conceptual underpinnings was described in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "Ravanbakhsh et al. (2016), however, similar to the above, it is only admissible in a semi-supervised", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 581, + 326, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 326, + 592 + ], + "score": 1.0, + "content": "scenario whereas our applies to the unsupervised case.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 547, + 505, + 592 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 619, + 253, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 254, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 254, + 632 + ], + "score": 1.0, + "content": "A.5 Softmax REPRESENTABILITY", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 646, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 645, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 104, + 645, + 141, + 661 + ], + "score": 1.0, + "content": "Let soft", + "type": "text" + }, + { + "bbox": [ + 141, + 645, + 269, + 659 + ], + "score": 0.81, + "content": "n a x ( V _ { i } ) = \\hat { V } \\in [ \\operatorname* { m i n } _ { V _ { i } } , \\operatorname* { m a x } _ { V _ { i } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 645, + 309, + 661 + ], + "score": 1.0, + "content": ". Also let", + "type": "text" + }, + { + "bbox": [ + 309, + 648, + 378, + 659 + ], + "score": 0.87, + "content": "a = a r g \\operatorname* { m i n } _ { i } V _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 645, + 382, + 661 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 383, + 648, + 451, + 659 + ], + "score": 0.9, + "content": "b = a r g \\operatorname* { m a x } _ { i } V _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 645, + 473, + 661 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 473, + 647, + 505, + 659 + ], + "score": 0.91, + "content": "\\gamma ( t ) =", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 107, + 659, + 196, + 670 + ], + "score": 0.89, + "content": "V ( ( 1 - t ) D _ { a } + t D _ { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 657, + 230, + 671 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 231, + 659, + 279, + 670 + ], + "score": 0.91, + "content": "\\nu ( 0 ) = V _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 657, + 299, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 300, + 658, + 347, + 670 + ], + "score": 0.92, + "content": "\\nu ( 1 ) = V _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 657, + 506, + 671 + ], + "score": 1.0, + "content": ". The softmax and minimax objective", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 147, + 681 + ], + "score": 0.9, + "content": "V ( D _ { i } , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "are both continuous in their inputs, so by the intermediate value theorem, we have that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 680, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 154, + 694 + ], + "score": 0.85, + "content": "\\exists { \\hat { t } } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 680, + 172, + 695 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 173, + 680, + 216, + 694 + ], + "score": 0.91, + "content": "\\mathcal { V } ( \\hat { t } ) = \\hat { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 680, + 281, + 695 + ], + "score": 1.0, + "content": ", which implies", + "type": "text" + }, + { + "bbox": [ + 281, + 680, + 321, + 693 + ], + "score": 0.87, + "content": "\\exists \\hat { D } \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 680, + 340, + 695 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 340, + 680, + 402, + 694 + ], + "score": 0.92, + "content": "V ( \\hat { D } , G ) = \\hat { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 680, + 506, + 695 + ], + "score": 1.0, + "content": ". This result implies that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 693, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 428, + 708 + ], + "score": 1.0, + "content": "the softmax (and any other continuous substitute) can be interpreted as returning", + "type": "text" + }, + { + "bbox": [ + 428, + 693, + 466, + 707 + ], + "score": 0.94, + "content": "V ( \\hat { D } , G )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 694, + 505, + 708 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 706, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 706, + 117, + 718 + ], + "score": 0.85, + "content": "\\hat { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 706, + 506, + 721 + ], + "score": 1.0, + "content": "selected by computing an another, unknown function over the space of the discriminators. This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 719, + 479, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 185, + 732 + ], + "score": 1.0, + "content": "result holds even if", + "type": "text" + }, + { + "bbox": [ + 185, + 719, + 195, + 730 + ], + "score": 0.87, + "content": "\\hat { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 721, + 397, + 732 + ], + "score": 1.0, + "content": "is not representable by the architecture chosen for", + "type": "text" + }, + { + "bbox": [ + 397, + 721, + 406, + 730 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 721, + 479, + 732 + ], + "score": 1.0, + "content": "’s neural network.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 645, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 275, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 277, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 277, + 95 + ], + "score": 1.0, + "content": "A.6 UNCONSTRAINED OPTIMIZATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "To convert GMAN∗ minimax formulation to an unconstrained minimax formulation, we introduce", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 495, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 193, + 127 + ], + "score": 1.0, + "content": "an auxiliary variable,", + "type": "text" + }, + { + "bbox": [ + 193, + 114, + 201, + 124 + ], + "score": 0.59, + "content": "\\Lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 114, + 232, + 127 + ], + "score": 1.0, + "content": ", define", + "type": "text" + }, + { + "bbox": [ + 232, + 114, + 315, + 126 + ], + "score": 0.94, + "content": "\\lambda ( \\Lambda ) = \\log ( 1 + e ^ { \\Lambda } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 114, + 463, + 127 + ], + "score": 1.0, + "content": ", and let the generator minimize over", + "type": "text" + }, + { + "bbox": [ + 463, + 114, + 490, + 124 + ], + "score": 0.89, + "content": "\\Lambda \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 114, + 495, + 127 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 138, + 261, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 263, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 263, + 151 + ], + "score": 1.0, + "content": "A.7 BOOSTING WITH AdaBoost.OL", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 159, + 506, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "AdaBoost.OL (Beygelzimer et al. 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(2015)).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 431, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 366, + 443 + ], + "score": 1.0, + "content": "We use convolutional transpose layers (Zeiler et al. (2010)) for", + "type": "text" + }, + { + "bbox": [ + 366, + 432, + 375, + 442 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 431, + 494, + 443 + ], + "score": 1.0, + "content": "and strided convolutions for", + "type": "text" + }, + { + "bbox": [ + 494, + 432, + 504, + 442 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 202, + 455 + ], + "score": 1.0, + "content": "except for the input of", + "type": "text" + }, + { + "bbox": [ + 203, + 443, + 212, + 453 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 442, + 300, + 455 + ], + "score": 1.0, + "content": "and the last layer of", + "type": "text" + }, + { + "bbox": [ + 300, + 443, + 310, + 453 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 442, + 505, + 455 + ], + "score": 1.0, + "content": ". We use the single step gradient method as in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "(Nowozin et al. (2016)), and batch normalization (Ioffe & Szegedy (2015)) was used in each of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "the generator layers. The different discriminators were trained with varying dropout rates from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "[0.3, 0.7]. Variations in the discriminators were effected in two ways. We varied the architecture by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "varying the number of filters in the discriminator layers (reduced by factors of 2, 4 and so on), as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "well as varying dropout rates. Secondly we also decorrelated the samples that the disriminators were", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "training on by splitting the minibatch across the discriminators. The code was written in Tensorflow", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "(Abadi et al. (2016)) and run on Nvidia GTX 980 GPUs. Code to reproduce experiments and plots", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 530, + 498, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 498, + 543 + ], + "score": 1.0, + "content": "is at https://github.com/iDurugkar/GMAN. 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That is,", + "type": "text" + }, + { + "bbox": [ + 307, + 609, + 489, + 621 + ], + "score": 0.87, + "content": "\\left( 3 2 , 3 2 , 1 \\right) , \\left( 1 6 , 1 6 , 1 6 \\right) , \\left( 8 , 8 , 3 2 \\right) , \\left( 4 , 4 , 6 4 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 143, + 619, + 322, + 632 + ], + "spans": [ + { + "bbox": [ + 143, + 619, + 318, + 632 + ], + "score": 0.9, + "content": "\\left( 3 2 , 3 2 , 1 \\right) , \\left( 1 6 , 1 6 , 8 \\right) , \\left( 8 , 8 , 1 6 \\right) , \\left( 4 , 4 , 3 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 619, + 322, + 632 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 136, + 633, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 136, + 633, + 504, + 647 + ], + "score": 1.0, + "content": "• ReLu activations for all the hidden units. 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ScoreVariantGMAN*GMAN-1GANGMAN-0GMAN-maxmod-GAN
←Feneg0.184 0.067GMAN*=-0.007-0.040-0.020-0.028-0.089
0.030GMAN-10.0071-0.008-0.008-0.021-0.037
GAN0.0400.008-0.002-0.018-0.058
0.005GMAN-00.0200.0080.002--0.013-0.018
-0.091GMAN-max0.0280.0210.0180.013--0.011
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ScoreVariantGMAN-0GMAN*GMAN-1mod-GAN
FFeeeg0.180GMAN-0=-0.008-0.041-0.132
0.122GMAN*0.008=-0.038-0.092
0.010GMAN-10.0410.038-0.089
-0.313mod-GAN0.1320.0920.0891
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ScoreVariantGMAN-0GMAN-1GMAN*mod-GAN
Beeneg0.172GMAN-01-0.022-0.062-0.088
0.050GMAN-10.0220.006-0.078
-0.055GMAN*0.062-0.006=-0.001
-0.167mod-GAN0.0880.0780.001=
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GMAN-1GMAN-0GMAN*mod-GAN
Score6.001± 0.1945.957 ± 0.1355.955 ± 0.1535.738 ± 0.176
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GMAN-0GMAN-1mod-GANGMAN*
Score5.878± 0.1935.765 ± 0.1685.738± 0.1765.539± 0.099
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/dev/null +++ b/parse/train/Hkxvl0EtDH/images/fd86cd4d6b23b74665f6c839fa13bb8af955691af08ec3b431789e71bda6b6d7.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2d7246e9868dcd53a575f098171bea3d583bf07f74eec6ee8f55dae29df8d489 +size 6960 diff --git a/parse/train/IBVBtz_sRSm/IBVBtz_sRSm.md b/parse/train/IBVBtz_sRSm/IBVBtz_sRSm.md new file mode 100644 index 0000000000000000000000000000000000000000..e9e9d1b38ef3cc543863be2d8999add129d103d5 --- /dev/null +++ b/parse/train/IBVBtz_sRSm/IBVBtz_sRSm.md @@ -0,0 +1,332 @@ +# ReAct: Out-of-distribution Detection With Rectified Activations + +Yiyou Sun +Department of Computer Sciences +University of Wisconsin-Madison sunyiyou@cs.wisc.edu + +Chuan Guo Facebook AI Research chuanguo@fb.com + +Yixuan Li +Department of Computer Sciences +University of Wisconsin-Madison sharonli@cs.wisc.edu + +# Abstract + +Out-of-distribution (OOD) detection has received much attention lately due to its practical importance in enhancing the safe deployment of neural networks. One of the primary challenges is that models often produce highly confident predictions on OOD data, which undermines the driving principle in OOD detection that the model should only be confident about in-distribution samples. In this work, we propose ReAct—a simple and effective technique for reducing model overconfidence on OOD data. Our method is motivated by novel analysis on internal activations of neural networks, which displays highly distinctive signature patterns for OOD distributions. Our method can generalize effectively to different network architectures and different OOD detection scores. We empirically demonstrate that ReAct achieves competitive detection performance on a comprehensive suite of benchmark datasets, and give theoretical explication for our method’s efficacy. On the ImageNet benchmark, ReAct reduces the false positive rate (FPR95) by $2 5 . 0 5 \%$ compared to the previous best method1. + +# 1 Introduction + +Neural networks deployed in real-world systems often encounter out-of-distribution (OOD) inputs— unknown samples that the network has not been exposed to during training. Identifying and handling these OOD inputs can be paramount in safety-critical applications such as autonomous driving [9] and health care. For example, an autonomous vehicle may fail to recognize objects on the road that do not appear in its object detection model’s training set, potentially leading to a crash. This can be prevented if the system identifies the unrecognized object as OOD and warns the driver in advance. + +A driving idea behind OOD detection is that the model should be much more uncertain about samples outside of its training distribution. However, $\mathrm { \Delta N g u y e n \ e t { a l . } } \rVert { 3 8 } \rVert$ revealed that modern neural networks can produce overconfident predictions on OOD inputs. This phenomenon renders the separation of in-distribution (ID) and OOD data a non-trivial task. Indeed, much of the prior work on OOD detection focused on defining more suitable measures of OOD uncertainty [17, 19, 28, 29, 31, 33]. Despite the improvement, it is arguable that continued research progress in OOD detection requires insights into the fundamental cause and mitigation of model overconfidence on OOD data. + +In this paper, we start by revealing an important observation that OOD data can trigger unit activation patterns that are significantly different from ID data. Figure 1(b) shows the distribution of activations in the penultimate layer of ResNet-50 trained on ImageNet [7]. Each point on the horizontal axis corresponds to a single unit. The mean and standard deviation are shown by the solid line and shaded area, respectively. The mean activation for ID data (blue) is well-behaved with a near-constant mean and standard deviation. In contrast, for OOD data (gray), the mean activation has significantly larger variations across units and is biased towards having sharp positive values (i.e., positively skewed). As a result, such high unit activation can undesirably manifest in model output, producing overconfident predictions on OOD data. A similar distributional property holds for other OOD datasets as well. + +![](images/24e9b1c612ee980b996a70ed12b719e19f1e79634c823067cead43b6f4764fd6.jpg) +Figure 1: Plots showing (a) the distribution of ID (ImageNet $\mathbb { U }$ ) and OOD (iNaturalist $\mathbb { I } 1 6 \mathbb { I }$ ) uncertainty scores before truncation, (b) the distribution of per-unit activations in the penultimate layer for ID and OOD data, and (c) the distribution of OOD uncertainty scores $\pmb { \Vert 3 3 } \Vert$ after rectification. Applying ReAct drastically improves the separation of ID and OOD data. See text for details. + +The above observation naturally inspires a simple yet surprisingly effective method—Rectified Activations (dubbed ReAct) for OOD detection. In particular, the outsized activation of a few selected hidden units can be attenuated by rectifying the activations at an upper limit $c > 0$ . Conveniently, this can be done on a pre-trained model without any modification to training. The dashed horizontal line in Figure 1(b) shows the cutoff point $c$ , and its effect on the OOD uncertainty score is shown in Figure 1(c). After rectification, the output distributions for ID and OOD data become much more well-separated and the false positive rate (FPR) is significantly reduced from $5 5 . 7 2 \%$ to $2 0 . 3 8 \%$ . Importantly, this truncation largely preserves the activation for in-distribution data, and therefore ensures the classification accuracy on the original task is largely comparable. + +We provide both empirical and theoretical insights, characterizing and explaining the mechanism by which ReAct improves OOD detection. We perform extensive evaluations and establish state-ofthe-art performance on a suite of common OOD detection benchmarks, including CIFAR-10 and CIFAR-100, as well as a large-scale ImageNet dataset $\mathbb { U }$ . ReAct outperforms the best baseline by a large margin, reducing the average FPR95 by up to $2 5 . 0 5 \%$ . We further analyze our method theoretically and show that ReAct is more beneficial when OOD activations are more chaotic (i.e., having a larger variance) and positively skewed compared to ID activations, a behavior that is typical of many OOD datasets (cf. Figure 1) . In summary, our key results and contributions are: + +1. We introduce ReAct —a simple and effective post hoc OOD detection approach that utilizes activation truncation. We show that ReAct can generalize effectively to different network architectures and works with different OOD detection methods including MSP [15], ODIN [31], and energy score $\mathbb { \lVert 3 3 \rVert }$ . + +2. We extensively evaluate ReAct on a suite of OOD detection tasks and establish a stateof-the-art performance among post hoc methods. Compared to the previous best method, ReAct achieves an FPR95 reduction of $2 5 . 0 5 \%$ on a large-scale ImageNet benchmark. + +3. We provide both empirical ablation and theoretical analysis, revealing important insights that abnormally high activations on OOD data can harm their detection and how ReAct effectively mitigates this issue. We hope that our insights inspire future research to further examine the internal mechanisms of neural networks for OOD detection. Code and dataset will be released for reproducible research. + +# 2 Background + +The problem of out-of-distribution detection for classification is defined using the following setup. Let $\mathcal { X }$ (resp. $\mathcal { V }$ ) be the input (resp. output) space and let $\mathcal { P }$ be a distribution over $\mathcal { X } \times \mathcal { V }$ . Let $f : \mathcal { X } \to \mathbb { R } ^ { | \mathcal { V } | }$ be a neural network trained on samples drawn from $\mathcal { P }$ to output a logit vector, which is used to predict the label of an input sample. Denote by $\mathcal { D } _ { \mathrm { i n } }$ the marginal distribution of $\mathcal { P }$ for $\mathcal { X }$ , which represents the distribution of in-distribution data. At test time, the environment can present a distribution $\mathcal { D } _ { \mathrm { o u t } }$ over $\mathcal { X }$ of out-of-distribution data. The goal of OOD detection is to define a decision function $G$ such that for a given test input $\mathbf { x } \in \mathcal { X }$ : + +$$ +\begin{array} { r } { G ( \mathbf { x } ; f ) = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f } \mathbf { x } \sim \mathcal { D } _ { \mathrm { o u t } } , } \\ { 1 } & { \mathrm { i f } \mathbf { x } \sim \mathcal { D } _ { \mathrm { i n } } . } \end{array} \right. } \end{array} +$$ + +The difficulty of OOD detection greatly depends on the separation between $\mathcal { D } _ { \mathrm { i n } }$ and $\mathcal { D } _ { \mathrm { o u t } }$ . In the extreme case, if $\mathcal { D } _ { \mathrm { i n } } = \mathcal { D } _ { \mathrm { o u t } }$ then the optimal decision function is a random coin flip. In practice, $\mathcal { D } _ { \mathrm { o u t } }$ is often defined by a distribution that simulates unknowns encountered during deployment time, such as samples from an irrelevant distribution whose label set has no intersection with $\mathcal { V }$ and therefore should not be predicted by the model. + +# 3 Method + +We introduce a simple and surprisingly effective technique, Rectified Activations (ReAct), for improving OOD detection performance. Our key idea is to perform post hoc modification to the unit activation, so to bring the overall activation pattern closer to the well-behaved case. Specifically, we consider a pre-trained neural network parameterized by $\theta$ , which encodes an input $\mathbf { x } \in \mathbb { R } ^ { d }$ to a feature space with dimension $m$ . We denote by $h ( \mathbf { x } ) \in \mathbb { R } ^ { \bar { m } }$ the feature vector from the penultimate layer of the network. A weight matrix $\mathbf { W } \in \mathbb { R } ^ { m \times K }$ connects the feature $h ( \mathbf { x } )$ to the output $f ( \mathbf { x } )$ , where $K$ is the total number of classes in $\mathcal { V } = \{ 1 , 2 , . . . , K \}$ . + +ReAct: Rectified Activation. We propose the ReAct operation, which is applied on the penultimate layer of a network: + +$$ +\begin{array} { r } { \bar { h } ( \mathbf { x } ) = \mathtt { R e A c t } ( h ( \mathbf { x } ) ; c ) , } \end{array} +$$ + +where $\mathtt { R e A c t } ( x ; c ) = \operatorname* { m i n } ( x , c )$ and is applied element-wise to the feature vector $h ( \mathbf { x } )$ . In effect, this operation truncates activations above $c$ to limit the effect of noise. The model output after rectified activation is given by: + +$$ +f ^ { \mathrm { R e A c t } } ( \mathbf { x } ; \boldsymbol { \theta } ) = \mathbf { W } ^ { \top } \bar { h } ( \mathbf { x } ) + \mathbf { b } , +$$ + +where $\mathbf { b } \in \mathbb { R } ^ { K }$ is the bias vector. A higher $c$ indicates a larger threshold of activation truncation. When $c = \infty$ , the output becomes equivalent to the original output $f ( \mathbf { x } ; \theta )$ without rectification, where $f ( \mathbf { x } ; \theta ) = \mathbf { W } ^ { \top } h ( \mathbf { x } ) + \mathbf { b }$ . Ideally, the rectification parameter $c$ should be chosen to sufficiently preserve the activations for ID data while rectifying that of OOD data. In practice, we set $c$ based on the $p$ -th percentile of activations estimated on the ID data. For example, when $p = 9 0$ , it indicates that $90 \%$ percent of the ID activations are less than the threshold $c$ . We discuss the effect of percentile in detail in Section 4. + +OOD detection with rectified activation. During test time, ReAct can be leveraged by a variety of downstream OOD scoring functions relying on $f ^ { \widetilde { \mathrm { R e A c t } } } ( \mathbf { x } ; \theta )$ : + +$$ +G _ { \lambda } ( \mathbf { x } ; f ^ { \mathrm { R e A c t } } ) = \left\{ { \begin{array} { l l } { \mathrm { i n } } & { S ( \mathbf { x } ; f ^ { \mathrm { R e A c t } } ) \geq \lambda } \\ { \mathrm { o u t } } & { S ( \mathbf { x } ; f ^ { \mathrm { R e A c t } } ) < \lambda } \end{array} } , \right. +$$ + +where a thresholding mechanism is exercised to distinguish between $\mathrm { I D }$ and OOD during test time. To align with the convention, samples with higher scores $S ( \mathbf { x } ; f )$ are classified as ID and vice versa. The threshold $\lambda$ is typically chosen so that a high fraction of ID data (e.g., $9 5 \%$ ) is correctly classified. ReAct can be compatible with several commonly used OOD scoring functions derived from the model output $f ( \mathbf { x } ; \theta )$ , including the softmax confidence $\mathbb { \left. \boldsymbol { \cdot } \boldsymbol { \cdot } \right. }$ , ODIN score $\pmb { \mathbb { B } } \pmb { \mathbb { 1 } }$ , and the energy score $\mathbb { \lVert 3 3 \rVert }$ . For readers’ convenience, we provide an expansive description of these methods in Appendix $\boxed { \mathbf { B } }$ In Section $^ { 4 , }$ we default to using the energy score (since it is hyperparameter-free and does not require fine-tuning), but demonstrate the benefit of using ReAct with other OOD scoring functions too. + +
ModelMethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95AUROCFPR95AUROC 个FPR95AUROCFPR95AUROCFPR95 AUROC
ResNetMSP 园↓ 54.99→ 87.74√ 70.8380.86↓ 73.99个 79.76↓ 68.00↑ 79.61√ 66.95→ 81.99
ODIN 白47.6689.6660.1584.5967.8981.7850.2385.6256.4885.41
Mahalanobis [ 2997.0052.6598.5042.4198.4041.7955.8085.0187.4355.47
Energy 国55.7289.9559.2685.8964.9282.8653.7285.9958.4186.17
ReAct (Ours)20.3896.2224.2094.2033.8591.5847.3089.8031.4392.95
77.0279.2376.2777.3073.5179.00
MobileNetMSP 固64.2985.32 87.6254.0777.10 85.8873.5185.0354.2085.81
ODIN 白 四55.39 62.1181.0047.8286.3357.36 52.0984.71 83.6349.96 92.3833.0663.6071.01
Mahalanobis59.5088.9162.6584.5069.3781.1958.0585.0362.3984.91
Energy 国 ReAct (Ours)42.4091.5347.6988.1651.5686.6438.4291.5345.0289.47
+ +Table 1: Main results. Comparison with competitive post hoc out-of-distribution detection methods. All methods are based on a model trained on ID data only (ImageNet-1k), without using any auxiliary outlier data. $\uparrow$ indicates larger values are better and $\downarrow$ indicates smaller values are better. All values are percentages. + +# 4 Experiments + +In this section, we evaluate ReAct on a suite of OOD detection tasks. We first evaluate a on large-scale OOD detection benchmark based on ImageNet [18] ( Section $\boxed { 4 . 1 }$ , and then proceed in Section 4.2 with CIFAR benchmarks $ { \mathbb { \left[ \left[ 2 7 \right] \right] } }$ . + +# 4.1 Evaluation on Large-scale ImageNet Classification Networks + +We first evaluate ReAct on a large-scale OOD detection benchmark developed in [18]. Compared to the CIFAR benchmarks that are routinely used in literature, the ImageNet benchmark is more challenging due to a larger label space $( K = 1 , 0 0 0 )$ . Moreover, such large-scale evaluation is more relevant to real-world applications, where the deployed models often operate on images that have high resolution and contain more classes than the CIFAR benchmarks. + +Setup. We use a pre-trained ResNet-50 model $[ \mathbb { 1 2 } ]$ for ImageNet-1k. At test time, all images are resized to $2 2 4 \times 2 2 4$ . We evaluate on four test OOD datasets from (subsets of) Places365 $\lVert \rVert \mathbf { 6 } 2 \rVert$ , Textures [5], iNaturalist $\mathbb { I } \mathbb { 1 6 } ]$ , and SUN $\mathbb { \left[ \left. 5 5 \right] \right. }$ with non-overlapping categories w.r.t ImageNet. Our evaluations span a diverse range of domains including fine-grained images, scene images, and textural images (see Appendix $\boxed { \mathbf { C } }$ for details). We use a validation set of Gaussian noise images, which are generated by sampling from $\mathcal { N } ( 0 , 1 )$ for each pixel location. To ensure validity, we further verify the activation pattern under Gaussian noise, which exhibits a similar distributional trend with positive skewness and chaoticness; see Figure $^ { 7 }$ in Appendix $\mathbf { G }$ for details. We select $p$ from $\{ 1 0 , \bar { 6 } 5 , 8 0 , 8 5 , 9 0 , 9 5 , 9 9 \}$ based on the FPR95 performance. The optimal $p$ is 90. All experiments are based on the hardware described in Appendix D. + +Comparison with competitive OOD detection methods. In Table 1, we compare ReAct with OOD detection methods that are competitive in the literature, where ReAct establishes the state-of-the-art performance. For a fair comparison, all methods use the pre-trained networks post hoc. We report performance for each OOD test dataset, as well as the average of the four. ReAct outperforms all baselines considered, including Maximum Softmax Probability $\mathbb { \lVert 1 5 \rVert }$ , ODIN $\pmb { \mathbb { B } } \mathbf { \mathbb { 1 } }$ , Mahalanobis distance $\mathbb { \left. 2 9 \right. }$ , and energy score $\bar { \big \| } \bar { 3 3 } \big \|$ . Noticeably, ReAct reduces the FPR95 by $2 5 . 0 5 \%$ compared to the best baseline $\pmb { \mathbb { B } } \mathbf { \mathbb { 1 } }$ on ResNet. Note that Mahalanobis requires training a separate binary classifier, and displays limiting performance since the increased size of label space makes the class-conditional Gaussian density estimation less viable. In contrast, ReAct is much easier to use in practice, and can be implemented through a simple post hoc activation rectification. + +Effect of rectification threshold $c$ . We now characterize the effect of the rectification parameter $c$ , which can be modulated by the percentile $p$ described in Section $3 .$ In Table $^ { 2 , }$ we summarize the OOD detection performance, where we vary $p = \{ 1 0 , 6 5 , 8 0 , 8 5 , 9 0 , 9 5 , 9 9 \}$ . This ablation confirms that over-activation does compromise the ability to detect OOD data, and ReAct can effectively alleviate this problem. Moreover, when $p$ is sufficiently large, ReAct can improve OOD detection while maintaining a comparable ID classification accuracy. Alternatively, once a sample is detected to be ID, one can always use the original activation $h ( \dot { \bf x } )$ , which is guaranteed to give identical classification accuracy. When $p$ is too small, OOD performance starts to degrade as expected. + +
Rectification percentileFPR95 √AUROC →AUPR 个ID ACC. 个Rectification threshold c
No ReAct 358.4186.1796.8875.088
p=9944.5790.4597.9675.122.25
p=9535.3992.3998.3774.761.50
p= 9031.4392.9598.5073.751.00
p=8534.0892.0598.3572.910.84
p=8041.5189.5497.9171.930.72
p=6574.6274.1494.3967.140.50
p=1074.7057.5586.061.220.06
+ +Table 2: Effect of rectification threshold for inference. Model is trained on ImageNet using ResNet-50 [11]. All numbers are percentages and are averaged over 4 OOD test datasets. + +Effect on other network architectures. We show that ReAct is effective on a different architecture in Table $\mathbb { A }$ In particular, we consider a lightweight model MobileNet-v2 [44], which can be suitable for OOD detection in on-device mobile applications. Same as before, we apply ReAct on the output of the penultimate layer, with the rectification threshold chosen based on the 90-th percentile. Our method reduces the FPR95 by $9 . 1 8 \%$ compared to the best baseline [31]. + +What about applying ReAct on other layers? Our results suggest that applying ReAct on the penultimate layer is the most effective, since the activation patterns are most distinctive. To see this, we provide the activation and performance study for intermediate layers in Appendix E (see Figure 6 and Table $^ { 6 ) }$ . Interestingly, early layers display less distinctive signatures between ID and OOD data. This is expected because neural networks generally capture lower-level features in early layers (such as Gabor filters $\left[ \left[ 6 1 \right] \right]$ in layer 1), whose activations can be very similar between ID and OOD. The semantic-level features only emerge as with deeper layers, where ReAct is the most effective. + +# 4.2 Evaluation on CIFAR Benchmarks + +Datasets. We evaluate on CIFAR-10 and CIFAR-100 [27] datasets as in-distribution data, using the standard split with 50,000 training images and 10,000 test images. For OOD data, we consider six common benchmark datasets: Textures [5], SVHN [37], Places365 [62], LSUN-Crop [60], LSUN-Resize $\pmb { \mathbb { H } }$ , and iSUN [57]. + +Experimental details. We train a standard ResNet-18 [11] model on in-distribution data. The feature dimension of the penultimate layer is 512. For both CIFAR-10 and CIFAR-100, the models are trained for 100 epochs. The start learning rate is 0.1 and decays by a factor of 10 at epochs 50, 75, and 90. For threshold $c$ , we use the 90-th percentile of activations estimated on the ID data. + +ReAct is compatible with various OOD scoring functions. We show in Table $\textcircled { 3 }$ that ReAct is a flexible method that is compatible with alternative scoring functions $S ( \mathbf { x } ; f ^ { \mathrm { R e A c t } } )$ . To see this, we consider commonly used scoring functions, and compare the performance both with and without using ReAct respectively. In particular, we consider softmax confidence $\mathbb { \lVert 1 5 \rVert }$ , ODIN score $\textcircled { \scriptsize { 1 3 1 } }$ as well as energy score [33]— all of which derive OOD scores directly from the output $f ( \mathbf { x } )$ . In particular, using ReAct on energy score yields the best performance, which is desirable as energy is a hyperparameter-free OOD score and is easy to compute in practice. Note that Mahalanobis $\overline { { \| 2 9 } }$ estimates OOD score using feature representations instead of the model output $f ( \mathbf { x } )$ , hence is less compatible with ReAct. On all three in-distribution datasets, using ReAct consistently outperforms the counterpart without rectification. Results in Table $3$ are based on the average across multiple OOD test datasets. Detailed performance for each OOD test dataset is provided in Appendix, Table 7. + +
MethodCIFAR-10CIFAR-100ImageNet
FPR95 √AUROC →AUPR →FPR95 ←AUROC →AUPR 个FPR95 ←AUROC →AUPR →
Softmax score 园56.7191.1779.1180.7276.8378.4166.9581.9995.76
Softmax score + ReAct53.8191.7092.1175.4580.4084.2858.2887.0697.22
Energy B335.6093.5795.0171.9382.8286.2858.4186.1796.88
Energy+ReAct32.9194.2795.5359.6187.4889.63 31.4392.9598.50
ODIN B131.1093.7994.9566.2182.8886.2556.4885.4196.61
ODIN+ReAct28.8194.0494.8259.9185.2387.5344.1090.7098.04
+ +Table 3: Ablation results. ReAct is compatible with different OOD scoring functions. For each ID dataset, we use the same model and compare the performance with and without ReAct respectively. $\uparrow$ indicates larger values are better and $\downarrow$ indicates smaller values are better. All values are percentages and are averaged over multiple OOD test datasets. Detailed performance for each OOD test dataset is available in Table $^ { 7 }$ in Appendix. + +![](images/73ef4dbf0548b1c453a52b9baca2581dc1a02715e7389b0b1368b2ed7449f176.jpg) +Figure 2: Positively skewed distribution of $\mu _ { i }$ (mean of each unit) in the penultimate layer for four OOD datasets (iNaturalist [16], SUN [55], Places $\pmb { \| 6 2 \| }$ , Textures $\pmb { \| 5 \| }$ ). Model is trained on ImageNet. Note that the right tail has a higher density than the left tail. By left and right tail we refer to the samples that are to the left and right of the median, which is close to the mode in the case of skewed distribution. + +# 5 Theoretical Analysis + +To better understand the effect of ReAct, we mathematically model the ID and OOD activations as rectified Gaussian distributions and derive their respective distributions after applying ReAct. These modeling assumptions are based on the activation statistics observed on ImageNet in Figure $^ { 1 } .$ In the following analysis, we show that ReAct reduces mean OOD activations more than ID activations since OOD activations are more positively skewed (see Appendix A for derivations). + +ID activations. Let $h ( \mathbf { x } ) = ( z _ { 1 } , \ldots , z _ { m } ) = : \mathbf { z }$ be the activations for the penultimate layer. We assume each $z _ { i } \sim \mathcal { N } ^ { R } ( \mu , \sigma _ { \mathrm { i n } } ^ { 2 } )$ for some $\sigma _ { \mathrm { i n } } > 0$ . Here $\mathcal N ^ { R } ( \mu , \sigma _ { \mathrm { i n } } ^ { 2 } ) = \operatorname* { m a x } ( 0 , \mathcal N ( \mu , \sigma _ { \mathrm { i n } } ^ { 2 } ) )$ denotes the rectified Gaussian distribution, which reflects the fact that activations after ReLU have no negative components. Before truncation with ReAct, the expectation of $z _ { i }$ is given by: + +$$ +\mathbb { E } _ { \mathrm { i n } } [ z _ { i } ] = \left[ 1 - \Phi \left( \frac { - \mu } { \sigma _ { \mathrm { i n } } } \right) \right] \cdot \mu + \phi \left( \frac { - \mu } { \sigma _ { \mathrm { i n } } } \right) \cdot \sigma _ { \mathrm { i n } } , +$$ + +where $\Phi$ and $\phi$ denote the cdf and pdf of the standard normal distribution, respectively. After rectification with ReAct, the expectation of $\bar { z } _ { i } = \operatorname* { m i n } ( z _ { i } , c )$ is: + +$$ +\mathcal { \bar { c } } _ { \mathrm { i n } } \big [ \bar { z } _ { \mathrm { i } } \big ] = \left[ \Phi \left( \frac { c - \mu } { \sigma _ { \mathrm { i n } } } \right) - \Phi \left( \frac { - \mu } { \sigma _ { \mathrm { i n } } } \right) \right] \cdot \mu + \left[ 1 - \Phi \left( \frac { c - \mu } { \sigma _ { \mathrm { i n } } } \right) \right] \cdot c + \left[ \phi \left( \frac { - \mu } { \sigma _ { \mathrm { i n } } } \right) - \phi \left( \frac { c - \mu } { \sigma _ { \mathrm { i n } } } \right) \right] \cdot \sigma _ { \mathrm { i n } } , +$$ + +The reduction in activation after ReAct is: + +$$ +\mathbb { E } _ { \mathrm { i n } } [ z _ { i } - \bar { z } _ { i } ] = \phi \left( \frac { c - \mu } { \sigma _ { \mathrm { i n } } } \right) \cdot \sigma _ { \mathrm { i n } } - \left[ 1 - \Phi \left( \frac { c - \mu } { \sigma _ { \mathrm { i n } } } \right) \right] \cdot ( c - \mu ) +$$ + +OOD activations. We model OOD activations as being generated by a two-stage process: Each OOD distribution defines a set of $\mu _ { i }$ ’s that represent the mode of the activation distribution for unit $i$ , and the activations $z _ { i }$ given $\mu _ { i }$ is represented by $z _ { i } | \mu _ { i } \sim \mathcal { N } ^ { R } ( \mu _ { i } , \tau ^ { 2 } )$ with $\tau > 0$ . For instance, the dark gray line in Figure $^ 1$ shows the set $\mu _ { i }$ ’s on the iNaturalist dataset, and the light gray area depicts the distribution of $z _ { i } | \mu _ { i }$ . One commonality across different OOD datasets is that the distribution of $\mu _ { i }$ is positively skewed. The assumption of positive skewness is motivated by our observation on real OOD data. Indeed, Figure $2$ shows the empirical distribution of $\mu _ { i }$ on an ImageNet pre-trained model for four OOD datasets, all of which display strong positive-skewness, i.e., the right tail has a much higher density than the left tail. This observation is surprisingly consistent across datasets and model architectures. Although a more in-depth understanding of the fundamental cause of positive skewness is important, for this work, we chose to rely on this empirically verifiable assumption and instead focus on analyzing our method ReAct. + +Utilizing the positive-skewness property of $\mu _ { i }$ , we analyze the distribution of $z _ { i }$ after marginalizing out $\mu _ { i }$ , which corresponds to averaging across different $\mu _ { i }$ ’s induced by various OOD distributions. Let $\dot { x } _ { i } | \mu _ { i } \sim \mathcal { N } ( \mu _ { i } , \tau ^ { 2 } )$ so that $z _ { i } | \mu _ { i } = \operatorname* { m a x } ( x _ { i } | \mu _ { i } , 0 )$ . Since $x _ { i } | \mu _ { i }$ is symmetric and $\mu _ { i }$ is positivelyskewed, the marginal distribution of $x _ { i }$ is also positively-skewed $\bigstar$ which we model with the epsilonskew-normal (ESN) distribution $\begin{array} { r l } { { \bigl \| \overline { { 3 5 } } \bigr \| } } & { { } } \end{array}$ . Specifically, we assume that $x _ { i } \sim \mathrm { E S N } ( \mu , \sigma _ { \mathrm { o u t } } ^ { 2 } , \epsilon )$ , which has + +![](images/bd8aa52dd90e0f12755c850a4b9ed932d05a82092ca47ec30f0e979709fce31b.jpg) +Figure 3: Plot showing the relationship between the skewness parameter $\epsilon$ and the chaotic-ness parameter $\sigma$ on activation reduction after applying ReAct. The function $\mathbb { E } [ z _ { i } - \bar { z } _ { i } ]$ is increasing in both $- \epsilon$ and $\sigma$ , which suggests that ReAct has a greater reduction effect for activation distributions with positive skewness $( \epsilon < 0 )$ ) and chaoticness—two signature characteristics of OOD activation. + +the following density function: + +$$ +q ( x ) = \left\{ \begin{array} { l l } { \phi ( ( x - \mu ) / \sigma _ { \mathrm { o u t } } ( 1 + \epsilon ) ) / \sigma _ { \mathrm { o u t } } } & { \mathrm { i f ~ } x < \mu , } \\ { \phi ( ( x - \mu ) / \sigma _ { \mathrm { o u t } } ( 1 - \epsilon ) ) / \sigma _ { \mathrm { o u t } } } & { \mathrm { i f ~ } x \geq \mu . } \end{array} \right. +$$ + +with $\epsilon \in [ - 1 , 1 ]$ controlling the skewness. In particular, the ESN distribution is positively-skewed when $\epsilon < 0$ . It follows that $z _ { i } = \operatorname* { m a x } ( x _ { i } , 0 )$ , with expectation: + +$$ +\mathbb { E } _ { \mathrm { o u t } } [ z _ { i } ] = \mu - ( 1 + \epsilon ) \Phi \left( \frac { - \mu } { ( 1 + \epsilon ) \sigma _ { \mathrm { o u t } } } \right) \cdot \mu + ( 1 + \epsilon ) ^ { 2 } \phi \left( \frac { - \mu } { ( 1 + \epsilon ) \sigma _ { \mathrm { o u t } } } \right) \cdot \sigma _ { \mathrm { o u t } } - \frac { 4 \epsilon } { \sqrt { 2 \pi } } \cdot \sigma _ { \mathrm { o u t } } . +$$ + +Expectation after applying ReAct becomes: + +$$ +\begin{array} { r } { \mathbb { E } _ { \mathrm { o u t } } [ \bar { z } _ { i } ] = \mu - ( 1 + \epsilon ) \Phi \left( \frac { - \mu } { ( 1 + \epsilon ) \sigma _ { \mathrm { o u t } } } \right) \cdot \mu + ( 1 - \epsilon ) \left[ 1 - \Phi \left( \frac { c - \mu } { ( 1 - \epsilon ) \sigma _ { \mathrm { o u t } } } \right) \right] \cdot ( c - \mu ) } \\ { + \left[ ( 1 + \epsilon ) ^ { 2 } \phi \left( \frac { - \mu } { ( 1 + \epsilon ) \sigma _ { \mathrm { o u t } } } \right) - ( 1 - \epsilon ) ^ { 2 } \phi \left( \frac { c - \mu } { ( 1 - \epsilon ) \sigma _ { \mathrm { o u t } } } \right) - \frac { 4 \epsilon } { \sqrt { 2 \pi } } \right] \cdot \sigma _ { \mathrm { o u t } } , } \end{array} +$$ + +Hence: + +$\mathbb { E } _ { \mathrm { o u t } } [ z _ { i } - \bar { z } _ { i } ] = ( 1 - \epsilon ) ^ { 2 } \phi \left( \frac { c - \mu } { ( 1 - \epsilon ) \sigma _ { \mathrm { o u t } } } \right) \cdot \sigma _ { \mathrm { o u t } } - ( 1 - \epsilon ) \left[ 1 - \Phi \left( \frac { c - \mu } { ( 1 - \epsilon ) \sigma _ { \mathrm { o u t } } } \right) \right] \cdot ( c - \mu ) ,$ which recovers Equation 4 when $\epsilon = 0$ and $\sigma _ { \mathrm { o u t } } = \sigma _ { \mathrm { i n } }$ . + +Remark 1: Activation reduction on OOD is more than ID. Figure 3 shows a plot of $\mathbb { E } [ z _ { i } - \bar { z } _ { i } ]$ for $\mu = 0 . 5$ and $c = 1$ . Observe that decreasing $\epsilon$ (more positive-skewness) or increasing $\sigma$ (more chaotic-ness) leads to a larger reduction in the mean activation after applying ReAct. For example, under the same $\sigma$ , a larger $\mathbb { E } _ { \mathrm { o u t } } [ z _ { i } - \bar { z } _ { i } ] - \mathbb { E } _ { \mathrm { i n } } [ z _ { i } - \bar { z } _ { i } ]$ can be observed by the gap of $z$ -axis value between $\epsilon = 0$ and $\epsilon < 0$ (e.g., $\epsilon = - 0 . 4 )$ . This suggests that rectification on average affects OOD activations more severely compared to ID activations. + +Remark 2: Output reduction on OOD is more than ID. To derive the effect on the distribution of model output, consider output logits $f ( \mathbf { z } ) = W \mathbf { z } + \mathbf { b }$ and assume without loss of generality that $W \mathbf { 1 } > 0$ element-wise. This can be achieved by adding a positive constant to $W$ without changing the output probabilities or classification decision. Let $\begin{array} { r } { \bar { \boldsymbol \delta } = \mathbf { \dot { \mathbb { E } } _ { \mathrm { o u t } } } [ \mathbf { z } - \bar { \mathbf { z } } ] - \mathbb { E } _ { \mathrm { i n } } [ \mathbf { z } - \bar { \mathbf { z } } ] > 0 . } \end{array}$ Then: + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \mathrm { o u t } } [ f ( \mathbf { z } ) - f ( \bar { \mathbf { z } } ) ] = \mathbb { E } _ { \mathrm { o u t } } [ W ( \mathbf { z } - \bar { \mathbf { z } } ) ] = W \mathbb { E } _ { \mathrm { o u t } } [ \mathbf { z } - \bar { \mathbf { z } } ] = W \left( \mathbb { E } _ { \mathrm { i n } } [ \mathbf { z } - \bar { \mathbf { z } } ] + \delta \mathbf { 1 } \right) } \\ & { \qquad = \mathbb { E } _ { \mathrm { i n } } [ W ( \mathbf { z } - \bar { \mathbf { z } } ) ] + \delta W \mathbf { 1 } } \\ & { \qquad > \mathbb { E } _ { \mathrm { i n } } [ f ( \mathbf { z } ) - f ( \bar { \mathbf { z } } ) ] . } \end{array} +$$ + +Hence the increased separation between OOD and $\mathrm { I D }$ activations transfers to the output space as well. Note that the condition of $W \mathbf { 1 } > 0$ is sufficient but not necessary for this result to hold. In fact, our experiments in Section $\textcircled { 4 } \mathrm { d o }$ not require this condition. However, we verified empirically that ensuring $W \mathbf { 1 } > 0$ by adding a positive constant to $W$ and applying ReAct does confer benefits to OOD detection, which validates our theoretical analysis. + +Why ReAct improves the OOD scoring functions? Our theoretical analysis above shows that ReAct suppresses logit output for OOD data more so than for ID data. This means that for detection scores depending on the logit output (e.g., energy score $\mathbb { \lVert 3 3 \rVert }$ ), the gap between OOD and ID score will be enlarged after applying ReAct, which makes thresholding more capable of separating OOD and ID inputs; see Figure 1(a) and (c) for a concrete example showing this effect. + +# 6 Discussion and Further Analysis + +Why do OOD samples trigger abnormal unit activation patterns? So far we have shown that OOD data can trigger unit activation patterns that are significantly different from ID data, and that ReAct can effectively alleviate this issue (empirically in Section $^ 4$ and theoretically in Section 5). Yet a question left in mystery is why such a pattern occurs in modern neural networks? Answering this question requires carefully examining the internal mechanism by which the network is trained and evaluated. Here we provide one plausible explanation for the activation patterns observed in Figure 1, with the hope of shedding light for future research. + +Intriguingly, our analysis reveals an important insight that batch normalization (BatchNorm) $[ [ 2 2 ] ] { - \mathsf { a } }$ common technique employed during model training—is in fact both a blessing (for ID classification) and a curse (for OOD detection). Specifically, for a unit activation denoted by $z$ , the network estimates the running mean $\mathbb { E } _ { \mathrm { i n } } ( z )$ and variance $\mathrm { V a r } _ { \mathrm { i n } } ( z )$ , over the entire ID training set during training. During inference time, the network applies BatchNorm statistics $\mathbb { E } _ { \mathrm { i n } } ( z )$ and $\mathrm { V a r } _ { \mathrm { i n } } ( z )$ , which helps normalize the activations for the test data with the same distribution $\mathcal { D } _ { \mathrm { i n } }$ : + +$$ +\mathrm { B a t c h N o r m } ( z ; \gamma , \beta , \epsilon ) = \frac { z - \mathbb { E } _ { \mathrm { i n } } [ z ] } { \sqrt { \mathrm { V a r } _ { \mathrm { i n } } [ z ] + \epsilon } } \cdot \gamma + \beta +$$ + +However, our key observation is that using mismatched BatchNorm statistics—that are estimated on $\mathcal { D } _ { \mathrm { i n } }$ yet blindly applied to the OOD $\mathcal { D } _ { \mathrm { o u t } }$ —can trigger abnormally high unit activations. As a thought experiment, we instead apply the true BatchNorm statistics estimated on a batch of OOD images and we observe well-behaved activation patterns with near-constant mean and standard deviations—just like the ones observed on the ID data (see Figure $\mathbb { E } ,$ top). Our study therefore reveals one of the fundamental causes for neural networks to produce overconfident predictions for OOD data. After applying the true statistics (estimated on OOD), the output distributions between ID and OOD data become much more separable. While this thought experiment has shed some guiding light, the solution of estimating BatchNorm statistics on a batch of OOD data is not at all satisfactory and realistic. Arguably, it poses a strong and impractical assumption of having access to a batch of OOD data during test time. Despite its limitation, we view it as an oracle, which serves as an upper bound on performance for ReAct. + +![](images/ce8b1cf28903076ecbaa2d2a5e4c5ec6c6bec4e39ce57f4b41bcff3d3ce6683a.jpg) +Figure 4: The distribution of per-unit activations in the penultimate layer for OOD data (iNaturalist) by using true (top) vs. mismatched (bottom) BatchNorm statistics for OOD data. + +
MethodiNaturalistPlacesSUNTextures
Oracle (batch OOD for estimating BN statistics)99.5999.0998.3291.43
ReAct (single OOD)96.2294.2091.5889.80
No ReAct B389.9585.8982.8685.99
+ +Table 4: Comparison with oracle using OOD BN statistics. Model is trained on ImageNet (see Section 4.1) Values are AUROC. + +In particular, results in Table $\boxed { \sharp }$ suggest that our method favorably matches the oracle performance using the ground truth BN statistics. This is encouraging as our method does not impose any batch assumption and can be feasible for single-input testing scenarios. + +What about networks trained with different normalization mechanisms? Going beyond batch normalization $\pmb { \left. 2 2 \right. }$ , we further investigate (1) whether networks trained with alternative normalization approaches exhibit similar activation patterns, and (2) whether ReAct is helpful there. To answer this question, we additionally evaluate networks trained with WeightNorm $\underline { { \hat { \vert 4 3 \vert } } }$ and GroupNorm [54]— two other well-known normalization methods. As shown in Figure $\triangledown$ the unit activations also display highly distinctive signature patterns between ID and OOD data, with more chaos on OOD data. In all cases, the networks are trained to adapt to the ID data, resulting in abnormal activation signatures on OOD data in testing. Unlike BatchNorm, there is no easy oracle (e.g., re-estimating the statistics on OOD data) to counteract the ill-fated normalizations. + +We apply ReAct on models trained with WeightNorm and GroupNorm, and report results in Table 5. Our results suggest that ReAct is consistently effective under various normalization schemes. For example, ReAct reduces the average FPR95 by $2 3 . 5 4 \%$ and $1 4 . 7 \%$ respectively. Overall, ReAct has shown broad efficacy and compatibility with different OOD scoring functions (Section $4 . 2 )$ + +![](images/5317d04dc38b221b3468d7e8c83dadf02fc27f7e464b69af03aba71d821889fc.jpg) +Model: WeightNormFigure 5: The distribution of per-unit activations in the penultimate layer for ID (ImageNet $\pmb { \mathbb { Z } } \mathbf { I }$ ) and OOD (iNaturalist $\pm \boxed { 1 6 }$ ) on model trained with weight normalization [43]. +Table 5: Effectiveness of ReAct for different normalization methods. ReAct consistently improves OOD detection performance for the model trained with GroupNorm and WeightNorm. In-distribution is ImageNet-1k dataset. Bold numbers are superior results. + +
MethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95 ↓AUROC 个FPR95 ↓AUROC 个FPR95AUROCFPR95AUROC 个FPR95 ↓AUROC 个
GroupNormw.o.ReAct65.3888.4565.1185.52↓ 65.46个 84.34↓ 69.1783.2266.2885.38
w/ReAct39.4592.9551.5787.9052.7887.3262.5081.7651.5887.48
WeightNormw.o. ReAct40.7192.5248.0789.3950.9287.8761.6580.7150.3487.62
w/ReAct19.7395.9131.3993.2142.3488.9413.7496.9826.8093.76
+ +# 7 Related Work + +OOD uncertainty estimation with discriminative models. The problem of classification with rejection can date back to early works on abstention [4, 10], which considered simple model families such as SVMs $\pmb { \Vert 6 \Vert }$ . A comprehensive survey on OOD detection can be found in $\left[ \left[ 5 8 \right] \right]$ . The phenomenon of neural networks’ overconfidence in out-of-distribution data is first revealed by Nguyen et al. [38]. Early works attempted to improve the OOD uncertainty estimation by proposing the ODIN score [17, 31], OpenMax score $\pmb { \mathbb { B } } \|$ , and Mahalanobis distance-based score $\pmb { \mathbb { Z } } \pmb { \mathcal { 0 } }$ . Recent work by Liu et al. [33] proposed using an energy score for OOD uncertainty estimation, which can be easily derived from a discriminative classifier and demonstrated advantages over the softmax confidence score both empirically and theoretically. Wang et al. [51] further showed an energy-based approach can improve OOD uncertainty estimation for multi-label classification networks. Huang and Li [18] revealed that approaches developed for common CIFAR benchmarks might not translate effectively into a large-scale ImageNet benchmark, highlighting the need to evaluate OOD uncertainty estimation in a large-scale real-world setting. Previous methods primarily derive OOD scores using original activations. In contrast, our work is motivated by a novel analysis of internal activations, and shows that rectifying activations is a surprisingly effective approach for OOD detection. + +Neural network activation analysis. Neural networks have been studied at the granularity of the activation of individual layers [21, 34, 59, 63], or individual networks $\textcircled { \lvert 3 0 \rvert }$ . In particular, Li et al. [30] studied the similarity of activation space between two independently trained neural networks. + +Previously, $\underline { { \mathrm { f l e i n ~ e t ~ a l . } } } \underline { { \mathbb { ( } \mathbb { 1 3 } } } \|$ showed that neural networks with ReLU activation can lead to arbitrary high activation for inputs far away from the training data. We show that using ReAct could efficiently alleviate this undesirable phenomenon. ReAct does not rely on auxiliary data and can be conveniently used for pre-trained models. The idea of rectifying unit activation $\dot { \mathbb { Z } } \dot { 6 } \mathbb { I }$ , which is known as Relu6, was used to facilitate the learning of sparse features. In this paper, we first show that rectifying activation can drastically alleviate the overconfidence issue for OOD data, and as a result, improve OOD detection. + +OOD uncertainty estimation with generative models. Alternative approaches for detecting OOD inputs resort to generative models that directly estimate density [8, 20, 23, 41, 49, 50]. An input is deemed as OOD if it lies in the low-likelihood regions. A plethora of literature has emerged to utilize generative models for OOD detection [24, 40, 45, 46, 52, 56]. Interestingly, Nalisnick et al. [36] showed that deep generative models can assign a high likelihood to OOD data. Moreover, generative models can be prohibitively challenging to train and optimize, and the performance can often lag behind the discriminative counterpart. In contrast, our method relies on a discriminative classifier, which is easier to optimize and achieves stronger performance. + +Distributional shifts. Distributional shifts have attracted increasing research interests $\mathbb { \left[ \left[ 2 5 \right] \right] }$ . It is important to recognize and differentiate various types of distributional shift problems. Literature in OOD detection is commonly concerned about model reliability and detection of label-space shifts, where the OOD inputs have disjoint labels w.r.t. ID data and therefore should not be predicted by the model. Meanwhile, some works considered label distribution shift [1, 2, 32, 42, 47, 53], where the label space is common between ID and OOD but the marginal label distribution changes, as well as covariate shift in the input space [14, 39], where inputs can be corruption-shifted or domainshifted $ { \mathbb { I } } { \mathrm { I Z } } , { \mathbb { H } } { \mathrm { 8 l } }$ . It is important to note that our work focuses on the detection of shifts where the label space $\mathcal { V }$ is different between ID and OOD data and hence the model should not make any prediction, instead of label distribution shift and covariate shift where the model is expected to generalize. + +# 8 Conclusion + +This paper provides a simple activation rectification strategy termed ReAct, which truncates the high activations during test time for OOD detection. We provide both empirical and theoretical insights characterizing and explaining the mechanism by which ReAct improves OOD uncertainty estimation. By rectifying the activations, the outsized contribution of hidden units on OOD output can be attenuated, resulting in a stronger separability from ID data. Extensive experiments show ReAct can significantly improve the performance of OOD detection on both common benchmarks and large-scale image classification models. Applications of multi-class classification can benefit from our method, and we anticipate further research in OOD detection to extend this work to tasks beyond image classification. We hope that our insights inspire future research to further examine the internal mechanisms of neural networks for OOD detection. + +# 9 Societal Impact + +Our project aims to improve the dependability and trustworthiness of modern machine learning models. This stands to benefit a wide range of fields and societal activities. We believe out-ofdistribution uncertainty estimation is an increasingly critical component of systems that range from consumer and business applications (e.g., digital content understanding) to transportation (e.g., driver assistance systems and autonomous vehicles), and to health care (e.g., unseen disease identification). Many of these applications require classification models in operation. Through this work and by releasing our code, we hope to provide machine learning researchers with a new methodological perspective and offer machine learning practitioners a plug-and-play tool that renders safety against OOD data in the real world. While we do not anticipate any negative consequences to our work, we hope to continue to build on our method in future work. + +# Acknowledgement + +YS and YL are supported by the Office of the Vice Chancellor for Research and Graduate Education (OVCRGE) with funding from the Wisconsin Alumni Research Foundation (WARF). + +References +[1] Amr Alexandari, Anshul Kundaje, and Avanti Shrikumar. Maximum likelihood with biascorrected calibration is hard-to-beat at label shift adaptation. In International Conference on Machine Learning, pages 222–232. PMLR, 2020. +[2] Kamyar Azizzadenesheli, Anqi Liu, Fanny Yang, and Animashree Anandkumar. Regularized learning for domain adaptation under label shifts. 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Revisiting the importance of individual units in cnns via ablation. arXiv preprint arXiv:1806.02891, 2018. \ No newline at end of file diff --git a/parse/train/IBVBtz_sRSm/IBVBtz_sRSm_content_list.json b/parse/train/IBVBtz_sRSm/IBVBtz_sRSm_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..cb5bfaa0aed8d7427441fed4a2c7a19b95ca2fdc --- /dev/null +++ b/parse/train/IBVBtz_sRSm/IBVBtz_sRSm_content_list.json @@ -0,0 +1,1547 @@ +[ + { + "type": "text", + "text": "ReAct: Out-of-distribution Detection With Rectified Activations ", + "text_level": 1, + "bbox": [ + 240, + 122, + 758, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yiyou Sun \nDepartment of Computer Sciences \nUniversity of Wisconsin-Madison sunyiyou@cs.wisc.edu ", + "bbox": [ + 263, + 222, + 490, + 276 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Chuan Guo Facebook AI Research chuanguo@fb.com ", + "bbox": [ + 584, + 222, + 735, + 262 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yixuan Li \nDepartment of Computer Sciences \nUniversity of Wisconsin-Madison sharonli@cs.wisc.edu ", + "bbox": [ + 385, + 296, + 612, + 352 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 387, + 535, + 404 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Out-of-distribution (OOD) detection has received much attention lately due to its practical importance in enhancing the safe deployment of neural networks. One of the primary challenges is that models often produce highly confident predictions on OOD data, which undermines the driving principle in OOD detection that the model should only be confident about in-distribution samples. In this work, we propose ReAct—a simple and effective technique for reducing model overconfidence on OOD data. Our method is motivated by novel analysis on internal activations of neural networks, which displays highly distinctive signature patterns for OOD distributions. Our method can generalize effectively to different network architectures and different OOD detection scores. We empirically demonstrate that ReAct achieves competitive detection performance on a comprehensive suite of benchmark datasets, and give theoretical explication for our method’s efficacy. On the ImageNet benchmark, ReAct reduces the false positive rate (FPR95) by $2 5 . 0 5 \\%$ compared to the previous best method1. ", + "bbox": [ + 233, + 417, + 766, + 608 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 631, + 310, + 647 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural networks deployed in real-world systems often encounter out-of-distribution (OOD) inputs— unknown samples that the network has not been exposed to during training. Identifying and handling these OOD inputs can be paramount in safety-critical applications such as autonomous driving [9] and health care. For example, an autonomous vehicle may fail to recognize objects on the road that do not appear in its object detection model’s training set, potentially leading to a crash. This can be prevented if the system identifies the unrecognized object as OOD and warns the driver in advance. ", + "bbox": [ + 174, + 661, + 825, + 743 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A driving idea behind OOD detection is that the model should be much more uncertain about samples outside of its training distribution. However, $\\mathrm { \\Delta N g u y e n \\ e t { a l . } } \\rVert { 3 8 } \\rVert$ revealed that modern neural networks can produce overconfident predictions on OOD inputs. This phenomenon renders the separation of in-distribution (ID) and OOD data a non-trivial task. Indeed, much of the prior work on OOD detection focused on defining more suitable measures of OOD uncertainty [17, 19, 28, 29, 31, 33]. Despite the improvement, it is arguable that continued research progress in OOD detection requires insights into the fundamental cause and mitigation of model overconfidence on OOD data. ", + "bbox": [ + 174, + 748, + 825, + 844 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we start by revealing an important observation that OOD data can trigger unit activation patterns that are significantly different from ID data. Figure 1(b) shows the distribution of activations in the penultimate layer of ResNet-50 trained on ImageNet [7]. Each point on the horizontal axis corresponds to a single unit. The mean and standard deviation are shown by the solid line and shaded area, respectively. The mean activation for ID data (blue) is well-behaved with a near-constant mean and standard deviation. In contrast, for OOD data (gray), the mean activation has significantly larger variations across units and is biased towards having sharp positive values (i.e., positively skewed). As a result, such high unit activation can undesirably manifest in model output, producing overconfident predictions on OOD data. A similar distributional property holds for other OOD datasets as well. ", + "bbox": [ + 173, + 852, + 821, + 878 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/24e9b1c612ee980b996a70ed12b719e19f1e79634c823067cead43b6f4764fd6.jpg", + "image_caption": [ + "Figure 1: Plots showing (a) the distribution of ID (ImageNet $\\mathbb { U }$ ) and OOD (iNaturalist $\\mathbb { I } 1 6 \\mathbb { I }$ ) uncertainty scores before truncation, (b) the distribution of per-unit activations in the penultimate layer for ID and OOD data, and (c) the distribution of OOD uncertainty scores $\\pmb { \\Vert 3 3 } \\Vert$ after rectification. Applying ReAct drastically improves the separation of ID and OOD data. See text for details. " + ], + "image_footnote": [], + "bbox": [ + 176, + 90, + 820, + 233 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 343, + 825, + 439 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The above observation naturally inspires a simple yet surprisingly effective method—Rectified Activations (dubbed ReAct) for OOD detection. In particular, the outsized activation of a few selected hidden units can be attenuated by rectifying the activations at an upper limit $c > 0$ . Conveniently, this can be done on a pre-trained model without any modification to training. The dashed horizontal line in Figure 1(b) shows the cutoff point $c$ , and its effect on the OOD uncertainty score is shown in Figure 1(c). After rectification, the output distributions for ID and OOD data become much more well-separated and the false positive rate (FPR) is significantly reduced from $5 5 . 7 2 \\%$ to $2 0 . 3 8 \\%$ . Importantly, this truncation largely preserves the activation for in-distribution data, and therefore ensures the classification accuracy on the original task is largely comparable. ", + "bbox": [ + 173, + 445, + 825, + 568 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We provide both empirical and theoretical insights, characterizing and explaining the mechanism by which ReAct improves OOD detection. We perform extensive evaluations and establish state-ofthe-art performance on a suite of common OOD detection benchmarks, including CIFAR-10 and CIFAR-100, as well as a large-scale ImageNet dataset $\\mathbb { U }$ . ReAct outperforms the best baseline by a large margin, reducing the average FPR95 by up to $2 5 . 0 5 \\%$ . We further analyze our method theoretically and show that ReAct is more beneficial when OOD activations are more chaotic (i.e., having a larger variance) and positively skewed compared to ID activations, a behavior that is typical of many OOD datasets (cf. Figure 1) . In summary, our key results and contributions are: ", + "bbox": [ + 173, + 574, + 825, + 683 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We introduce ReAct —a simple and effective post hoc OOD detection approach that utilizes activation truncation. We show that ReAct can generalize effectively to different network architectures and works with different OOD detection methods including MSP [15], ODIN [31], and energy score $\\mathbb { \\lVert 3 3 \\rVert }$ . ", + "bbox": [ + 214, + 702, + 825, + 758 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2. We extensively evaluate ReAct on a suite of OOD detection tasks and establish a stateof-the-art performance among post hoc methods. Compared to the previous best method, ReAct achieves an FPR95 reduction of $2 5 . 0 5 \\%$ on a large-scale ImageNet benchmark. ", + "bbox": [ + 214, + 779, + 826, + 820 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3. We provide both empirical ablation and theoretical analysis, revealing important insights that abnormally high activations on OOD data can harm their detection and how ReAct effectively mitigates this issue. We hope that our insights inspire future research to further examine the internal mechanisms of neural networks for OOD detection. Code and dataset will be released for reproducible research. ", + "bbox": [ + 214, + 843, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Background ", + "text_level": 1, + "bbox": [ + 173, + 89, + 308, + 106 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The problem of out-of-distribution detection for classification is defined using the following setup. Let $\\mathcal { X }$ (resp. $\\mathcal { V }$ ) be the input (resp. output) space and let $\\mathcal { P }$ be a distribution over $\\mathcal { X } \\times \\mathcal { V }$ . Let $f : \\mathcal { X } \\to \\mathbb { R } ^ { | \\mathcal { V } | }$ be a neural network trained on samples drawn from $\\mathcal { P }$ to output a logit vector, which is used to predict the label of an input sample. Denote by $\\mathcal { D } _ { \\mathrm { i n } }$ the marginal distribution of $\\mathcal { P }$ for $\\mathcal { X }$ , which represents the distribution of in-distribution data. At test time, the environment can present a distribution $\\mathcal { D } _ { \\mathrm { o u t } }$ over $\\mathcal { X }$ of out-of-distribution data. The goal of OOD detection is to define a decision function $G$ such that for a given test input $\\mathbf { x } \\in \\mathcal { X }$ : ", + "bbox": [ + 173, + 119, + 826, + 218 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/42d3d579f85c89b33dd42312ede29cfb5d7e2481b0535a56aff68c98199f3c72.jpg", + "text": "$$\n\\begin{array} { r } { G ( \\mathbf { x } ; f ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { i f } \\mathbf { x } \\sim \\mathcal { D } _ { \\mathrm { o u t } } , } \\\\ { 1 } & { \\mathrm { i f } \\mathbf { x } \\sim \\mathcal { D } _ { \\mathrm { i n } } . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 398, + 224, + 594, + 261 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The difficulty of OOD detection greatly depends on the separation between $\\mathcal { D } _ { \\mathrm { i n } }$ and $\\mathcal { D } _ { \\mathrm { o u t } }$ . In the extreme case, if $\\mathcal { D } _ { \\mathrm { i n } } = \\mathcal { D } _ { \\mathrm { o u t } }$ then the optimal decision function is a random coin flip. In practice, $\\mathcal { D } _ { \\mathrm { o u t } }$ is often defined by a distribution that simulates unknowns encountered during deployment time, such as samples from an irrelevant distribution whose label set has no intersection with $\\mathcal { V }$ and therefore should not be predicted by the model. ", + "bbox": [ + 173, + 267, + 826, + 337 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Method ", + "text_level": 1, + "bbox": [ + 174, + 356, + 271, + 372 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We introduce a simple and surprisingly effective technique, Rectified Activations (ReAct), for improving OOD detection performance. Our key idea is to perform post hoc modification to the unit activation, so to bring the overall activation pattern closer to the well-behaved case. Specifically, we consider a pre-trained neural network parameterized by $\\theta$ , which encodes an input $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ to a feature space with dimension $m$ . We denote by $h ( \\mathbf { x } ) \\in \\mathbb { R } ^ { \\bar { m } }$ the feature vector from the penultimate layer of the network. A weight matrix $\\mathbf { W } \\in \\mathbb { R } ^ { m \\times K }$ connects the feature $h ( \\mathbf { x } )$ to the output $f ( \\mathbf { x } )$ , where $K$ is the total number of classes in $\\mathcal { V } = \\{ 1 , 2 , . . . , K \\}$ . ", + "bbox": [ + 173, + 386, + 826, + 484 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "ReAct: Rectified Activation. We propose the ReAct operation, which is applied on the penultimate layer of a network: ", + "bbox": [ + 171, + 489, + 823, + 517 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c152f7201cbd38d79f716c32a0de75aba4d7d25ec556bbec2a585d2ac3b20bb5.jpg", + "text": "$$\n\\begin{array} { r } { \\bar { h } ( \\mathbf { x } ) = \\mathtt { R e A c t } ( h ( \\mathbf { x } ) ; c ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 416, + 523, + 580, + 541 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathtt { R e A c t } ( x ; c ) = \\operatorname* { m i n } ( x , c )$ and is applied element-wise to the feature vector $h ( \\mathbf { x } )$ . In effect, this operation truncates activations above $c$ to limit the effect of noise. The model output after rectified activation is given by: ", + "bbox": [ + 174, + 547, + 825, + 590 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0510be9e39488ca2db4c7e360cf9d165b95e022c458e8ef4bd15524883d0386c.jpg", + "text": "$$\nf ^ { \\mathrm { R e A c t } } ( \\mathbf { x } ; \\boldsymbol { \\theta } ) = \\mathbf { W } ^ { \\top } \\bar { h } ( \\mathbf { x } ) + \\mathbf { b } ,\n$$", + "text_format": "latex", + "bbox": [ + 398, + 597, + 598, + 616 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { b } \\in \\mathbb { R } ^ { K }$ is the bias vector. A higher $c$ indicates a larger threshold of activation truncation. When $c = \\infty$ , the output becomes equivalent to the original output $f ( \\mathbf { x } ; \\theta )$ without rectification, where $f ( \\mathbf { x } ; \\theta ) = \\mathbf { W } ^ { \\top } h ( \\mathbf { x } ) + \\mathbf { b }$ . Ideally, the rectification parameter $c$ should be chosen to sufficiently preserve the activations for ID data while rectifying that of OOD data. In practice, we set $c$ based on the $p$ -th percentile of activations estimated on the ID data. For example, when $p = 9 0$ , it indicates that $90 \\%$ percent of the ID activations are less than the threshold $c$ . We discuss the effect of percentile in detail in Section 4. ", + "bbox": [ + 173, + 625, + 825, + 723 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "OOD detection with rectified activation. During test time, ReAct can be leveraged by a variety of downstream OOD scoring functions relying on $f ^ { \\widetilde { \\mathrm { R e A c t } } } ( \\mathbf { x } ; \\theta )$ : ", + "bbox": [ + 171, + 728, + 823, + 757 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d16773bbd6a0fba09297450364b444e408810814f7fe9c5988863e1d150c9616.jpg", + "text": "$$\nG _ { \\lambda } ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) = \\left\\{ { \\begin{array} { l l } { \\mathrm { i n } } & { S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) \\geq \\lambda } \\\\ { \\mathrm { o u t } } & { S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) < \\lambda } \\end{array} } , \\right.\n$$", + "text_format": "latex", + "bbox": [ + 356, + 763, + 642, + 799 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where a thresholding mechanism is exercised to distinguish between $\\mathrm { I D }$ and OOD during test time. To align with the convention, samples with higher scores $S ( \\mathbf { x } ; f )$ are classified as ID and vice versa. The threshold $\\lambda$ is typically chosen so that a high fraction of ID data (e.g., $9 5 \\%$ ) is correctly classified. ReAct can be compatible with several commonly used OOD scoring functions derived from the model output $f ( \\mathbf { x } ; \\theta )$ , including the softmax confidence $\\mathbb { \\left. \\boldsymbol { \\cdot } \\boldsymbol { \\cdot } \\right. }$ , ODIN score $\\pmb { \\mathbb { B } } \\pmb { \\mathbb { 1 } }$ , and the energy score $\\mathbb { \\lVert 3 3 \\rVert }$ . For readers’ convenience, we provide an expansive description of these methods in Appendix $\\boxed { \\mathbf { B } }$ In Section $^ { 4 , }$ we default to using the energy score (since it is hyperparameter-free and does not require fine-tuning), but demonstrate the benefit of using ReAct with other OOD scoring functions too. ", + "bbox": [ + 173, + 806, + 826, + 916 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/f19809875f47af69b8c7910d0c1c9fdcc4ce30edec26a24dfdf16f7c7145e9e9.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Main results. Comparison with competitive post hoc out-of-distribution detection methods. All methods are based on a model trained on ID data only (ImageNet-1k), without using any auxiliary outlier data. $\\uparrow$ indicates larger values are better and $\\downarrow$ indicates smaller values are better. All values are percentages. " + ], + "table_body": "
ModelMethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95AUROCFPR95AUROC 个FPR95AUROCFPR95AUROCFPR95 AUROC
ResNetMSP 园↓ 54.99→ 87.74√ 70.8380.86↓ 73.99个 79.76↓ 68.00↑ 79.61√ 66.95→ 81.99
ODIN 白47.6689.6660.1584.5967.8981.7850.2385.6256.4885.41
Mahalanobis [ 2997.0052.6598.5042.4198.4041.7955.8085.0187.4355.47
Energy 国55.7289.9559.2685.8964.9282.8653.7285.9958.4186.17
ReAct (Ours)20.3896.2224.2094.2033.8591.5847.3089.8031.4392.95
77.0279.2376.2777.3073.5179.00
MobileNetMSP 固64.2985.32 87.6254.0777.10 85.8873.5185.0354.2085.81
ODIN 白 四55.39 62.1181.0047.8286.3357.36 52.0984.71 83.6349.96 92.3833.0663.6071.01
Mahalanobis59.5088.9162.6584.5069.3781.1958.0585.0362.3984.91
Energy 国 ReAct (Ours)42.4091.5347.6988.1651.5686.6438.4291.5345.0289.47
", + "bbox": [ + 174, + 88, + 830, + 242 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 313, + 312, + 330 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we evaluate ReAct on a suite of OOD detection tasks. We first evaluate a on large-scale OOD detection benchmark based on ImageNet [18] ( Section $\\boxed { 4 . 1 }$ , and then proceed in Section 4.2 with CIFAR benchmarks $ { \\mathbb { \\left[ \\left[ 2 7 \\right] \\right] } }$ . ", + "bbox": [ + 174, + 348, + 826, + 391 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 Evaluation on Large-scale ImageNet Classification Networks ", + "text_level": 1, + "bbox": [ + 174, + 415, + 633, + 430 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We first evaluate ReAct on a large-scale OOD detection benchmark developed in [18]. Compared to the CIFAR benchmarks that are routinely used in literature, the ImageNet benchmark is more challenging due to a larger label space $( K = 1 , 0 0 0 )$ . Moreover, such large-scale evaluation is more relevant to real-world applications, where the deployed models often operate on images that have high resolution and contain more classes than the CIFAR benchmarks. ", + "bbox": [ + 174, + 444, + 825, + 512 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Setup. We use a pre-trained ResNet-50 model $[ \\mathbb { 1 2 } ]$ for ImageNet-1k. At test time, all images are resized to $2 2 4 \\times 2 2 4$ . We evaluate on four test OOD datasets from (subsets of) Places365 $\\lVert \\rVert \\mathbf { 6 } 2 \\rVert$ , Textures [5], iNaturalist $\\mathbb { I } \\mathbb { 1 6 } ]$ , and SUN $\\mathbb { \\left[ \\left. 5 5 \\right] \\right. }$ with non-overlapping categories w.r.t ImageNet. Our evaluations span a diverse range of domains including fine-grained images, scene images, and textural images (see Appendix $\\boxed { \\mathbf { C } }$ for details). We use a validation set of Gaussian noise images, which are generated by sampling from $\\mathcal { N } ( 0 , 1 )$ for each pixel location. To ensure validity, we further verify the activation pattern under Gaussian noise, which exhibits a similar distributional trend with positive skewness and chaoticness; see Figure $^ { 7 }$ in Appendix $\\mathbf { G }$ for details. We select $p$ from $\\{ 1 0 , \\bar { 6 } 5 , 8 0 , 8 5 , 9 0 , 9 5 , 9 9 \\}$ based on the FPR95 performance. The optimal $p$ is 90. All experiments are based on the hardware described in Appendix D. ", + "bbox": [ + 173, + 517, + 825, + 656 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Comparison with competitive OOD detection methods. In Table 1, we compare ReAct with OOD detection methods that are competitive in the literature, where ReAct establishes the state-of-the-art performance. For a fair comparison, all methods use the pre-trained networks post hoc. We report performance for each OOD test dataset, as well as the average of the four. ReAct outperforms all baselines considered, including Maximum Softmax Probability $\\mathbb { \\lVert 1 5 \\rVert }$ , ODIN $\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }$ , Mahalanobis distance $\\mathbb { \\left. 2 9 \\right. }$ , and energy score $\\bar { \\big \\| } \\bar { 3 3 } \\big \\|$ . Noticeably, ReAct reduces the FPR95 by $2 5 . 0 5 \\%$ compared to the best baseline $\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }$ on ResNet. Note that Mahalanobis requires training a separate binary classifier, and displays limiting performance since the increased size of label space makes the class-conditional Gaussian density estimation less viable. In contrast, ReAct is much easier to use in practice, and can be implemented through a simple post hoc activation rectification. ", + "bbox": [ + 173, + 660, + 825, + 796 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Effect of rectification threshold $c$ . We now characterize the effect of the rectification parameter $c$ , which can be modulated by the percentile $p$ described in Section $3 .$ In Table $^ { 2 , }$ we summarize the OOD detection performance, where we vary $p = \\{ 1 0 , 6 5 , 8 0 , 8 5 , 9 0 , 9 5 , 9 9 \\}$ . This ablation confirms that over-activation does compromise the ability to detect OOD data, and ReAct can effectively alleviate this problem. Moreover, when $p$ is sufficiently large, ReAct can improve OOD detection while maintaining a comparable ID classification accuracy. Alternatively, once a sample is detected to be ID, one can always use the original activation $h ( \\dot { \\bf x } )$ , which is guaranteed to give identical classification accuracy. When $p$ is too small, OOD performance starts to degrade as expected. ", + "bbox": [ + 173, + 803, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/35ef878190a234d26fd6b05ccb702ea7a99c5b28ee21df5b118cb46e19627e46.jpg", + "table_caption": [], + "table_footnote": [ + "Table 2: Effect of rectification threshold for inference. Model is trained on ImageNet using ResNet-50 [11]. All numbers are percentages and are averaged over 4 OOD test datasets. " + ], + "table_body": "
Rectification percentileFPR95 √AUROC →AUPR 个ID ACC. 个Rectification threshold c
No ReAct 358.4186.1796.8875.088
p=9944.5790.4597.9675.122.25
p=9535.3992.3998.3774.761.50
p= 9031.4392.9598.5073.751.00
p=8534.0892.0598.3572.910.84
p=8041.5189.5497.9171.930.72
p=6574.6274.1494.3967.140.50
p=1074.7057.5586.061.220.06
", + "bbox": [ + 202, + 82, + 789, + 191 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Effect on other network architectures. We show that ReAct is effective on a different architecture in Table $\\mathbb { A }$ In particular, we consider a lightweight model MobileNet-v2 [44], which can be suitable for OOD detection in on-device mobile applications. Same as before, we apply ReAct on the output of the penultimate layer, with the rectification threshold chosen based on the 90-th percentile. Our method reduces the FPR95 by $9 . 1 8 \\%$ compared to the best baseline [31]. ", + "bbox": [ + 173, + 243, + 825, + 311 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "What about applying ReAct on other layers? Our results suggest that applying ReAct on the penultimate layer is the most effective, since the activation patterns are most distinctive. To see this, we provide the activation and performance study for intermediate layers in Appendix E (see Figure 6 and Table $^ { 6 ) }$ . Interestingly, early layers display less distinctive signatures between ID and OOD data. This is expected because neural networks generally capture lower-level features in early layers (such as Gabor filters $\\left[ \\left[ 6 1 \\right] \\right]$ in layer 1), whose activations can be very similar between ID and OOD. The semantic-level features only emerge as with deeper layers, where ReAct is the most effective. ", + "bbox": [ + 173, + 318, + 826, + 414 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 Evaluation on CIFAR Benchmarks ", + "text_level": 1, + "bbox": [ + 176, + 421, + 457, + 436 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Datasets. We evaluate on CIFAR-10 and CIFAR-100 [27] datasets as in-distribution data, using the standard split with 50,000 training images and 10,000 test images. For OOD data, we consider six common benchmark datasets: Textures [5], SVHN [37], Places365 [62], LSUN-Crop [60], LSUN-Resize $\\pmb { \\mathbb { H } }$ , and iSUN [57]. ", + "bbox": [ + 174, + 446, + 825, + 502 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Experimental details. We train a standard ResNet-18 [11] model on in-distribution data. The feature dimension of the penultimate layer is 512. For both CIFAR-10 and CIFAR-100, the models are trained for 100 epochs. The start learning rate is 0.1 and decays by a factor of 10 at epochs 50, 75, and 90. For threshold $c$ , we use the 90-th percentile of activations estimated on the ID data. ", + "bbox": [ + 174, + 507, + 825, + 563 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "ReAct is compatible with various OOD scoring functions. We show in Table $\\textcircled { 3 }$ that ReAct is a flexible method that is compatible with alternative scoring functions $S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } )$ . To see this, we consider commonly used scoring functions, and compare the performance both with and without using ReAct respectively. In particular, we consider softmax confidence $\\mathbb { \\lVert 1 5 \\rVert }$ , ODIN score $\\textcircled { \\scriptsize { 1 3 1 } }$ as well as energy score [33]— all of which derive OOD scores directly from the output $f ( \\mathbf { x } )$ . In particular, using ReAct on energy score yields the best performance, which is desirable as energy is a hyperparameter-free OOD score and is easy to compute in practice. Note that Mahalanobis $\\overline { { \\| 2 9 } }$ estimates OOD score using feature representations instead of the model output $f ( \\mathbf { x } )$ , hence is less compatible with ReAct. On all three in-distribution datasets, using ReAct consistently outperforms the counterpart without rectification. Results in Table $3$ are based on the average across multiple OOD test datasets. Detailed performance for each OOD test dataset is provided in Appendix, Table 7. ", + "bbox": [ + 173, + 569, + 825, + 718 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/d1041be9ec01148bf1be938f9e8934b94e7526c64bbb5b9c8a2487eb575421e8.jpg", + "table_caption": [], + "table_footnote": [ + "Table 3: Ablation results. ReAct is compatible with different OOD scoring functions. For each ID dataset, we use the same model and compare the performance with and without ReAct respectively. $\\uparrow$ indicates larger values are better and $\\downarrow$ indicates smaller values are better. All values are percentages and are averaged over multiple OOD test datasets. Detailed performance for each OOD test dataset is available in Table $^ { 7 }$ in Appendix. " + ], + "table_body": "
MethodCIFAR-10CIFAR-100ImageNet
FPR95 √AUROC →AUPR →FPR95 ←AUROC →AUPR 个FPR95 ←AUROC →AUPR →
Softmax score 园56.7191.1779.1180.7276.8378.4166.9581.9995.76
Softmax score + ReAct53.8191.7092.1175.4580.4084.2858.2887.0697.22
Energy B335.6093.5795.0171.9382.8286.2858.4186.1796.88
Energy+ReAct32.9194.2795.5359.6187.4889.63 31.4392.9598.50
ODIN B131.1093.7994.9566.2182.8886.2556.4885.4196.61
ODIN+ReAct28.8194.0494.8259.9185.2387.5344.1090.7098.04
", + "bbox": [ + 186, + 741, + 812, + 858 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/73ef4dbf0548b1c453a52b9baca2581dc1a02715e7389b0b1368b2ed7449f176.jpg", + "image_caption": [ + "Figure 2: Positively skewed distribution of $\\mu _ { i }$ (mean of each unit) in the penultimate layer for four OOD datasets (iNaturalist [16], SUN [55], Places $\\pmb { \\| 6 2 \\| }$ , Textures $\\pmb { \\| 5 \\| }$ ). Model is trained on ImageNet. Note that the right tail has a higher density than the left tail. By left and right tail we refer to the samples that are to the left and right of the median, which is close to the mode in the case of skewed distribution. " + ], + "image_footnote": [], + "bbox": [ + 207, + 88, + 784, + 194 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Theoretical Analysis ", + "text_level": 1, + "bbox": [ + 174, + 273, + 377, + 291 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To better understand the effect of ReAct, we mathematically model the ID and OOD activations as rectified Gaussian distributions and derive their respective distributions after applying ReAct. These modeling assumptions are based on the activation statistics observed on ImageNet in Figure $^ { 1 } .$ In the following analysis, we show that ReAct reduces mean OOD activations more than ID activations since OOD activations are more positively skewed (see Appendix A for derivations). ", + "bbox": [ + 173, + 304, + 825, + 375 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "ID activations. Let $h ( \\mathbf { x } ) = ( z _ { 1 } , \\ldots , z _ { m } ) = : \\mathbf { z }$ be the activations for the penultimate layer. We assume each $z _ { i } \\sim \\mathcal { N } ^ { R } ( \\mu , \\sigma _ { \\mathrm { i n } } ^ { 2 } )$ for some $\\sigma _ { \\mathrm { i n } } > 0$ . Here $\\mathcal N ^ { R } ( \\mu , \\sigma _ { \\mathrm { i n } } ^ { 2 } ) = \\operatorname* { m a x } ( 0 , \\mathcal N ( \\mu , \\sigma _ { \\mathrm { i n } } ^ { 2 } ) )$ denotes the rectified Gaussian distribution, which reflects the fact that activations after ReLU have no negative components. Before truncation with ReAct, the expectation of $z _ { i }$ is given by: ", + "bbox": [ + 174, + 387, + 825, + 445 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/455ae3a2e14ef7c9973312688aab34a1172f3bd0bc64e2f124f2d5bb39941888.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathrm { i n } } [ z _ { i } ] = \\left[ 1 - \\Phi \\left( \\frac { - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) \\right] \\cdot \\mu + \\phi \\left( \\frac { - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) \\cdot \\sigma _ { \\mathrm { i n } } ,\n$$", + "text_format": "latex", + "bbox": [ + 338, + 452, + 656, + 487 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\Phi$ and $\\phi$ denote the cdf and pdf of the standard normal distribution, respectively. After rectification with ReAct, the expectation of $\\bar { z } _ { i } = \\operatorname* { m i n } ( z _ { i } , c )$ is: ", + "bbox": [ + 171, + 492, + 823, + 522 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ba9ca98aedf1dd73a7c16db6ae5ffbc38e8e0d71d32586108cc8df641599b413.jpg", + "text": "$$\n\\mathcal { \\bar { c } } _ { \\mathrm { i n } } \\big [ \\bar { z } _ { \\mathrm { i } } \\big ] = \\left[ \\Phi \\left( \\frac { c - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) - \\Phi \\left( \\frac { - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) \\right] \\cdot \\mu + \\left[ 1 - \\Phi \\left( \\frac { c - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) \\right] \\cdot c + \\left[ \\phi \\left( \\frac { - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) - \\phi \\left( \\frac { c - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) \\right] \\cdot \\sigma _ { \\mathrm { i n } } ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 527, + 825, + 564 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The reduction in activation after ReAct is: ", + "bbox": [ + 174, + 569, + 450, + 583 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/a2fa543ead7e387c18ccc0cd2f8dacf1000201b2dce389fc18d4ff27ba967ce6.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathrm { i n } } [ z _ { i } - \\bar { z } _ { i } ] = \\phi \\left( \\frac { c - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) \\cdot \\sigma _ { \\mathrm { i n } } - \\left[ 1 - \\Phi \\left( \\frac { c - \\mu } { \\sigma _ { \\mathrm { i n } } } \\right) \\right] \\cdot ( c - \\mu )\n$$", + "text_format": "latex", + "bbox": [ + 290, + 587, + 707, + 622 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "OOD activations. We model OOD activations as being generated by a two-stage process: Each OOD distribution defines a set of $\\mu _ { i }$ ’s that represent the mode of the activation distribution for unit $i$ , and the activations $z _ { i }$ given $\\mu _ { i }$ is represented by $z _ { i } | \\mu _ { i } \\sim \\mathcal { N } ^ { R } ( \\mu _ { i } , \\tau ^ { 2 } )$ with $\\tau > 0$ . For instance, the dark gray line in Figure $^ 1$ shows the set $\\mu _ { i }$ ’s on the iNaturalist dataset, and the light gray area depicts the distribution of $z _ { i } | \\mu _ { i }$ . One commonality across different OOD datasets is that the distribution of $\\mu _ { i }$ is positively skewed. The assumption of positive skewness is motivated by our observation on real OOD data. Indeed, Figure $2$ shows the empirical distribution of $\\mu _ { i }$ on an ImageNet pre-trained model for four OOD datasets, all of which display strong positive-skewness, i.e., the right tail has a much higher density than the left tail. This observation is surprisingly consistent across datasets and model architectures. Although a more in-depth understanding of the fundamental cause of positive skewness is important, for this work, we chose to rely on this empirically verifiable assumption and instead focus on analyzing our method ReAct. ", + "bbox": [ + 173, + 635, + 826, + 797 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Utilizing the positive-skewness property of $\\mu _ { i }$ , we analyze the distribution of $z _ { i }$ after marginalizing out $\\mu _ { i }$ , which corresponds to averaging across different $\\mu _ { i }$ ’s induced by various OOD distributions. Let $\\dot { x } _ { i } | \\mu _ { i } \\sim \\mathcal { N } ( \\mu _ { i } , \\tau ^ { 2 } )$ so that $z _ { i } | \\mu _ { i } = \\operatorname* { m a x } ( x _ { i } | \\mu _ { i } , 0 )$ . Since $x _ { i } | \\mu _ { i }$ is symmetric and $\\mu _ { i }$ is positivelyskewed, the marginal distribution of $x _ { i }$ is also positively-skewed $\\bigstar$ which we model with the epsilonskew-normal (ESN) distribution $\\begin{array} { r l } { { \\bigl \\| \\overline { { 3 5 } } \\bigr \\| } } & { { } } \\end{array}$ . Specifically, we assume that $x _ { i } \\sim \\mathrm { E S N } ( \\mu , \\sigma _ { \\mathrm { o u t } } ^ { 2 } , \\epsilon )$ , which has ", + "bbox": [ + 173, + 804, + 825, + 875 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/bd8aa52dd90e0f12755c850a4b9ed932d05a82092ca47ec30f0e979709fce31b.jpg", + "image_caption": [ + "Figure 3: Plot showing the relationship between the skewness parameter $\\epsilon$ and the chaotic-ness parameter $\\sigma$ on activation reduction after applying ReAct. The function $\\mathbb { E } [ z _ { i } - \\bar { z } _ { i } ]$ is increasing in both $- \\epsilon$ and $\\sigma$ , which suggests that ReAct has a greater reduction effect for activation distributions with positive skewness $( \\epsilon < 0 )$ ) and chaoticness—two signature characteristics of OOD activation. " + ], + "image_footnote": [], + "bbox": [ + 200, + 130, + 570, + 342 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "the following density function: ", + "bbox": [ + 173, + 359, + 379, + 373 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/2adc9d14984eb7525ef0519801625bdba5cdbca948d5d8bd9d3b73a480bb576e.jpg", + "text": "$$\nq ( x ) = \\left\\{ \\begin{array} { l l } { \\phi ( ( x - \\mu ) / \\sigma _ { \\mathrm { o u t } } ( 1 + \\epsilon ) ) / \\sigma _ { \\mathrm { o u t } } } & { \\mathrm { i f ~ } x < \\mu , } \\\\ { \\phi ( ( x - \\mu ) / \\sigma _ { \\mathrm { o u t } } ( 1 - \\epsilon ) ) / \\sigma _ { \\mathrm { o u t } } } & { \\mathrm { i f ~ } x \\geq \\mu . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 331, + 376, + 665, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "with $\\epsilon \\in [ - 1 , 1 ]$ controlling the skewness. In particular, the ESN distribution is positively-skewed when $\\epsilon < 0$ . It follows that $z _ { i } = \\operatorname* { m a x } ( x _ { i } , 0 )$ , with expectation: ", + "bbox": [ + 171, + 412, + 823, + 440 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/22bcc29d0c859395f9d300bcbfa3ab28e9cc2d3394441570930b193f3c08c774.jpg", + "text": "$$\n\\mathbb { E } _ { \\mathrm { o u t } } [ z _ { i } ] = \\mu - ( 1 + \\epsilon ) \\Phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\mu + ( 1 + \\epsilon ) ^ { 2 } \\phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\sigma _ { \\mathrm { o u t } } - \\frac { 4 \\epsilon } { \\sqrt { 2 \\pi } } \\cdot \\sigma _ { \\mathrm { o u t } } .\n$$", + "text_format": "latex", + "bbox": [ + 184, + 443, + 790, + 477 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Expectation after applying ReAct becomes: ", + "bbox": [ + 176, + 478, + 462, + 492 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/70e8a59dc9c7a7fea0b398b204cf4f2edcf97911ff361f7388c9cfde2c32f277.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\mathrm { o u t } } [ \\bar { z } _ { i } ] = \\mu - ( 1 + \\epsilon ) \\Phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\mu + ( 1 - \\epsilon ) \\left[ 1 - \\Phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\right] \\cdot ( c - \\mu ) } \\\\ { + \\left[ ( 1 + \\epsilon ) ^ { 2 } \\phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) - ( 1 - \\epsilon ) ^ { 2 } \\phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) - \\frac { 4 \\epsilon } { \\sqrt { 2 \\pi } } \\right] \\cdot \\sigma _ { \\mathrm { o u t } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 209, + 492, + 787, + 565 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Hence: ", + "bbox": [ + 173, + 563, + 222, + 577 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "$\\mathbb { E } _ { \\mathrm { o u t } } [ z _ { i } - \\bar { z } _ { i } ] = ( 1 - \\epsilon ) ^ { 2 } \\phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\sigma _ { \\mathrm { o u t } } - ( 1 - \\epsilon ) \\left[ 1 - \\Phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\right] \\cdot ( c - \\mu ) ,$ which recovers Equation 4 when $\\epsilon = 0$ and $\\sigma _ { \\mathrm { o u t } } = \\sigma _ { \\mathrm { i n } }$ . ", + "bbox": [ + 174, + 575, + 790, + 627 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Remark 1: Activation reduction on OOD is more than ID. Figure 3 shows a plot of $\\mathbb { E } [ z _ { i } - \\bar { z } _ { i } ]$ for $\\mu = 0 . 5$ and $c = 1$ . Observe that decreasing $\\epsilon$ (more positive-skewness) or increasing $\\sigma$ (more chaotic-ness) leads to a larger reduction in the mean activation after applying ReAct. For example, under the same $\\sigma$ , a larger $\\mathbb { E } _ { \\mathrm { o u t } } [ z _ { i } - \\bar { z } _ { i } ] - \\mathbb { E } _ { \\mathrm { i n } } [ z _ { i } - \\bar { z } _ { i } ]$ can be observed by the gap of $z$ -axis value between $\\epsilon = 0$ and $\\epsilon < 0$ (e.g., $\\epsilon = - 0 . 4 )$ . This suggests that rectification on average affects OOD activations more severely compared to ID activations. ", + "bbox": [ + 173, + 637, + 826, + 722 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Remark 2: Output reduction on OOD is more than ID. To derive the effect on the distribution of model output, consider output logits $f ( \\mathbf { z } ) = W \\mathbf { z } + \\mathbf { b }$ and assume without loss of generality that $W \\mathbf { 1 } > 0$ element-wise. This can be achieved by adding a positive constant to $W$ without changing the output probabilities or classification decision. Let $\\begin{array} { r } { \\bar { \\boldsymbol \\delta } = \\mathbf { \\dot { \\mathbb { E } } _ { \\mathrm { o u t } } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] - \\mathbb { E } _ { \\mathrm { i n } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] > 0 . } \\end{array}$ Then: ", + "bbox": [ + 176, + 733, + 825, + 789 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/ec951ae179c2fdcf1f1758def8e1b9799c8f6b29c0cce6bfe9202e0591544dc9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\mathrm { o u t } } [ f ( \\mathbf { z } ) - f ( \\bar { \\mathbf { z } } ) ] = \\mathbb { E } _ { \\mathrm { o u t } } [ W ( \\mathbf { z } - \\bar { \\mathbf { z } } ) ] = W \\mathbb { E } _ { \\mathrm { o u t } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] = W \\left( \\mathbb { E } _ { \\mathrm { i n } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] + \\delta \\mathbf { 1 } \\right) } \\\\ & { \\qquad = \\mathbb { E } _ { \\mathrm { i n } } [ W ( \\mathbf { z } - \\bar { \\mathbf { z } } ) ] + \\delta W \\mathbf { 1 } } \\\\ & { \\qquad > \\mathbb { E } _ { \\mathrm { i n } } [ f ( \\mathbf { z } ) - f ( \\bar { \\mathbf { z } } ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 789, + 756, + 843 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Hence the increased separation between OOD and $\\mathrm { I D }$ activations transfers to the output space as well. Note that the condition of $W \\mathbf { 1 } > 0$ is sufficient but not necessary for this result to hold. In fact, our experiments in Section $\\textcircled { 4 } \\mathrm { d o }$ not require this condition. However, we verified empirically that ensuring $W \\mathbf { 1 } > 0$ by adding a positive constant to $W$ and applying ReAct does confer benefits to OOD detection, which validates our theoretical analysis. ", + "bbox": [ + 173, + 843, + 826, + 911 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Why ReAct improves the OOD scoring functions? Our theoretical analysis above shows that ReAct suppresses logit output for OOD data more so than for ID data. This means that for detection scores depending on the logit output (e.g., energy score $\\mathbb { \\lVert 3 3 \\rVert }$ ), the gap between OOD and ID score will be enlarged after applying ReAct, which makes thresholding more capable of separating OOD and ID inputs; see Figure 1(a) and (c) for a concrete example showing this effect. ", + "bbox": [ + 174, + 90, + 825, + 160 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 Discussion and Further Analysis ", + "text_level": 1, + "bbox": [ + 173, + 181, + 478, + 199 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Why do OOD samples trigger abnormal unit activation patterns? So far we have shown that OOD data can trigger unit activation patterns that are significantly different from ID data, and that ReAct can effectively alleviate this issue (empirically in Section $^ 4$ and theoretically in Section 5). Yet a question left in mystery is why such a pattern occurs in modern neural networks? Answering this question requires carefully examining the internal mechanism by which the network is trained and evaluated. Here we provide one plausible explanation for the activation patterns observed in Figure 1, with the hope of shedding light for future research. ", + "bbox": [ + 173, + 214, + 825, + 310 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Intriguingly, our analysis reveals an important insight that batch normalization (BatchNorm) $[ [ 2 2 ] ] { - \\mathsf { a } }$ common technique employed during model training—is in fact both a blessing (for ID classification) and a curse (for OOD detection). Specifically, for a unit activation denoted by $z$ , the network estimates the running mean $\\mathbb { E } _ { \\mathrm { i n } } ( z )$ and variance $\\mathrm { V a r } _ { \\mathrm { i n } } ( z )$ , over the entire ID training set during training. During inference time, the network applies BatchNorm statistics $\\mathbb { E } _ { \\mathrm { i n } } ( z )$ and $\\mathrm { V a r } _ { \\mathrm { i n } } ( z )$ , which helps normalize the activations for the test data with the same distribution $\\mathcal { D } _ { \\mathrm { i n } }$ : ", + "bbox": [ + 173, + 315, + 825, + 397 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/b0099b7328cf980e9ce000affce9b7dce0fc72bba5cebd90e736f2bb13760ade.jpg", + "text": "$$\n\\mathrm { B a t c h N o r m } ( z ; \\gamma , \\beta , \\epsilon ) = \\frac { z - \\mathbb { E } _ { \\mathrm { i n } } [ z ] } { \\sqrt { \\mathrm { V a r } _ { \\mathrm { i n } } [ z ] + \\epsilon } } \\cdot \\gamma + \\beta\n$$", + "text_format": "latex", + "bbox": [ + 338, + 406, + 658, + 443 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "However, our key observation is that using mismatched BatchNorm statistics—that are estimated on $\\mathcal { D } _ { \\mathrm { i n } }$ yet blindly applied to the OOD $\\mathcal { D } _ { \\mathrm { o u t } }$ —can trigger abnormally high unit activations. As a thought experiment, we instead apply the true BatchNorm statistics estimated on a batch of OOD images and we observe well-behaved activation patterns with near-constant mean and standard deviations—just like the ones observed on the ID data (see Figure $\\mathbb { E } ,$ top). Our study therefore reveals one of the fundamental causes for neural networks to produce overconfident predictions for OOD data. After applying the true statistics (estimated on OOD), the output distributions between ID and OOD data become much more separable. While this thought experiment has shed some guiding light, the solution of estimating BatchNorm statistics on a batch of OOD data is not at all satisfactory and realistic. Arguably, it poses a strong and impractical assumption of having access to a batch of OOD data during test time. Despite its limitation, we view it as an oracle, which serves as an upper bound on performance for ReAct. ", + "bbox": [ + 173, + 450, + 825, + 613 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/ce8b1cf28903076ecbaa2d2a5e4c5ec6c6bec4e39ce57f4b41bcff3d3ce6683a.jpg", + "image_caption": [ + "Figure 4: The distribution of per-unit activations in the penultimate layer for OOD data (iNaturalist) by using true (top) vs. mismatched (bottom) BatchNorm statistics for OOD data. " + ], + "image_footnote": [], + "bbox": [ + 210, + 622, + 790, + 768 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/00b41d3529c5a9fb6db8ae55248b7fb35bea6ee1816256212a2ef4bc9dbe7783.jpg", + "table_caption": [], + "table_footnote": [ + "Table 4: Comparison with oracle using OOD BN statistics. Model is trained on ImageNet (see Section 4.1) Values are AUROC. " + ], + "table_body": "
MethodiNaturalistPlacesSUNTextures
Oracle (batch OOD for estimating BN statistics)99.5999.0998.3291.43
ReAct (single OOD)96.2294.2091.5889.80
No ReAct B389.9585.8982.8685.99
", + "bbox": [ + 215, + 827, + 784, + 885 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In particular, results in Table $\\boxed { \\sharp }$ suggest that our method favorably matches the oracle performance using the ground truth BN statistics. This is encouraging as our method does not impose any batch assumption and can be feasible for single-input testing scenarios. ", + "bbox": [ + 173, + 90, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "What about networks trained with different normalization mechanisms? Going beyond batch normalization $\\pmb { \\left. 2 2 \\right. }$ , we further investigate (1) whether networks trained with alternative normalization approaches exhibit similar activation patterns, and (2) whether ReAct is helpful there. To answer this question, we additionally evaluate networks trained with WeightNorm $\\underline { { \\hat { \\vert 4 3 \\vert } } }$ and GroupNorm [54]— two other well-known normalization methods. As shown in Figure $\\triangledown$ the unit activations also display highly distinctive signature patterns between ID and OOD data, with more chaos on OOD data. In all cases, the networks are trained to adapt to the ID data, resulting in abnormal activation signatures on OOD data in testing. Unlike BatchNorm, there is no easy oracle (e.g., re-estimating the statistics on OOD data) to counteract the ill-fated normalizations. ", + "bbox": [ + 173, + 147, + 826, + 268 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We apply ReAct on models trained with WeightNorm and GroupNorm, and report results in Table 5. Our results suggest that ReAct is consistently effective under various normalization schemes. For example, ReAct reduces the average FPR95 by $2 3 . 5 4 \\%$ and $1 4 . 7 \\%$ respectively. Overall, ReAct has shown broad efficacy and compatibility with different OOD scoring functions (Section $4 . 2 )$ ", + "bbox": [ + 174, + 275, + 825, + 330 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/5317d04dc38b221b3468d7e8c83dadf02fc27f7e464b69af03aba71d821889fc.jpg", + "image_caption": [ + "Model: WeightNormFigure 5: The distribution of per-unit activations in the penultimate layer for ID (ImageNet $\\pmb { \\mathbb { Z } } \\mathbf { I }$ ) and OOD (iNaturalist $\\pm \\boxed { 1 6 }$ ) on model trained with weight normalization [43]. ", + "Table 5: Effectiveness of ReAct for different normalization methods. ReAct consistently improves OOD detection performance for the model trained with GroupNorm and WeightNorm. In-distribution is ImageNet-1k dataset. Bold numbers are superior results. " + ], + "image_footnote": [], + "bbox": [ + 204, + 343, + 794, + 412 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/1e6fbd06627a8775795b484d3ce3fb048d607fe780f7a0d98fd97f7252cb4837.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95 ↓AUROC 个FPR95 ↓AUROC 个FPR95AUROCFPR95AUROC 个FPR95 ↓AUROC 个
GroupNormw.o.ReAct65.3888.4565.1185.52↓ 65.46个 84.34↓ 69.1783.2266.2885.38
w/ReAct39.4592.9551.5787.9052.7887.3262.5081.7651.5887.48
WeightNormw.o. ReAct40.7192.5248.0789.3950.9287.8761.6580.7150.3487.62
w/ReAct19.7395.9131.3993.2142.3488.9413.7496.9826.8093.76
", + "bbox": [ + 174, + 462, + 825, + 564 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 628, + 323, + 645 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "OOD uncertainty estimation with discriminative models. The problem of classification with rejection can date back to early works on abstention [4, 10], which considered simple model families such as SVMs $\\pmb { \\Vert 6 \\Vert }$ . A comprehensive survey on OOD detection can be found in $\\left[ \\left[ 5 8 \\right] \\right]$ . The phenomenon of neural networks’ overconfidence in out-of-distribution data is first revealed by Nguyen et al. [38]. Early works attempted to improve the OOD uncertainty estimation by proposing the ODIN score [17, 31], OpenMax score $\\pmb { \\mathbb { B } } \\|$ , and Mahalanobis distance-based score $\\pmb { \\mathbb { Z } } \\pmb { \\mathcal { 0 } }$ . Recent work by Liu et al. [33] proposed using an energy score for OOD uncertainty estimation, which can be easily derived from a discriminative classifier and demonstrated advantages over the softmax confidence score both empirically and theoretically. Wang et al. [51] further showed an energy-based approach can improve OOD uncertainty estimation for multi-label classification networks. Huang and Li [18] revealed that approaches developed for common CIFAR benchmarks might not translate effectively into a large-scale ImageNet benchmark, highlighting the need to evaluate OOD uncertainty estimation in a large-scale real-world setting. Previous methods primarily derive OOD scores using original activations. In contrast, our work is motivated by a novel analysis of internal activations, and shows that rectifying activations is a surprisingly effective approach for OOD detection. ", + "bbox": [ + 173, + 651, + 825, + 856 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Neural network activation analysis. Neural networks have been studied at the granularity of the activation of individual layers [21, 34, 59, 63], or individual networks $\\textcircled { \\lvert 3 0 \\rvert }$ . In particular, Li et al. [30] studied the similarity of activation space between two independently trained neural networks. ", + "bbox": [ + 174, + 871, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Previously, $\\underline { { \\mathrm { f l e i n ~ e t ~ a l . } } } \\underline { { \\mathbb { ( } \\mathbb { 1 3 } } } \\|$ showed that neural networks with ReLU activation can lead to arbitrary high activation for inputs far away from the training data. We show that using ReAct could efficiently alleviate this undesirable phenomenon. ReAct does not rely on auxiliary data and can be conveniently used for pre-trained models. The idea of rectifying unit activation $\\dot { \\mathbb { Z } } \\dot { 6 } \\mathbb { I }$ , which is known as Relu6, was used to facilitate the learning of sparse features. In this paper, we first show that rectifying activation can drastically alleviate the overconfidence issue for OOD data, and as a result, improve OOD detection. ", + "bbox": [ + 174, + 90, + 825, + 185 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "OOD uncertainty estimation with generative models. Alternative approaches for detecting OOD inputs resort to generative models that directly estimate density [8, 20, 23, 41, 49, 50]. An input is deemed as OOD if it lies in the low-likelihood regions. A plethora of literature has emerged to utilize generative models for OOD detection [24, 40, 45, 46, 52, 56]. Interestingly, Nalisnick et al. [36] showed that deep generative models can assign a high likelihood to OOD data. Moreover, generative models can be prohibitively challenging to train and optimize, and the performance can often lag behind the discriminative counterpart. In contrast, our method relies on a discriminative classifier, which is easier to optimize and achieves stronger performance. ", + "bbox": [ + 173, + 200, + 825, + 310 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Distributional shifts. Distributional shifts have attracted increasing research interests $\\mathbb { \\left[ \\left[ 2 5 \\right] \\right] }$ . It is important to recognize and differentiate various types of distributional shift problems. Literature in OOD detection is commonly concerned about model reliability and detection of label-space shifts, where the OOD inputs have disjoint labels w.r.t. ID data and therefore should not be predicted by the model. Meanwhile, some works considered label distribution shift [1, 2, 32, 42, 47, 53], where the label space is common between ID and OOD but the marginal label distribution changes, as well as covariate shift in the input space [14, 39], where inputs can be corruption-shifted or domainshifted $ { \\mathbb { I } } { \\mathrm { I Z } } , { \\mathbb { H } } { \\mathrm { 8 l } }$ . It is important to note that our work focuses on the detection of shifts where the label space $\\mathcal { V }$ is different between ID and OOD data and hence the model should not make any prediction, instead of label distribution shift and covariate shift where the model is expected to generalize. ", + "bbox": [ + 174, + 324, + 825, + 460 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 479, + 299, + 496 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This paper provides a simple activation rectification strategy termed ReAct, which truncates the high activations during test time for OOD detection. We provide both empirical and theoretical insights characterizing and explaining the mechanism by which ReAct improves OOD uncertainty estimation. By rectifying the activations, the outsized contribution of hidden units on OOD output can be attenuated, resulting in a stronger separability from ID data. Extensive experiments show ReAct can significantly improve the performance of OOD detection on both common benchmarks and large-scale image classification models. Applications of multi-class classification can benefit from our method, and we anticipate further research in OOD detection to extend this work to tasks beyond image classification. We hope that our insights inspire future research to further examine the internal mechanisms of neural networks for OOD detection. ", + "bbox": [ + 174, + 511, + 825, + 646 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "9 Societal Impact ", + "text_level": 1, + "bbox": [ + 174, + 665, + 334, + 683 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our project aims to improve the dependability and trustworthiness of modern machine learning models. This stands to benefit a wide range of fields and societal activities. We believe out-ofdistribution uncertainty estimation is an increasingly critical component of systems that range from consumer and business applications (e.g., digital content understanding) to transportation (e.g., driver assistance systems and autonomous vehicles), and to health care (e.g., unseen disease identification). Many of these applications require classification models in operation. Through this work and by releasing our code, we hope to provide machine learning researchers with a new methodological perspective and offer machine learning practitioners a plug-and-play tool that renders safety against OOD data in the real world. While we do not anticipate any negative consequences to our work, we hope to continue to build on our method in future work. ", + "bbox": [ + 174, + 695, + 825, + 832 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgement ", + "text_level": 1, + "bbox": [ + 176, + 852, + 330, + 868 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "YS and YL are supported by the Office of the Vice Chancellor for Research and Graduate Education (OVCRGE) with funding from the Wisconsin Alumni Research Foundation (WARF). ", + "bbox": [ + 174, + 882, + 821, + 910 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References \n[1] Amr Alexandari, Anshul Kundaje, and Avanti Shrikumar. Maximum likelihood with biascorrected calibration is hard-to-beat at label shift adaptation. In International Conference on Machine Learning, pages 222–232. PMLR, 2020. \n[2] Kamyar Azizzadenesheli, Anqi Liu, Fanny Yang, and Animashree Anandkumar. Regularized learning for domain adaptation under label shifts. In International Conference on Learning Representations, 2019. \n[3] Abhijit Bendale and Terrance E Boult. 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On", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 460, + 469, + 472 + ], + "spans": [ + { + "bbox": [ + 142, + 460, + 437, + 472 + ], + "score": 1.0, + "content": "the ImageNet benchmark, ReAct reduces the false positive rate (FPR95) by", + "type": "text" + }, + { + "bbox": [ + 438, + 460, + 469, + 470 + ], + "score": 0.86, + "content": "2 5 . 0 5 \\%", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 469, + 303, + 484 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 303, + 484 + ], + "score": 1.0, + "content": "compared to the previous best method1.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20.5, + "bbox_fs": [ + 141, + 331, + 471, + 484 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 190, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 192, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 192, + 516 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 499, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 499, + 536 + ], + "score": 1.0, + "content": "Neural networks deployed in real-world systems often encounter out-of-distribution (OOD) inputs—", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "score": 1.0, + "content": "unknown samples that the network has not been exposed to during training. Identifying and handling", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 545, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 504, + 558 + ], + "score": 1.0, + "content": "these OOD inputs can be paramount in safety-critical applications such as autonomous driving [9]", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "score": 1.0, + "content": "and health care. For example, an autonomous vehicle may fail to recognize objects on the road that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "do not appear in its object detection model’s training set, potentially leading to a crash. This can be", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "prevented if the system identifies the unrecognized object as OOD and warns the driver in advance.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 524, + 505, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 593, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "A driving idea behind OOD detection is that the model should be much more uncertain about samples", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 281, + 617 + ], + "score": 1.0, + "content": "outside of its training distribution. However,", + "type": "text" + }, + { + "bbox": [ + 281, + 604, + 354, + 616 + ], + "score": 0.29, + "content": "\\mathrm { \\Delta N g u y e n \\ e t { a l . } } \\rVert { 3 8 } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "revealed that modern neural networks", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "can produce overconfident predictions on OOD inputs. This phenomenon renders the separation", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 637 + ], + "score": 1.0, + "content": "of in-distribution (ID) and OOD data a non-trivial task. Indeed, much of the prior work on OOD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 636, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 457, + 649 + ], + "score": 1.0, + "content": "detection focused on defining more suitable measures of OOD uncertainty [17, 19, 28,", + "type": "text" + }, + { + "bbox": [ + 458, + 637, + 504, + 648 + ], + "score": 1.0, + "content": "29, 31, 33].", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 647, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 506, + 659 + ], + "score": 1.0, + "content": "Despite the improvement, it is arguable that continued research progress in OOD detection requires", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 658, + 469, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 469, + 670 + ], + "score": 1.0, + "content": "insights into the fundamental cause and mitigation of model overconfidence on OOD data.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 593, + 506, + 670 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 675, + 503, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 674, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 505, + 685 + ], + "score": 1.0, + "content": "In this paper, we start by revealing an important observation that OOD data can trigger unit activation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "patterns that are significantly different from ID data. Figure 1(b) shows the distribution of activations", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "in the penultimate layer of ResNet-50 trained on ImageNet [7]. Each point on the horizontal axis", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "corresponds to a single unit. The mean and standard deviation are shown by the solid line and shaded", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "area, respectively. The mean activation for ID data (blue) is well-behaved with a near-constant mean", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "and standard deviation. In contrast, for OOD data (gray), the mean activation has significantly larger", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "variations across units and is biased towards having sharp positive values (i.e., positively skewed). As", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "a result, such high unit activation can undesirably manifest in model output, producing overconfident", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 337, + 495, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 495, + 349 + ], + "score": 1.0, + "content": "predictions on OOD data. 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Applying ReAct drastically improves the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 227, + 295, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 295, + 239 + ], + "score": 1.0, + "content": "separation of ID and OOD data. See text for details.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "in the penultimate layer of ResNet-50 trained on ImageNet [7]. Each point on the horizontal axis", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "corresponds to a single unit. 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As", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "a result, such high unit activation can undesirably manifest in model output, producing overconfident", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 337, + 495, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 495, + 349 + ], + "score": 1.0, + "content": "predictions on OOD data. A similar distributional property holds for other OOD datasets as well.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "The above observation naturally inspires a simple yet surprisingly effective method—Rectified", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "Activations (dubbed ReAct) for OOD detection. In particular, the outsized activation of a few selected", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 418, + 387 + ], + "score": 1.0, + "content": "hidden units can be attenuated by rectifying the activations at an upper limit", + "type": "text" + }, + { + "bbox": [ + 419, + 375, + 443, + 385 + ], + "score": 0.89, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 374, + 506, + 387 + ], + "score": 1.0, + "content": ". Conveniently,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "this can be done on a pre-trained model without any modification to training. The dashed horizontal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 272, + 408 + ], + "score": 1.0, + "content": "line in Figure 1(b) shows the cutoff point", + "type": "text" + }, + { + "bbox": [ + 272, + 398, + 278, + 406 + ], + "score": 0.56, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 396, + 506, + 408 + ], + "score": 1.0, + "content": ", and its effect on the OOD uncertainty score is shown in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "Figure 1(c). After rectification, the output distributions for ID and OOD data become much more", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 425, + 429 + ], + "score": 1.0, + "content": "well-separated and the false positive rate (FPR) is significantly reduced from", + "type": "text" + }, + { + "bbox": [ + 426, + 417, + 458, + 428 + ], + "score": 0.89, + "content": "5 5 . 7 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 417, + 471, + 429 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 471, + 417, + 503, + 428 + ], + "score": 0.9, + "content": "2 0 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 417, + 506, + 429 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "Importantly, this truncation largely preserves the activation for in-distribution data, and therefore", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 438, + 416, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 416, + 451 + ], + "score": 1.0, + "content": "ensures the classification accuracy on the original task is largely comparable.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "We provide both empirical and theoretical insights, characterizing and explaining the mechanism", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 466, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "by which ReAct improves OOD detection. We perform extensive evaluations and establish state-of-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "the-art performance on a suite of common OOD detection benchmarks, including CIFAR-10 and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 333, + 499 + ], + "score": 1.0, + "content": "CIFAR-100, as well as a large-scale ImageNet dataset", + "type": "text" + }, + { + "bbox": [ + 333, + 487, + 346, + 498 + ], + "score": 0.68, + "content": "\\mathbb { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 487, + 506, + 499 + ], + "score": 1.0, + "content": ". ReAct outperforms the best baseline", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 338, + 510 + ], + "score": 1.0, + "content": "by a large margin, reducing the average FPR95 by up to", + "type": "text" + }, + { + "bbox": [ + 338, + 498, + 370, + 508 + ], + "score": 0.87, + "content": "2 5 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 498, + 506, + 510 + ], + "score": 1.0, + "content": ". 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During test time, ReAct can be leveraged by a variety of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 350, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 296, + 601 + ], + "score": 1.0, + "content": "downstream OOD scoring functions relying on", + "type": "text" + }, + { + "bbox": [ + 297, + 587, + 344, + 600 + ], + "score": 0.93, + "content": "f ^ { \\widetilde { \\mathrm { R e A c t } } } ( \\mathbf { x } ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 585, + 350, + 601 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 605, + 393, + 633 + ], + "lines": [ + { + "bbox": [ + 218, + 605, + 393, + 633 + ], + "spans": [ + { + "bbox": [ + 218, + 605, + 393, + 633 + ], + "score": 0.94, + "content": "G _ { \\lambda } ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) = \\left\\{ { \\begin{array} { l l } { \\mathrm { i n } } & { S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) \\geq \\lambda } \\\\ { \\mathrm { o u t } } & { S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) < \\lambda } \\end{array} } , \\right.", + "type": "interline_equation", + "image_path": "d16773bbd6a0fba09297450364b444e408810814f7fe9c5988863e1d150c9616.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 605, + 393, + 619.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 218, + 619.0, + 393, + 633.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 506, + 726 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 507, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 384, + 652 + ], + "score": 1.0, + "content": "where a thresholding mechanism is exercised to distinguish between", + "type": "text" + }, + { + "bbox": [ + 384, + 640, + 396, + 650 + ], + "score": 0.32, + "content": "\\mathrm { I D }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 639, + 507, + 652 + ], + "score": 1.0, + "content": "and OOD during test time.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 334, + 663 + ], + "score": 1.0, + "content": "To align with the convention, samples with higher scores", + "type": "text" + }, + { + "bbox": [ + 334, + 650, + 366, + 662 + ], + "score": 0.94, + "content": "S ( \\mathbf { x } ; 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In the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 507, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 173, + 237 + ], + "score": 1.0, + "content": "extreme case, if", + "type": "text" + }, + { + "bbox": [ + 174, + 223, + 223, + 234 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { i n } } = \\mathcal { D } _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 222, + 507, + 237 + ], + "score": 1.0, + "content": "then the optimal decision function is a random coin flip. 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Our key idea is to perform post hoc modification to the unit", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 326, + 507, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 507, + 342 + ], + "score": 1.0, + "content": "activation, so to bring the overall activation pattern closer to the well-behaved case. Specifically,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 347, + 352 + ], + "score": 1.0, + "content": "we consider a pre-trained neural network parameterized by", + "type": "text" + }, + { + "bbox": [ + 348, + 339, + 353, + 349 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 338, + 455, + 352 + ], + "score": 1.0, + "content": ", which encodes an input", + "type": "text" + }, + { + "bbox": [ + 455, + 338, + 487, + 349 + ], + "score": 0.91, + "content": "\\mathbf { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "to a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 226, + 362 + ], + "score": 1.0, + "content": "feature space with dimension", + "type": "text" + }, + { + "bbox": [ + 226, + 351, + 236, + 360 + ], + "score": 0.7, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 348, + 297, + 362 + ], + "score": 1.0, + "content": ". We denote by", + "type": "text" + }, + { + "bbox": [ + 298, + 349, + 345, + 361 + ], + "score": 0.9, + "content": "h ( \\mathbf { x } ) \\in \\mathbb { R } ^ { \\bar { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "the feature vector from the penultimate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 358, + 507, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 265, + 376 + ], + "score": 1.0, + "content": "layer of the network. A weight matrix", + "type": "text" + }, + { + "bbox": [ + 265, + 361, + 319, + 371 + ], + "score": 0.91, + "content": "\\mathbf { W } \\in \\mathbb { R } ^ { m \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 358, + 405, + 376 + ], + "score": 1.0, + "content": "connects the feature", + "type": "text" + }, + { + "bbox": [ + 405, + 361, + 425, + 373 + ], + "score": 0.92, + "content": "h ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 358, + 482, + 376 + ], + "score": 1.0, + "content": "to the output", + "type": "text" + }, + { + "bbox": [ + 482, + 361, + 502, + 374 + ], + "score": 0.91, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 358, + 507, + 376 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 349, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 133, + 385 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 372, + 144, + 382 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 371, + 272, + 385 + ], + "score": 1.0, + "content": "is the total number of classes in", + "type": "text" + }, + { + "bbox": [ + 273, + 372, + 345, + 384 + ], + "score": 0.92, + "content": "\\mathcal { V } = \\{ 1 , 2 , . . . , K \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 371, + 349, + 385 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 306, + 507, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 388, + 504, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "score": 1.0, + "content": "ReAct: Rectified Activation. 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The model output after rectified", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 457, + 198, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 198, + 469 + ], + "score": 1.0, + "content": "activation is given by:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 434, + 505, + 469 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 473, + 366, + 488 + ], + "lines": [ + { + "bbox": [ + 244, + 473, + 366, + 488 + ], + "spans": [ + { + "bbox": [ + 244, + 473, + 366, + 488 + ], + "score": 0.92, + "content": "f ^ { \\mathrm { R e A c t } } ( \\mathbf { x } ; \\boldsymbol { \\theta } ) = \\mathbf { W } ^ { \\top } \\bar { h } ( \\mathbf { x } ) + \\mathbf { b } ,", + "type": "interline_equation", + "image_path": "0510be9e39488ca2db4c7e360cf9d165b95e022c458e8ef4bd15524883d0386c.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 244, + 473, + 366, + 488 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 133, + 509 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 495, + 169, + 506 + ], + "score": 0.91, + "content": "\\mathbf { b } \\in \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 494, + 286, + 509 + ], + "score": 1.0, + "content": "is the bias vector. 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During test time, ReAct can be leveraged by a variety of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 350, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 296, + 601 + ], + "score": 1.0, + "content": "downstream OOD scoring functions relying on", + "type": "text" + }, + { + "bbox": [ + 297, + 587, + 344, + 600 + ], + "score": 0.93, + "content": "f ^ { \\widetilde { \\mathrm { R e A c t } } } ( \\mathbf { x } ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 585, + 350, + 601 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 577, + 505, + 601 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 605, + 393, + 633 + ], + "lines": [ + { + "bbox": [ + 218, + 605, + 393, + 633 + ], + "spans": [ + { + "bbox": [ + 218, + 605, + 393, + 633 + ], + "score": 0.94, + "content": "G _ { \\lambda } ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) = \\left\\{ { \\begin{array} { l l } { \\mathrm { i n } } & { S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) \\geq \\lambda } \\\\ { \\mathrm { o u t } } & { S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } ) < \\lambda } \\end{array} } , \\right.", + "type": "interline_equation", + "image_path": "d16773bbd6a0fba09297450364b444e408810814f7fe9c5988863e1d150c9616.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 605, + 393, + 619.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 218, + 619.0, + 393, + 633.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 506, + 726 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 507, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 384, + 652 + ], + "score": 1.0, + "content": "where a thresholding mechanism is exercised to distinguish between", + "type": "text" + }, + { + "bbox": [ + 384, + 640, + 396, + 650 + ], + "score": 0.32, + "content": "\\mathrm { I D }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 639, + 507, + 652 + ], + "score": 1.0, + "content": "and OOD during test time.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 334, + 663 + ], + "score": 1.0, + "content": "To align with the convention, samples with higher scores", + "type": "text" + }, + { + "bbox": [ + 334, + 650, + 366, + 662 + ], + "score": 0.94, + "content": "S ( \\mathbf { x } ; 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ModelMethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95AUROCFPR95AUROC 个FPR95AUROCFPR95AUROCFPR95 AUROC
ResNetMSP 园↓ 54.99→ 87.74√ 70.8380.86↓ 73.99个 79.76↓ 68.00↑ 79.61√ 66.95→ 81.99
ODIN 白47.6689.6660.1584.5967.8981.7850.2385.6256.4885.41
Mahalanobis [ 2997.0052.6598.5042.4198.4041.7955.8085.0187.4355.47
Energy 国55.7289.9559.2685.8964.9282.8653.7285.9958.4186.17
ReAct (Ours)20.3896.2224.2094.2033.8591.5847.3089.8031.4392.95
77.0279.2376.2777.3073.5179.00
MobileNetMSP 固64.2985.32 87.6254.0777.10 85.8873.5185.0354.2085.81
ODIN 白 四55.39 62.1181.0047.8286.3357.36 52.0984.71 83.6349.96 92.3833.0663.6071.01
Mahalanobis59.5088.9162.6584.5069.3781.1958.0585.0362.3984.91
Energy 国 ReAct (Ours)42.4091.5347.6988.1651.5686.6438.4291.5345.0289.47
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All", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "score": 1.0, + "content": "methods are based on a model trained on ID data only (ImageNet-1k), without using any auxiliary outlier data.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 212, + 478, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 113, + 223 + ], + "score": 0.41, + "content": "\\uparrow", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 212, + 246, + 225 + ], + "score": 1.0, + "content": "indicates larger values are better and", + "type": "text" + }, + { + "bbox": [ + 246, + 213, + 253, + 223 + ], + "score": 0.37, + "content": "\\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 212, + 478, + 225 + ], + "score": 1.0, + "content": "indicates smaller values are better. All values are percentages.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 248, + 191, + 262 + ], + "lines": [ + { + "bbox": [ + 104, + 246, + 193, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 246, + 193, + 264 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 506, + 310 + ], + "lines": [ + { + "bbox": [ + 104, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "In this section, we evaluate ReAct on a suite of OOD detection tasks. We first evaluate a on large-scale", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 353, + 300 + ], + "score": 1.0, + "content": "OOD detection benchmark based on ImageNet [18] ( Section", + "type": "text" + }, + { + "bbox": [ + 354, + 287, + 372, + 300 + ], + "score": 0.37, + "content": "\\boxed { 4 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 286, + 505, + 300 + ], + "score": 1.0, + "content": ", and then proceed in Section 4.2", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 298, + 228, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 208, + 311 + ], + "score": 1.0, + "content": "with CIFAR benchmarks", + "type": "text" + }, + { + "bbox": [ + 208, + 298, + 226, + 310 + ], + "score": 0.53, + "content": " { \\mathbb { \\left[ \\left[ 2 7 \\right] \\right] } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 298, + 228, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 329, + 388, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 389, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 389, + 343 + ], + "score": 1.0, + "content": "4.1 Evaluation on Large-scale ImageNet Classification Networks", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "We first evaluate ReAct on a large-scale OOD detection benchmark developed in [18]. Compared", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "to the CIFAR benchmarks that are routinely used in literature, the ImageNet benchmark is more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 263, + 387 + ], + "score": 1.0, + "content": "challenging due to a larger label space", + "type": "text" + }, + { + "bbox": [ + 263, + 374, + 314, + 385 + ], + "score": 0.83, + "content": "( K = 1 , 0 0 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 372, + 506, + 387 + ], + "score": 1.0, + "content": ". Moreover, such large-scale evaluation is more", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "relevant to real-world applications, where the deployed models often operate on images that have", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 395, + 389, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 389, + 408 + ], + "score": 1.0, + "content": "high resolution and contain more classes than the CIFAR benchmarks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 299, + 424 + ], + "score": 1.0, + "content": "Setup. We use a pre-trained ResNet-50 model", + "type": "text" + }, + { + "bbox": [ + 299, + 411, + 316, + 422 + ], + "score": 0.77, + "content": "[ \\mathbb { 1 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "for ImageNet-1k. At test time, all images are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 149, + 434 + ], + "score": 1.0, + "content": "resized to", + "type": "text" + }, + { + "bbox": [ + 149, + 422, + 194, + 432 + ], + "score": 0.89, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 421, + 484, + 434 + ], + "score": 1.0, + "content": ". We evaluate on four test OOD datasets from (subsets of) Places365", + "type": "text" + }, + { + "bbox": [ + 485, + 421, + 502, + 433 + ], + "score": 0.31, + "content": "\\lVert \\rVert \\mathbf { 6 } 2 \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 421, + 506, + 434 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 228, + 445 + ], + "score": 1.0, + "content": "Textures [5], iNaturalist", + "type": "text" + }, + { + "bbox": [ + 229, + 432, + 246, + 444 + ], + "score": 0.67, + "content": "\\mathbb { I } \\mathbb { 1 6 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 432, + 287, + 445 + ], + "score": 1.0, + "content": ", and SUN", + "type": "text" + }, + { + "bbox": [ + 287, + 432, + 305, + 444 + ], + "score": 0.81, + "content": "\\mathbb { \\left[ \\left. 5 5 \\right] \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "with non-overlapping categories w.r.t ImageNet.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "Our evaluations span a diverse range of domains including fine-grained images, scene images, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 233, + 467 + ], + "score": 1.0, + "content": "textural images (see Appendix", + "type": "text" + }, + { + "bbox": [ + 234, + 453, + 245, + 466 + ], + "score": 0.82, + "content": "\\boxed { \\mathbf { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "for details). We use a validation set of Gaussian noise images,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 260, + 478 + ], + "score": 1.0, + "content": "which are generated by sampling from", + "type": "text" + }, + { + "bbox": [ + 260, + 465, + 293, + 477 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "for each pixel location. To ensure validity, we further", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "verify the activation pattern under Gaussian noise, which exhibits a similar distributional trend", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 315, + 499 + ], + "score": 1.0, + "content": "with positive skewness and chaoticness; see Figure", + "type": "text" + }, + { + "bbox": [ + 315, + 486, + 324, + 498 + ], + "score": 0.73, + "content": "^ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 485, + 375, + 499 + ], + "score": 1.0, + "content": "in Appendix", + "type": "text" + }, + { + "bbox": [ + 375, + 486, + 387, + 498 + ], + "score": 0.49, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 485, + 475, + 499 + ], + "score": 1.0, + "content": "for details. We select", + "type": "text" + }, + { + "bbox": [ + 475, + 488, + 482, + 497 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 214, + 509 + ], + "score": 0.86, + "content": "\\{ 1 0 , \\bar { 6 } 5 , 8 0 , 8 5 , 9 0 , 9 5 , 9 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 496, + 406, + 509 + ], + "score": 1.0, + "content": "based on the FPR95 performance. The optimal", + "type": "text" + }, + { + "bbox": [ + 406, + 498, + 412, + 508 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "is 90. All experiments", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 507, + 315, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 315, + 520 + ], + "score": 1.0, + "content": "are based on the hardware described in Appendix D.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "Comparison with competitive OOD detection methods. In Table 1, we compare ReAct with OOD", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "detection methods that are competitive in the literature, where ReAct establishes the state-of-the-art", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "performance. For a fair comparison, all methods use the pre-trained networks post hoc. We report", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "performance for each OOD test dataset, as well as the average of the four. ReAct outperforms", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 379, + 578 + ], + "score": 1.0, + "content": "all baselines considered, including Maximum Softmax Probability", + "type": "text" + }, + { + "bbox": [ + 379, + 566, + 396, + 577 + ], + "score": 0.56, + "content": "\\mathbb { \\lVert 1 5 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 566, + 428, + 578 + ], + "score": 1.0, + "content": ", ODIN", + "type": "text" + }, + { + "bbox": [ + 429, + 566, + 446, + 577 + ], + "score": 0.33, + "content": "\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 566, + 505, + 578 + ], + "score": 1.0, + "content": ", Mahalanobis", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 141, + 589 + ], + "score": 1.0, + "content": "distance", + "type": "text" + }, + { + "bbox": [ + 142, + 577, + 159, + 588 + ], + "score": 0.72, + "content": "\\mathbb { \\left. 2 9 \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 577, + 232, + 589 + ], + "score": 1.0, + "content": ", and energy score", + "type": "text" + }, + { + "bbox": [ + 232, + 577, + 250, + 588 + ], + "score": 0.84, + "content": "\\bar { \\big \\| } \\bar { 3 3 } \\big \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 577, + 420, + 589 + ], + "score": 1.0, + "content": ". Noticeably, ReAct reduces the FPR95 by", + "type": "text" + }, + { + "bbox": [ + 420, + 577, + 452, + 587 + ], + "score": 0.87, + "content": "2 5 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 173, + 600 + ], + "score": 1.0, + "content": "the best baseline", + "type": "text" + }, + { + "bbox": [ + 174, + 587, + 191, + 599 + ], + "score": 0.71, + "content": "\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "on ResNet. Note that Mahalanobis requires training a separate binary classifier,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "and displays limiting performance since the increased size of label space makes the class-conditional", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "Gaussian density estimation less viable. In contrast, ReAct is much easier to use in practice, and can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 620, + 371, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 371, + 631 + ], + "score": 1.0, + "content": "be implemented through a simple post hoc activation rectification.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 241, + 648 + ], + "score": 1.0, + "content": "Effect of rectification threshold", + "type": "text" + }, + { + "bbox": [ + 241, + 638, + 247, + 646 + ], + "score": 0.3, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 635, + 497, + 648 + ], + "score": 1.0, + "content": ". We now characterize the effect of the rectification parameter", + "type": "text" + }, + { + "bbox": [ + 497, + 639, + 502, + 646 + ], + "score": 0.46, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 635, + 506, + 648 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 278, + 659 + ], + "score": 1.0, + "content": "which can be modulated by the percentile", + "type": "text" + }, + { + "bbox": [ + 278, + 648, + 285, + 658 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 646, + 370, + 659 + ], + "score": 1.0, + "content": "described in Section", + "type": "text" + }, + { + "bbox": [ + 370, + 646, + 380, + 659 + ], + "score": 0.76, + "content": "3 .", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 646, + 417, + 659 + ], + "score": 1.0, + "content": "In Table", + "type": "text" + }, + { + "bbox": [ + 417, + 646, + 427, + 659 + ], + "score": 0.7, + "content": "^ { 2 , }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "we summarize the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 284, + 670 + ], + "score": 1.0, + "content": "OOD detection performance, where we vary", + "type": "text" + }, + { + "bbox": [ + 284, + 658, + 410, + 669 + ], + "score": 0.88, + "content": "p = \\{ 1 0 , 6 5 , 8 0 , 8 5 , 9 0 , 9 5 , 9 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 657, + 506, + 670 + ], + "score": 1.0, + "content": ". This ablation confirms", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "score": 1.0, + "content": "that over-activation does compromise the ability to detect OOD data, and ReAct can effectively", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 268, + 692 + ], + "score": 1.0, + "content": "alleviate this problem. Moreover, when", + "type": "text" + }, + { + "bbox": [ + 269, + 680, + 276, + 690 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 678, + 505, + 692 + ], + "score": 1.0, + "content": "is sufficiently large, ReAct can improve OOD detection", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "while maintaining a comparable ID classification accuracy. 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When", + "type": "text" + }, + { + "bbox": [ + 230, + 713, + 236, + 722 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 711, + 483, + 724 + ], + "score": 1.0, + "content": "is too small, OOD performance starts to degrade as expected.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 70, + 508, + 192 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 70, + 508, + 192 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 70, + 508, + 192 + ], + "spans": [ + { + "bbox": [ + 107, + 70, + 508, + 192 + ], + "score": 0.984, + "html": "
ModelMethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95AUROCFPR95AUROC 个FPR95AUROCFPR95AUROCFPR95 AUROC
ResNetMSP 园↓ 54.99→ 87.74√ 70.8380.86↓ 73.99个 79.76↓ 68.00↑ 79.61√ 66.95→ 81.99
ODIN 白47.6689.6660.1584.5967.8981.7850.2385.6256.4885.41
Mahalanobis [ 2997.0052.6598.5042.4198.4041.7955.8085.0187.4355.47
Energy 国55.7289.9559.2685.8964.9282.8653.7285.9958.4186.17
ReAct (Ours)20.3896.2224.2094.2033.8591.5847.3089.8031.4392.95
77.0279.2376.2777.3073.5179.00
MobileNetMSP 固64.2985.32 87.6254.0777.10 85.8873.5185.0354.2085.81
ODIN 白 四55.39 62.1181.0047.8286.3357.36 52.0984.71 83.6349.96 92.3833.0663.6071.01
Mahalanobis59.5088.9162.6584.5069.3781.1958.0585.0362.3984.91
Energy 国 ReAct (Ours)42.4091.5347.6988.1651.5686.6438.4291.5345.0289.47
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All", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "score": 1.0, + "content": "methods are based on a model trained on ID data only (ImageNet-1k), without using any auxiliary outlier data.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 212, + 478, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 113, + 223 + ], + "score": 0.41, + "content": "\\uparrow", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 212, + 246, + 225 + ], + "score": 1.0, + "content": "indicates larger values are better and", + "type": "text" + }, + { + "bbox": [ + 246, + 213, + 253, + 223 + ], + "score": 0.37, + "content": "\\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 212, + 478, + 225 + ], + "score": 1.0, + "content": "indicates smaller values are better. All values are percentages.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 248, + 191, + 262 + ], + "lines": [ + { + "bbox": [ + 104, + 246, + 193, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 246, + 193, + 264 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 506, + 310 + ], + "lines": [ + { + "bbox": [ + 104, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "In this section, we evaluate ReAct on a suite of OOD detection tasks. We first evaluate a on large-scale", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 353, + 300 + ], + "score": 1.0, + "content": "OOD detection benchmark based on ImageNet [18] ( Section", + "type": "text" + }, + { + "bbox": [ + 354, + 287, + 372, + 300 + ], + "score": 0.37, + "content": "\\boxed { 4 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 286, + 505, + 300 + ], + "score": 1.0, + "content": ", and then proceed in Section 4.2", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 298, + 228, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 208, + 311 + ], + "score": 1.0, + "content": "with CIFAR benchmarks", + "type": "text" + }, + { + "bbox": [ + 208, + 298, + 226, + 310 + ], + "score": 0.53, + "content": " { \\mathbb { \\left[ \\left[ 2 7 \\right] \\right] } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 298, + 228, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 275, + 506, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 329, + 388, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 389, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 389, + 343 + ], + "score": 1.0, + "content": "4.1 Evaluation on Large-scale ImageNet Classification Networks", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "We first evaluate ReAct on a large-scale OOD detection benchmark developed in [18]. Compared", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "to the CIFAR benchmarks that are routinely used in literature, the ImageNet benchmark is more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 263, + 387 + ], + "score": 1.0, + "content": "challenging due to a larger label space", + "type": "text" + }, + { + "bbox": [ + 263, + 374, + 314, + 385 + ], + "score": 0.83, + "content": "( K = 1 , 0 0 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 372, + 506, + 387 + ], + "score": 1.0, + "content": ". Moreover, such large-scale evaluation is more", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "relevant to real-world applications, where the deployed models often operate on images that have", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 395, + 389, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 389, + 408 + ], + "score": 1.0, + "content": "high resolution and contain more classes than the CIFAR benchmarks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 351, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 299, + 424 + ], + "score": 1.0, + "content": "Setup. We use a pre-trained ResNet-50 model", + "type": "text" + }, + { + "bbox": [ + 299, + 411, + 316, + 422 + ], + "score": 0.77, + "content": "[ \\mathbb { 1 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "for ImageNet-1k. At test time, all images are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 149, + 434 + ], + "score": 1.0, + "content": "resized to", + "type": "text" + }, + { + "bbox": [ + 149, + 422, + 194, + 432 + ], + "score": 0.89, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 421, + 484, + 434 + ], + "score": 1.0, + "content": ". We evaluate on four test OOD datasets from (subsets of) Places365", + "type": "text" + }, + { + "bbox": [ + 485, + 421, + 502, + 433 + ], + "score": 0.31, + "content": "\\lVert \\rVert \\mathbf { 6 } 2 \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 421, + 506, + 434 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 228, + 445 + ], + "score": 1.0, + "content": "Textures [5], iNaturalist", + "type": "text" + }, + { + "bbox": [ + 229, + 432, + 246, + 444 + ], + "score": 0.67, + "content": "\\mathbb { I } \\mathbb { 1 6 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 432, + 287, + 445 + ], + "score": 1.0, + "content": ", and SUN", + "type": "text" + }, + { + "bbox": [ + 287, + 432, + 305, + 444 + ], + "score": 0.81, + "content": "\\mathbb { \\left[ \\left. 5 5 \\right] \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "with non-overlapping categories w.r.t ImageNet.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "Our evaluations span a diverse range of domains including fine-grained images, scene images, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 233, + 467 + ], + "score": 1.0, + "content": "textural images (see Appendix", + "type": "text" + }, + { + "bbox": [ + 234, + 453, + 245, + 466 + ], + "score": 0.82, + "content": "\\boxed { \\mathbf { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "for details). We use a validation set of Gaussian noise images,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 260, + 478 + ], + "score": 1.0, + "content": "which are generated by sampling from", + "type": "text" + }, + { + "bbox": [ + 260, + 465, + 293, + 477 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "for each pixel location. To ensure validity, we further", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "verify the activation pattern under Gaussian noise, which exhibits a similar distributional trend", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 315, + 499 + ], + "score": 1.0, + "content": "with positive skewness and chaoticness; see Figure", + "type": "text" + }, + { + "bbox": [ + 315, + 486, + 324, + 498 + ], + "score": 0.73, + "content": "^ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 485, + 375, + 499 + ], + "score": 1.0, + "content": "in Appendix", + "type": "text" + }, + { + "bbox": [ + 375, + 486, + 387, + 498 + ], + "score": 0.49, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 485, + 475, + 499 + ], + "score": 1.0, + "content": "for details. We select", + "type": "text" + }, + { + "bbox": [ + 475, + 488, + 482, + 497 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 214, + 509 + ], + "score": 0.86, + "content": "\\{ 1 0 , \\bar { 6 } 5 , 8 0 , 8 5 , 9 0 , 9 5 , 9 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 496, + 406, + 509 + ], + "score": 1.0, + "content": "based on the FPR95 performance. The optimal", + "type": "text" + }, + { + "bbox": [ + 406, + 498, + 412, + 508 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "is 90. All experiments", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 507, + 315, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 315, + 520 + ], + "score": 1.0, + "content": "are based on the hardware described in Appendix D.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 410, + 506, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "Comparison with competitive OOD detection methods. In Table 1, we compare ReAct with OOD", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "detection methods that are competitive in the literature, where ReAct establishes the state-of-the-art", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "performance. For a fair comparison, all methods use the pre-trained networks post hoc. We report", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "performance for each OOD test dataset, as well as the average of the four. ReAct outperforms", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 379, + 578 + ], + "score": 1.0, + "content": "all baselines considered, including Maximum Softmax Probability", + "type": "text" + }, + { + "bbox": [ + 379, + 566, + 396, + 577 + ], + "score": 0.56, + "content": "\\mathbb { \\lVert 1 5 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 566, + 428, + 578 + ], + "score": 1.0, + "content": ", ODIN", + "type": "text" + }, + { + "bbox": [ + 429, + 566, + 446, + 577 + ], + "score": 0.33, + "content": "\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 566, + 505, + 578 + ], + "score": 1.0, + "content": ", Mahalanobis", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 141, + 589 + ], + "score": 1.0, + "content": "distance", + "type": "text" + }, + { + "bbox": [ + 142, + 577, + 159, + 588 + ], + "score": 0.72, + "content": "\\mathbb { \\left. 2 9 \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 577, + 232, + 589 + ], + "score": 1.0, + "content": ", and energy score", + "type": "text" + }, + { + "bbox": [ + 232, + 577, + 250, + 588 + ], + "score": 0.84, + "content": "\\bar { \\big \\| } \\bar { 3 3 } \\big \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 577, + 420, + 589 + ], + "score": 1.0, + "content": ". Noticeably, ReAct reduces the FPR95 by", + "type": "text" + }, + { + "bbox": [ + 420, + 577, + 452, + 587 + ], + "score": 0.87, + "content": "2 5 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 173, + 600 + ], + "score": 1.0, + "content": "the best baseline", + "type": "text" + }, + { + "bbox": [ + 174, + 587, + 191, + 599 + ], + "score": 0.71, + "content": "\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "on ResNet. Note that Mahalanobis requires training a separate binary classifier,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "and displays limiting performance since the increased size of label space makes the class-conditional", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "Gaussian density estimation less viable. In contrast, ReAct is much easier to use in practice, and can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 620, + 371, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 371, + 631 + ], + "score": 1.0, + "content": "be implemented through a simple post hoc activation rectification.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 523, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 241, + 648 + ], + "score": 1.0, + "content": "Effect of rectification threshold", + "type": "text" + }, + { + "bbox": [ + 241, + 638, + 247, + 646 + ], + "score": 0.3, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 635, + 497, + 648 + ], + "score": 1.0, + "content": ". 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This ablation confirms", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "score": 1.0, + "content": "that over-activation does compromise the ability to detect OOD data, and ReAct can effectively", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 268, + 692 + ], + "score": 1.0, + "content": "alleviate this problem. Moreover, when", + "type": "text" + }, + { + "bbox": [ + 269, + 680, + 276, + 690 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 678, + 505, + 692 + ], + "score": 1.0, + "content": "is sufficiently large, ReAct can improve OOD detection", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "while maintaining a comparable ID classification accuracy. 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Rectification percentileFPR95 √AUROC →AUPR 个ID ACC. 个Rectification threshold c
No ReAct 358.4186.1796.8875.088
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All", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 162, + 354, + 174 + ], + "spans": [ + { + "bbox": [ + 107, + 162, + 354, + 174 + ], + "score": 1.0, + "content": "numbers are percentages and are averaged over 4 OOD test datasets.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 193, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "Effect on other network architectures. We show that ReAct is effective on a different architecture", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 140, + 215 + ], + "score": 1.0, + "content": "in Table", + "type": "text" + }, + { + "bbox": [ + 140, + 203, + 150, + 216 + ], + "score": 0.41, + "content": "\\mathbb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "In particular, we consider a lightweight model MobileNet-v2 [44], which can be suitable", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "for OOD detection in on-device mobile applications. Same as before, we apply ReAct on the output", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "of the penultimate layer, with the rectification threshold chosen based on the 90-th percentile. Our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 235, + 398, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 228, + 248 + ], + "score": 1.0, + "content": "method reduces the FPR95 by", + "type": "text" + }, + { + "bbox": [ + 229, + 236, + 256, + 247 + ], + "score": 0.83, + "content": "9 . 1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 235, + 398, + 248 + ], + "score": 1.0, + "content": "compared to the best baseline [31].", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 506, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "What about applying ReAct on other layers? Our results suggest that applying ReAct on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "score": 1.0, + "content": "penultimate layer is the most effective, since the activation patterns are most distinctive. To see this,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "we provide the activation and performance study for intermediate layers in Appendix E (see Figure 6", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 146, + 297 + ], + "score": 1.0, + "content": "and Table", + "type": "text" + }, + { + "bbox": [ + 146, + 284, + 156, + 297 + ], + "score": 0.5, + "content": "^ { 6 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 284, + 506, + 297 + ], + "score": 1.0, + "content": ". Interestingly, early layers display less distinctive signatures between ID and OOD data.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "score": 1.0, + "content": "This is expected because neural networks generally capture lower-level features in early layers (such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 171, + 317 + ], + "score": 1.0, + "content": "as Gabor filters", + "type": "text" + }, + { + "bbox": [ + 172, + 305, + 189, + 317 + ], + "score": 0.74, + "content": "\\left[ \\left[ 6 1 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "in layer 1), whose activations can be very similar between ID and OOD. The", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 316, + 482, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 482, + 328 + ], + "score": 1.0, + "content": "semantic-level features only emerge as with deeper layers, where ReAct is the most effective.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 334, + 280, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 280, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 280, + 347 + ], + "score": 1.0, + "content": "4.2 Evaluation on CIFAR Benchmarks", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "score": 1.0, + "content": "Datasets. We evaluate on CIFAR-10 and CIFAR-100 [27] datasets as in-distribution data, using", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "the standard split with 50,000 training images and 10,000 test images. For OOD data, we consider", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 374, + 507, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 507, + 389 + ], + "score": 1.0, + "content": "six common benchmark datasets: Textures [5], SVHN [37], Places365 [62], LSUN-Crop [60],", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 386, + 249, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 167, + 399 + ], + "score": 1.0, + "content": "LSUN-Resize", + "type": "text" + }, + { + "bbox": [ + 167, + 386, + 184, + 398 + ], + "score": 0.41, + "content": "\\pmb { \\mathbb { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 386, + 249, + 399 + ], + "score": 1.0, + "content": ", and iSUN [57].", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Experimental details. We train a standard ResNet-18 [11] model on in-distribution data. The feature", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "dimension of the penultimate layer is 512. For both CIFAR-10 and CIFAR-100, the models are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "trained for 100 epochs. The start learning rate is 0.1 and decays by a factor of 10 at epochs 50, 75,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 435, + 471, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 194, + 447 + ], + "score": 1.0, + "content": "and 90. For threshold", + "type": "text" + }, + { + "bbox": [ + 195, + 437, + 200, + 445 + ], + "score": 0.44, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 435, + 471, + 447 + ], + "score": 1.0, + "content": ", we use the 90-th percentile of activations estimated on the ID data.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 432, + 463 + ], + "score": 1.0, + "content": "ReAct is compatible with various OOD scoring functions. We show in Table", + "type": "text" + }, + { + "bbox": [ + 432, + 451, + 441, + 462 + ], + "score": 0.75, + "content": "\\textcircled { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "that ReAct is a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 459, + 507, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 387, + 476 + ], + "score": 1.0, + "content": "flexible method that is compatible with alternative scoring functions", + "type": "text" + }, + { + "bbox": [ + 387, + 462, + 437, + 474 + ], + "score": 0.91, + "content": "S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 459, + 507, + 476 + ], + "score": 1.0, + "content": ". To see this, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "consider commonly used scoring functions, and compare the performance both with and without", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 504, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 409, + 496 + ], + "score": 1.0, + "content": "using ReAct respectively. In particular, we consider softmax confidence", + "type": "text" + }, + { + "bbox": [ + 410, + 483, + 428, + 495 + ], + "score": 0.75, + "content": "\\mathbb { \\lVert 1 5 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 483, + 486, + 496 + ], + "score": 1.0, + "content": ", ODIN score", + "type": "text" + }, + { + "bbox": [ + 486, + 483, + 504, + 494 + ], + "score": 0.66, + "content": "\\textcircled { \\scriptsize { 1 3 1 } }", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 494, + 504, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 224, + 506 + ], + "score": 1.0, + "content": "as well as energy score [33]—", + "type": "text" + }, + { + "bbox": [ + 226, + 495, + 467, + 506 + ], + "score": 1.0, + "content": "all of which derive OOD scores directly from the output", + "type": "text" + }, + { + "bbox": [ + 467, + 494, + 488, + 506 + ], + "score": 0.91, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 495, + 504, + 506 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "particular, using ReAct on energy score yields the best performance, which is desirable as energy is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 515, + 504, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 486, + 528 + ], + "score": 1.0, + "content": "a hyperparameter-free OOD score and is easy to compute in practice. Note that Mahalanobis", + "type": "text" + }, + { + "bbox": [ + 486, + 515, + 504, + 527 + ], + "score": 0.81, + "content": "\\overline { { \\| 2 9 } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 427, + 538 + ], + "score": 1.0, + "content": "estimates OOD score using feature representations instead of the model output", + "type": "text" + }, + { + "bbox": [ + 428, + 526, + 448, + 538 + ], + "score": 0.92, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 526, + 505, + 538 + ], + "score": 1.0, + "content": ", hence is less", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "compatible with ReAct. On all three in-distribution datasets, using ReAct consistently outperforms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 327, + 561 + ], + "score": 1.0, + "content": "the counterpart without rectification. Results in Table", + "type": "text" + }, + { + "bbox": [ + 327, + 547, + 337, + 560 + ], + "score": 0.75, + "content": "3", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "are based on the average across multiple", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "OOD test datasets. Detailed performance for each OOD test dataset is provided in Appendix, Table 7.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31 + }, + { + "type": "table", + "bbox": [ + 114, + 587, + 497, + 680 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 114, + 587, + 497, + 680 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 114, + 587, + 497, + 680 + ], + "spans": [ + { + "bbox": [ + 114, + 587, + 497, + 680 + ], + "score": 0.981, + "html": "
MethodCIFAR-10CIFAR-100ImageNet
FPR95 √AUROC →AUPR →FPR95 ←AUROC →AUPR 个FPR95 ←AUROC →AUPR →
Softmax score 园56.7191.1779.1180.7276.8378.4166.9581.9995.76
Softmax score + ReAct53.8191.7092.1175.4580.4084.2858.2887.0697.22
Energy B335.6093.5795.0171.9382.8286.2858.4186.1796.88
Energy+ReAct32.9194.2795.5359.6187.4889.63 31.4392.9598.50
ODIN B131.1093.7994.9566.2182.8886.2556.4885.4196.61
ODIN+ReAct28.8194.0494.8259.9185.2387.5344.1090.7098.04
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Rectification percentileFPR95 √AUROC →AUPR 个ID ACC. 个Rectification threshold c
No ReAct 358.4186.1796.8875.088
p=9944.5790.4597.9675.122.25
p=9535.3992.3998.3774.761.50
p= 9031.4392.9598.5073.751.00
p=8534.0892.0598.3572.910.84
p=8041.5189.5497.9171.930.72
p=6574.6274.1494.3967.140.50
p=1074.7057.5586.061.220.06
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All", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 162, + 354, + 174 + ], + "spans": [ + { + "bbox": [ + 107, + 162, + 354, + 174 + ], + "score": 1.0, + "content": "numbers are percentages and are averaged over 4 OOD test datasets.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 193, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "Effect on other network architectures. We show that ReAct is effective on a different architecture", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 140, + 215 + ], + "score": 1.0, + "content": "in Table", + "type": "text" + }, + { + "bbox": [ + 140, + 203, + 150, + 216 + ], + "score": 0.41, + "content": "\\mathbb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "In particular, we consider a lightweight model MobileNet-v2 [44], which can be suitable", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "for OOD detection in on-device mobile applications. Same as before, we apply ReAct on the output", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "of the penultimate layer, with the rectification threshold chosen based on the 90-th percentile. Our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 235, + 398, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 228, + 248 + ], + "score": 1.0, + "content": "method reduces the FPR95 by", + "type": "text" + }, + { + "bbox": [ + 229, + 236, + 256, + 247 + ], + "score": 0.83, + "content": "9 . 1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 235, + 398, + 248 + ], + "score": 1.0, + "content": "compared to the best baseline [31].", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 192, + 505, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 506, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "What about applying ReAct on other layers? Our results suggest that applying ReAct on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "score": 1.0, + "content": "penultimate layer is the most effective, since the activation patterns are most distinctive. To see this,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "we provide the activation and performance study for intermediate layers in Appendix E (see Figure 6", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 146, + 297 + ], + "score": 1.0, + "content": "and Table", + "type": "text" + }, + { + "bbox": [ + 146, + 284, + 156, + 297 + ], + "score": 0.5, + "content": "^ { 6 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 284, + 506, + 297 + ], + "score": 1.0, + "content": ". Interestingly, early layers display less distinctive signatures between ID and OOD data.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "score": 1.0, + "content": "This is expected because neural networks generally capture lower-level features in early layers (such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 171, + 317 + ], + "score": 1.0, + "content": "as Gabor filters", + "type": "text" + }, + { + "bbox": [ + 172, + 305, + 189, + 317 + ], + "score": 0.74, + "content": "\\left[ \\left[ 6 1 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "in layer 1), whose activations can be very similar between ID and OOD. The", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 316, + 482, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 482, + 328 + ], + "score": 1.0, + "content": "semantic-level features only emerge as with deeper layers, where ReAct is the most effective.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 252, + 506, + 328 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 334, + 280, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 280, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 280, + 347 + ], + "score": 1.0, + "content": "4.2 Evaluation on CIFAR Benchmarks", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 506, + 368 + ], + "score": 1.0, + "content": "Datasets. We evaluate on CIFAR-10 and CIFAR-100 [27] datasets as in-distribution data, using", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "the standard split with 50,000 training images and 10,000 test images. For OOD data, we consider", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 374, + 507, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 507, + 389 + ], + "score": 1.0, + "content": "six common benchmark datasets: Textures [5], SVHN [37], Places365 [62], LSUN-Crop [60],", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 386, + 249, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 167, + 399 + ], + "score": 1.0, + "content": "LSUN-Resize", + "type": "text" + }, + { + "bbox": [ + 167, + 386, + 184, + 398 + ], + "score": 0.41, + "content": "\\pmb { \\mathbb { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 386, + 249, + 399 + ], + "score": 1.0, + "content": ", and iSUN [57].", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 352, + 507, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Experimental details. We train a standard ResNet-18 [11] model on in-distribution data. The feature", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "dimension of the penultimate layer is 512. For both CIFAR-10 and CIFAR-100, the models are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "trained for 100 epochs. The start learning rate is 0.1 and decays by a factor of 10 at epochs 50, 75,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 435, + 471, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 194, + 447 + ], + "score": 1.0, + "content": "and 90. For threshold", + "type": "text" + }, + { + "bbox": [ + 195, + 437, + 200, + 445 + ], + "score": 0.44, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 435, + 471, + 447 + ], + "score": 1.0, + "content": ", we use the 90-th percentile of activations estimated on the ID data.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 402, + 506, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 432, + 463 + ], + "score": 1.0, + "content": "ReAct is compatible with various OOD scoring functions. We show in Table", + "type": "text" + }, + { + "bbox": [ + 432, + 451, + 441, + 462 + ], + "score": 0.75, + "content": "\\textcircled { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "that ReAct is a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 459, + 507, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 387, + 476 + ], + "score": 1.0, + "content": "flexible method that is compatible with alternative scoring functions", + "type": "text" + }, + { + "bbox": [ + 387, + 462, + 437, + 474 + ], + "score": 0.91, + "content": "S ( \\mathbf { x } ; f ^ { \\mathrm { R e A c t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 459, + 507, + 476 + ], + "score": 1.0, + "content": ". To see this, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "consider commonly used scoring functions, and compare the performance both with and without", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 504, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 409, + 496 + ], + "score": 1.0, + "content": "using ReAct respectively. In particular, we consider softmax confidence", + "type": "text" + }, + { + "bbox": [ + 410, + 483, + 428, + 495 + ], + "score": 0.75, + "content": "\\mathbb { \\lVert 1 5 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 483, + 486, + 496 + ], + "score": 1.0, + "content": ", ODIN score", + "type": "text" + }, + { + "bbox": [ + 486, + 483, + 504, + 494 + ], + "score": 0.66, + "content": "\\textcircled { \\scriptsize { 1 3 1 } }", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 494, + 504, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 224, + 506 + ], + "score": 1.0, + "content": "as well as energy score [33]—", + "type": "text" + }, + { + "bbox": [ + 226, + 495, + 467, + 506 + ], + "score": 1.0, + "content": "all of which derive OOD scores directly from the output", + "type": "text" + }, + { + "bbox": [ + 467, + 494, + 488, + 506 + ], + "score": 0.91, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 495, + 504, + 506 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "particular, using ReAct on energy score yields the best performance, which is desirable as energy is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 515, + 504, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 486, + 528 + ], + "score": 1.0, + "content": "a hyperparameter-free OOD score and is easy to compute in practice. Note that Mahalanobis", + "type": "text" + }, + { + "bbox": [ + 486, + 515, + 504, + 527 + ], + "score": 0.81, + "content": "\\overline { { \\| 2 9 } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 427, + 538 + ], + "score": 1.0, + "content": "estimates OOD score using feature representations instead of the model output", + "type": "text" + }, + { + "bbox": [ + 428, + 526, + 448, + 538 + ], + "score": 0.92, + "content": "f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 526, + 505, + 538 + ], + "score": 1.0, + "content": ", hence is less", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "compatible with ReAct. On all three in-distribution datasets, using ReAct consistently outperforms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 327, + 561 + ], + "score": 1.0, + "content": "the counterpart without rectification. Results in Table", + "type": "text" + }, + { + "bbox": [ + 327, + 547, + 337, + 560 + ], + "score": 0.75, + "content": "3", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "are based on the average across multiple", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "OOD test datasets. Detailed performance for each OOD test dataset is provided in Appendix, Table 7.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 451, + 507, + 571 + ] + }, + { + "type": "table", + "bbox": [ + 114, + 587, + 497, + 680 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 114, + 587, + 497, + 680 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 114, + 587, + 497, + 680 + ], + "spans": [ + { + "bbox": [ + 114, + 587, + 497, + 680 + ], + "score": 0.981, + "html": "
MethodCIFAR-10CIFAR-100ImageNet
FPR95 √AUROC →AUPR →FPR95 ←AUROC →AUPR 个FPR95 ←AUROC →AUPR →
Softmax score 园56.7191.1779.1180.7276.8378.4166.9581.9995.76
Softmax score + ReAct53.8191.7092.1175.4580.4084.2858.2887.0697.22
Energy B335.6093.5795.0171.9382.8286.2858.4186.1796.88
Energy+ReAct32.9194.2795.5359.6187.4889.63 31.4392.9598.50
ODIN B131.1093.7994.9566.2182.8886.2556.4885.4196.61
ODIN+ReAct28.8194.0494.8259.9185.2387.5344.1090.7098.04
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For each ID dataset, we", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 417, + 702 + ], + "score": 1.0, + "content": "use the same model and compare the performance with and without ReAct respectively.", + "type": "text" + }, + { + "bbox": [ + 417, + 690, + 424, + 700 + ], + "score": 0.76, + "content": "\\uparrow", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 690, + 505, + 702 + ], + "score": 1.0, + "content": "indicates larger values", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 158, + 711 + ], + "score": 1.0, + "content": "are better and", + "type": "text" + }, + { + "bbox": [ + 158, + 700, + 165, + 710 + ], + "score": 0.76, + "content": "\\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "indicates smaller values are better. 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Model is trained on ImageNet. Note that the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "right tail has a higher density than the left tail. By left and right tail we refer to the samples that are to the left", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 188, + 416, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 416, + 199 + ], + "score": 1.0, + "content": "and right of the median, which is close to the mode in the case of skewed distribution.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 107, + 217, + 231, + 231 + ], + "lines": [ + { + "bbox": [ + 104, + 216, + 231, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 231, + 234 + ], + "score": 1.0, + "content": "5 Theoretical Analysis", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "To better understand the effect of ReAct, we mathematically model the ID and OOD activations as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "rectified Gaussian distributions and derive their respective distributions after applying ReAct. These", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 261, + 504, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 481, + 278 + ], + "score": 1.0, + "content": "modeling assumptions are based on the activation statistics observed on ImageNet in Figure", + "type": "text" + }, + { + "bbox": [ + 482, + 263, + 492, + 277 + ], + "score": 0.73, + "content": "^ { 1 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 261, + 504, + 278 + ], + "score": 1.0, + "content": "In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "the following analysis, we show that ReAct reduces mean OOD activations more than ID activations", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 284, + 446, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 446, + 299 + ], + "score": 1.0, + "content": "since OOD activations are more positively skewed (see Appendix A for derivations).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 195, + 321 + ], + "score": 1.0, + "content": "ID activations. Let", + "type": "text" + }, + { + "bbox": [ + 195, + 308, + 306, + 320 + ], + "score": 0.91, + "content": "h ( \\mathbf { x } ) = ( z _ { 1 } , \\ldots , z _ { m } ) = : \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "be the activations for the penultimate layer. We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 318, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 158, + 333 + ], + "score": 1.0, + "content": "assume each", + "type": "text" + }, + { + "bbox": [ + 158, + 320, + 226, + 332 + ], + "score": 0.92, + "content": "z _ { i } \\sim \\mathcal { N } ^ { R } ( \\mu , \\sigma _ { \\mathrm { i n } } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 318, + 264, + 333 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 265, + 321, + 296, + 331 + ], + "score": 0.89, + "content": "\\sigma _ { \\mathrm { i n } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 318, + 321, + 333 + ], + "score": 1.0, + "content": ". 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It follows that", + "type": "text" + }, + { + "bbox": [ + 216, + 338, + 283, + 350 + ], + "score": 0.92, + "content": "z _ { i } = \\operatorname* { m a x } ( x _ { i } , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 337, + 358, + 351 + ], + "score": 1.0, + "content": ", with expectation:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 326, + 505, + 351 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 351, + 484, + 378 + ], + "lines": [ + { + "bbox": [ + 113, + 351, + 484, + 378 + ], + "spans": [ + { + "bbox": [ + 113, + 351, + 484, + 378 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathrm { o u t } } [ z _ { i } ] = \\mu - ( 1 + \\epsilon ) \\Phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\mu + ( 1 + \\epsilon ) ^ { 2 } \\phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\sigma _ { \\mathrm { o u t } } - \\frac { 4 \\epsilon } { \\sqrt { 2 \\pi } } \\cdot \\sigma _ { \\mathrm { o u t } } .", + "type": "interline_equation", + "image_path": "22bcc29d0c859395f9d300bcbfa3ab28e9cc2d3394441570930b193f3c08c774.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 113, + 351, + 484, + 360.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 113, + 360.0, + 484, + 369.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 113, + 369.0, + 484, + 378.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 283, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 282, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 282, + 391 + ], + "score": 1.0, + "content": "Expectation after applying ReAct becomes:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 378, + 282, + 391 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 390, + 482, + 448 + ], + "lines": [ + { + "bbox": [ + 128, + 390, + 482, + 448 + ], + "spans": [ + { + "bbox": [ + 128, + 390, + 482, + 448 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathrm { o u t } } [ \\bar { z } _ { i } ] = \\mu - ( 1 + \\epsilon ) \\Phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\mu + ( 1 - \\epsilon ) \\left[ 1 - \\Phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\right] \\cdot ( c - \\mu ) } \\\\ { + \\left[ ( 1 + \\epsilon ) ^ { 2 } \\phi \\left( \\frac { - \\mu } { ( 1 + \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) - ( 1 - \\epsilon ) ^ { 2 } \\phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) - \\frac { 4 \\epsilon } { \\sqrt { 2 \\pi } } \\right] \\cdot \\sigma _ { \\mathrm { o u t } } , } \\end{array}", + "type": "interline_equation", + "image_path": "70e8a59dc9c7a7fea0b398b204cf4f2edcf97911ff361f7388c9cfde2c32f277.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 128, + 390, + 482, + 409.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 128, + 409.3333333333333, + 482, + 428.66666666666663 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 128, + 428.66666666666663, + 482, + 447.99999999999994 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 136, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 137, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 137, + 459 + ], + "score": 1.0, + "content": "Hence:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 445, + 137, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 484, + 497 + ], + "lines": [ + { + "bbox": [ + 114, + 456, + 483, + 484 + ], + "spans": [ + { + "bbox": [ + 114, + 456, + 483, + 484 + ], + "score": 0.88, + "content": "\\mathbb { E } _ { \\mathrm { o u t } } [ z _ { i } - \\bar { z } _ { i } ] = ( 1 - \\epsilon ) ^ { 2 } \\phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\cdot \\sigma _ { \\mathrm { o u t } } - ( 1 - \\epsilon ) \\left[ 1 - \\Phi \\left( \\frac { c - \\mu } { ( 1 - \\epsilon ) \\sigma _ { \\mathrm { o u t } } } \\right) \\right] \\cdot ( c - \\mu ) ,", + "type": "inline_equation", + "image_path": "e015c0e643d59a2f73aefe83df20706deda9888a0a4c7557d3bdbd0e9d96fb34.jpg" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 482, + 326, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 239, + 498 + ], + "score": 1.0, + "content": "which recovers Equation 4 when", + "type": "text" + }, + { + "bbox": [ + 239, + 485, + 263, + 495 + ], + "score": 0.87, + "content": "\\epsilon = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 482, + 280, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 486, + 322, + 495 + ], + "score": 0.89, + "content": "\\sigma _ { \\mathrm { o u t } } = \\sigma _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 482, + 326, + 498 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 456, + 483, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 506, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 354, + 518 + ], + "score": 1.0, + "content": "Remark 1: Activation reduction on OOD is more than ID.", + "type": "text" + }, + { + "bbox": [ + 361, + 505, + 462, + 519 + ], + "score": 1.0, + "content": "Figure 3 shows a plot of", + "type": "text" + }, + { + "bbox": [ + 462, + 506, + 504, + 518 + ], + "score": 0.92, + "content": "\\mathbb { E } [ z _ { i } - \\bar { z } _ { i } ]", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 121, + 530 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 518, + 154, + 528 + ], + "score": 0.9, + "content": "\\mu = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 516, + 173, + 530 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 518, + 197, + 527 + ], + "score": 0.9, + "content": "c = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 516, + 301, + 530 + ], + "score": 1.0, + "content": ". Observe that decreasing", + "type": "text" + }, + { + "bbox": [ + 302, + 519, + 307, + 527 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 516, + 469, + 530 + ], + "score": 1.0, + "content": "(more positive-skewness) or increasing", + "type": "text" + }, + { + "bbox": [ + 470, + 520, + 477, + 527 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 516, + 505, + 530 + ], + "score": 1.0, + "content": "(more", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "chaotic-ness) leads to a larger reduction in the mean activation after applying ReAct. 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This suggests that rectification on average affects OOD", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 560, + 323, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 323, + 573 + ], + "score": 1.0, + "content": "activations more severely compared to ID activations.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 505, + 506, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 581, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "Remark 2: Output reduction on OOD is more than ID. To derive the effect on the distribution", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 265, + 605 + ], + "score": 1.0, + "content": "of model output, consider output logits", + "type": "text" + }, + { + "bbox": [ + 266, + 592, + 333, + 605 + ], + "score": 0.92, + "content": "f ( \\mathbf { z } ) = W \\mathbf { z } + \\mathbf { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "and assume without loss of generality that", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 602, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 143, + 614 + ], + "score": 0.9, + "content": "W \\mathbf { 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 602, + 419, + 617 + ], + "score": 1.0, + "content": "element-wise. 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Let", + "type": "text" + }, + { + "bbox": [ + 321, + 614, + 456, + 626 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\bar { \\boldsymbol \\delta } = \\mathbf { \\dot { \\mathbb { E } } _ { \\mathrm { o u t } } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] - \\mathbb { E } _ { \\mathrm { i n } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] > 0 . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 614, + 486, + 626 + ], + "score": 1.0, + "content": "Then:", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 106, + 582, + 506, + 626 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 625, + 463, + 668 + ], + "lines": [ + { + "bbox": [ + 147, + 625, + 463, + 668 + ], + "spans": [ + { + "bbox": [ + 147, + 625, + 463, + 668 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\mathrm { o u t } } [ f ( \\mathbf { z } ) - f ( \\bar { \\mathbf { z } } ) ] = \\mathbb { E } _ { \\mathrm { o u t } } [ W ( \\mathbf { z } - \\bar { \\mathbf { z } } ) ] = W \\mathbb { E } _ { \\mathrm { o u t } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] = W \\left( \\mathbb { E } _ { \\mathrm { i n } } [ \\mathbf { z } - \\bar { \\mathbf { z } } ] + \\delta \\mathbf { 1 } \\right) } \\\\ & { \\qquad = \\mathbb { E } _ { \\mathrm { i n } } [ W ( \\mathbf { z } - \\bar { \\mathbf { z } } ) ] + \\delta W \\mathbf { 1 } } \\\\ & { \\qquad > \\mathbb { E } _ { \\mathrm { i n } } [ f ( \\mathbf { z } ) - f ( \\bar { \\mathbf { z } } ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "ec951ae179c2fdcf1f1758def8e1b9799c8f6b29c0cce6bfe9202e0591544dc9.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 147, + 625, + 463, + 639.3333333333334 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 147, + 639.3333333333334, + 463, + 653.6666666666667 + ], + "spans": [], + "index": 50 + }, + { + "bbox": [ + 147, + 653.6666666666667, + 463, + 668.0000000000001 + ], + "spans": [], + "index": 51 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 668, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 315, + 680 + ], + "score": 1.0, + "content": "Hence the increased separation between OOD and", + "type": "text" + }, + { + "bbox": [ + 316, + 668, + 328, + 678 + ], + "score": 0.29, + "content": "\\mathrm { I D }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "activations transfers to the output space as", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 232, + 691 + ], + "score": 1.0, + "content": "well. Note that the condition of", + "type": "text" + }, + { + "bbox": [ + 232, + 679, + 268, + 689 + ], + "score": 0.9, + "content": "W \\mathbf { 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "is sufficient but not necessary for this result to hold. In fact,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 217, + 702 + ], + "score": 1.0, + "content": "our experiments in Section", + "type": "text" + }, + { + "bbox": [ + 218, + 689, + 238, + 702 + ], + "score": 0.28, + "content": "\\textcircled { 4 } \\mathrm { d o }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "not require this condition. However, we verified empirically that", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 144, + 713 + ], + "score": 1.0, + "content": "ensuring", + "type": "text" + }, + { + "bbox": [ + 144, + 700, + 181, + 711 + ], + "score": 0.9, + "content": "W \\mathbf { 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 699, + 313, + 713 + ], + "score": 1.0, + "content": "by adding a positive constant to", + "type": "text" + }, + { + "bbox": [ + 313, + 700, + 325, + 710 + ], + "score": 0.8, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "and applying ReAct does confer benefits to", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 710, + 334, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 334, + 723 + ], + "score": 1.0, + "content": "OOD detection, which validates our theoretical analysis.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 54, + "bbox_fs": [ + 105, + 667, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "Why ReAct improves the OOD scoring functions? Our theoretical analysis above shows that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "ReAct suppresses logit output for OOD data more so than for ID data. This means that for detection", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 333, + 107 + ], + "score": 1.0, + "content": "scores depending on the logit output (e.g., energy score", + "type": "text" + }, + { + "bbox": [ + 333, + 93, + 352, + 105 + ], + "score": 0.83, + "content": "\\mathbb { \\lVert 3 3 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "), the gap between OOD and ID score", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "will be enlarged after applying ReAct, which makes thresholding more capable of separating OOD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 115, + 432, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 432, + 128 + ], + "score": 1.0, + "content": "and ID inputs; see Figure 1(a) and (c) for a concrete example showing this effect.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 106, + 144, + 293, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 293, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 293, + 160 + ], + "score": 1.0, + "content": "6 Discussion and Further Analysis", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 170, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 506, + 182 + ], + "score": 1.0, + "content": "Why do OOD samples trigger abnormal unit activation patterns? So far we have shown that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "OOD data can trigger unit activation patterns that are significantly different from ID data, and that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 357, + 205 + ], + "score": 1.0, + "content": "ReAct can effectively alleviate this issue (empirically in Section", + "type": "text" + }, + { + "bbox": [ + 358, + 191, + 367, + 204 + ], + "score": 0.33, + "content": "^ 4", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "and theoretically in Section 5). Yet", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 202, + 506, + 215 + ], + "score": 1.0, + "content": "a question left in mystery is why such a pattern occurs in modern neural networks? Answering this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "question requires carefully examining the internal mechanism by which the network is trained and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 504, + 236 + ], + "score": 1.0, + "content": "evaluated. Here we provide one plausible explanation for the activation patterns observed in Figure 1,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 311, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 311, + 247 + ], + "score": 1.0, + "content": "with the hope of shedding light for future research.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 472, + 262 + ], + "score": 1.0, + "content": "Intriguingly, our analysis reveals an important insight that batch normalization (BatchNorm)", + "type": "text" + }, + { + "bbox": [ + 472, + 250, + 505, + 262 + ], + "score": 0.29, + "content": "[ [ 2 2 ] ] { - \\mathsf { a } }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "common technique employed during model training—is in fact both a blessing (for ID classification)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 408, + 285 + ], + "score": 1.0, + "content": "and a curse (for OOD detection). Specifically, for a unit activation denoted by", + "type": "text" + }, + { + "bbox": [ + 408, + 274, + 415, + 282 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 271, + 506, + 285 + ], + "score": 1.0, + "content": ", the network estimates", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 177, + 296 + ], + "score": 1.0, + "content": "the running mean", + "type": "text" + }, + { + "bbox": [ + 177, + 282, + 204, + 294 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathrm { i n } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 281, + 256, + 296 + ], + "score": 1.0, + "content": "and variance", + "type": "text" + }, + { + "bbox": [ + 257, + 283, + 290, + 294 + ], + "score": 0.91, + "content": "\\mathrm { V a r } _ { \\mathrm { i n } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 281, + 506, + 296 + ], + "score": 1.0, + "content": ", over the entire ID training set during training. During", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 331, + 306 + ], + "score": 1.0, + "content": "inference time, the network applies BatchNorm statistics", + "type": "text" + }, + { + "bbox": [ + 331, + 293, + 358, + 305 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathrm { i n } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 292, + 376, + 306 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 293, + 410, + 305 + ], + "score": 0.93, + "content": "\\mathrm { V a r } _ { \\mathrm { i n } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 292, + 505, + 306 + ], + "score": 1.0, + "content": ", which helps normalize", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 356, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 336, + 317 + ], + "score": 1.0, + "content": "the activations for the test data with the same distribution", + "type": "text" + }, + { + "bbox": [ + 337, + 305, + 351, + 315 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 302, + 356, + 317 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 322, + 403, + 351 + ], + "lines": [ + { + "bbox": [ + 207, + 322, + 403, + 351 + ], + "spans": [ + { + "bbox": [ + 207, + 322, + 403, + 351 + ], + "score": 0.92, + "content": "\\mathrm { B a t c h N o r m } ( z ; \\gamma , \\beta , \\epsilon ) = \\frac { z - \\mathbb { E } _ { \\mathrm { i n } } [ z ] } { \\sqrt { \\mathrm { V a r } _ { \\mathrm { i n } } [ z ] + \\epsilon } } \\cdot \\gamma + \\beta", + "type": "interline_equation", + "image_path": "b0099b7328cf980e9ce000affce9b7dce0fc72bba5cebd90e736f2bb13760ade.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 322, + 403, + 336.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 207, + 336.5, + 403, + 351.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 504, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 504, + 370 + ], + "score": 1.0, + "content": "However, our key observation is that using mismatched BatchNorm statistics—that are estimated on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 368, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 122, + 380 + ], + "score": 0.87, + "content": "\\mathcal { D } _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 368, + 247, + 382 + ], + "score": 1.0, + "content": "yet blindly applied to the OOD", + "type": "text" + }, + { + "bbox": [ + 248, + 369, + 266, + 379 + ], + "score": 0.89, + "content": "\\mathcal { D } _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 368, + 506, + 382 + ], + "score": 1.0, + "content": "—can trigger abnormally high unit activations. As a thought", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "experiment, we instead apply the true BatchNorm statistics estimated on a batch of OOD images and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "we observe well-behaved activation patterns with near-constant mean and standard deviations—just", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 310, + 415 + ], + "score": 1.0, + "content": "like the ones observed on the ID data (see Figure", + "type": "text" + }, + { + "bbox": [ + 311, + 400, + 321, + 413 + ], + "score": 0.62, + "content": "\\mathbb { E } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 399, + 506, + 415 + ], + "score": 1.0, + "content": "top). Our study therefore reveals one of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "fundamental causes for neural networks to produce overconfident predictions for OOD data. After", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "applying the true statistics (estimated on OOD), the output distributions between ID and OOD data", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "become much more separable. 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MethodiNaturalistPlacesSUNTextures
Oracle (batch OOD for estimating BN statistics)99.5999.0998.3291.43
ReAct (single OOD)96.2294.2091.5889.80
No ReAct B389.9585.8982.8685.99
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Model is trained on ImageNet (see Section 4.1)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 711, + 181, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 181, + 722 + ], + "score": 1.0, + "content": "Values are AUROC.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "index": 40.25 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 487, + 702, + 504, + 713 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "Why ReAct improves the OOD scoring functions? Our theoretical analysis above shows that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "ReAct suppresses logit output for OOD data more so than for ID data. This means that for detection", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 333, + 107 + ], + "score": 1.0, + "content": "scores depending on the logit output (e.g., energy score", + "type": "text" + }, + { + "bbox": [ + 333, + 93, + 352, + 105 + ], + "score": 0.83, + "content": "\\mathbb { \\lVert 3 3 \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "), the gap between OOD and ID score", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "will be enlarged after applying ReAct, which makes thresholding more capable of separating OOD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 115, + 432, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 432, + 128 + ], + "score": 1.0, + "content": "and ID inputs; see Figure 1(a) and (c) for a concrete example showing this effect.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 72, + 506, + 128 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 144, + 293, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 293, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 293, + 160 + ], + "score": 1.0, + "content": "6 Discussion and Further Analysis", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 170, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 506, + 182 + ], + "score": 1.0, + "content": "Why do OOD samples trigger abnormal unit activation patterns? 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Yet", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 202, + 506, + 215 + ], + "score": 1.0, + "content": "a question left in mystery is why such a pattern occurs in modern neural networks? Answering this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "question requires carefully examining the internal mechanism by which the network is trained and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 504, + 236 + ], + "score": 1.0, + "content": "evaluated. Here we provide one plausible explanation for the activation patterns observed in Figure 1,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 311, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 311, + 247 + ], + "score": 1.0, + "content": "with the hope of shedding light for future research.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 170, + 506, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 472, + 262 + ], + "score": 1.0, + "content": "Intriguingly, our analysis reveals an important insight that batch normalization (BatchNorm)", + "type": "text" + }, + { + "bbox": [ + 472, + 250, + 505, + 262 + ], + "score": 0.29, + "content": "[ [ 2 2 ] ] { - \\mathsf { a } }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "common technique employed during model training—is in fact both a blessing (for ID classification)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 408, + 285 + ], + "score": 1.0, + "content": "and a curse (for OOD detection). 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MethodiNaturalistPlacesSUNTextures
Oracle (batch OOD for estimating BN statistics)99.5999.0998.3291.43
ReAct (single OOD)96.2294.2091.5889.80
No ReAct B389.9585.8982.8685.99
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MethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95 ↓AUROC 个FPR95 ↓AUROC 个FPR95AUROCFPR95AUROC 个FPR95 ↓AUROC 个
GroupNormw.o.ReAct65.3888.4565.1185.52↓ 65.46个 84.34↓ 69.1783.2266.2885.38
w/ReAct39.4592.9551.5787.9052.7887.3262.5081.7651.5887.48
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w/ReAct19.7395.9131.3993.2142.3488.9413.7496.9826.8093.76
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The problem of classification with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "rejection can date back to early works on abstention [4, 10], which considered simple model families", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 164, + 550 + ], + "score": 1.0, + "content": "such as SVMs", + "type": "text" + }, + { + "bbox": [ + 164, + 538, + 176, + 550 + ], + "score": 0.39, + "content": "\\pmb { \\Vert 6 \\Vert }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 538, + 413, + 550 + ], + "score": 1.0, + "content": ". A comprehensive survey on OOD detection can be found in", + "type": "text" + }, + { + "bbox": [ + 413, + 538, + 430, + 549 + ], + "score": 0.64, + "content": "\\left[ \\left[ 5 8 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 538, + 505, + 550 + ], + "score": 1.0, + "content": ". The phenomenon", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "of neural networks’ overconfidence in out-of-distribution data is first revealed by Nguyen et al. [38].", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 507, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 507, + 572 + ], + "score": 1.0, + "content": "Early works attempted to improve the OOD uncertainty estimation by proposing the ODIN score [17,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 568, + 504, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 190, + 583 + ], + "score": 1.0, + "content": "31], OpenMax score", + "type": "text" + }, + { + "bbox": [ + 190, + 570, + 203, + 581 + ], + "score": 0.74, + "content": "\\pmb { \\mathbb { B } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 568, + 361, + 583 + ], + "score": 1.0, + "content": ", and Mahalanobis distance-based score", + "type": "text" + }, + { + "bbox": [ + 361, + 570, + 379, + 582 + ], + "score": 0.49, + "content": "\\pmb { \\mathbb { Z } } \\pmb { \\mathcal { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 568, + 504, + 583 + ], + "score": 1.0, + "content": ". Recent work by Liu et al. [33]", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 581, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 593 + ], + "score": 1.0, + "content": "proposed using an energy score for OOD uncertainty estimation, which can be easily derived from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "a discriminative classifier and demonstrated advantages over the softmax confidence score both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "score": 1.0, + "content": "empirically and theoretically. Wang et al. 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To answer this", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 496, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 390, + 161 + ], + "score": 1.0, + "content": "question, we additionally evaluate networks trained with WeightNorm", + "type": "text" + }, + { + "bbox": [ + 390, + 149, + 408, + 160 + ], + "score": 0.78, + "content": "\\underline { { \\hat { \\vert 4 3 \\vert } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 149, + 496, + 161 + ], + "score": 1.0, + "content": "and GroupNorm [54]—", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 368, + 173 + ], + "score": 1.0, + "content": "two other well-known normalization methods. As shown in Figure", + "type": "text" + }, + { + "bbox": [ + 369, + 159, + 379, + 172 + ], + "score": 0.74, + "content": "\\triangledown", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "the unit activations also display", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "highly distinctive signature patterns between ID and OOD data, with more chaos on OOD data. In all", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "cases, the networks are trained to adapt to the ID data, resulting in abnormal activation signatures on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "OOD data in testing. Unlike BatchNorm, there is no easy oracle (e.g., re-estimating the statistics on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 202, + 320, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 320, + 214 + ], + "score": 1.0, + "content": "OOD data) to counteract the ill-fated normalizations.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 117, + 506, + 214 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 218, + 504, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 504, + 231 + ], + "score": 1.0, + "content": "We apply ReAct on models trained with WeightNorm and GroupNorm, and report results in Table 5.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 229, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 242 + ], + "score": 1.0, + "content": "Our results suggest that ReAct is consistently effective under various normalization schemes. For", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 295, + 252 + ], + "score": 1.0, + "content": "example, ReAct reduces the average FPR95 by", + "type": "text" + }, + { + "bbox": [ + 296, + 240, + 328, + 250 + ], + "score": 0.89, + "content": "2 3 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 240, + 345, + 252 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 346, + 240, + 373, + 250 + ], + "score": 0.86, + "content": "1 4 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "respectively. 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MethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95 ↓AUROC 个FPR95 ↓AUROC 个FPR95AUROCFPR95AUROC 个FPR95 ↓AUROC 个
GroupNormw.o.ReAct65.3888.4565.1185.52↓ 65.46个 84.34↓ 69.1783.2266.2885.38
w/ReAct39.4592.9551.5787.9052.7887.3262.5081.7651.5887.48
WeightNormw.o. ReAct40.7192.5248.0789.3950.9287.8761.6580.7150.3487.62
w/ReAct19.7395.9131.3993.2142.3488.9413.7496.9826.8093.76
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The problem of classification with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "rejection can date back to early works on abstention [4, 10], which considered simple model families", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 164, + 550 + ], + "score": 1.0, + "content": "such as SVMs", + "type": "text" + }, + { + "bbox": [ + 164, + 538, + 176, + 550 + ], + "score": 0.39, + "content": "\\pmb { \\Vert 6 \\Vert }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 538, + 413, + 550 + ], + "score": 1.0, + "content": ". A comprehensive survey on OOD detection can be found in", + "type": "text" + }, + { + "bbox": [ + 413, + 538, + 430, + 549 + ], + "score": 0.64, + "content": "\\left[ \\left[ 5 8 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 538, + 505, + 550 + ], + "score": 1.0, + "content": ". The phenomenon", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "of neural networks’ overconfidence in out-of-distribution data is first revealed by Nguyen et al. [38].", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 507, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 507, + 572 + ], + "score": 1.0, + "content": "Early works attempted to improve the OOD uncertainty estimation by proposing the ODIN score [17,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 568, + 504, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 190, + 583 + ], + "score": 1.0, + "content": "31], OpenMax score", + "type": "text" + }, + { + "bbox": [ + 190, + 570, + 203, + 581 + ], + "score": 0.74, + "content": "\\pmb { \\mathbb { B } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 568, + 361, + 583 + ], + "score": 1.0, + "content": ", and Mahalanobis distance-based score", + "type": "text" + }, + { + "bbox": [ + 361, + 570, + 379, + 582 + ], + "score": 0.49, + "content": "\\pmb { \\mathbb { Z } } \\pmb { \\mathcal { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 568, + 504, + 583 + ], + "score": 1.0, + "content": ". Recent work by Liu et al. [33]", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 581, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 593 + ], + "score": 1.0, + "content": "proposed using an energy score for OOD uncertainty estimation, which can be easily derived from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "a discriminative classifier and demonstrated advantages over the softmax confidence score both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 614 + ], + "score": 1.0, + "content": "empirically and theoretically. Wang et al. [51] further showed an energy-based approach can improve", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "OOD uncertainty estimation for multi-label classification networks. Huang and Li [18] revealed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "that approaches developed for common CIFAR benchmarks might not translate effectively into a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "large-scale ImageNet benchmark, highlighting the need to evaluate OOD uncertainty estimation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "in a large-scale real-world setting. Previous methods primarily derive OOD scores using original", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "activations. In contrast, our work is motivated by a novel analysis of internal activations, and shows", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 432, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 432, + 679 + ], + "score": 1.0, + "content": "that rectifying activations is a surprisingly effective approach for OOD detection.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 516, + 507, + 679 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 690, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "Neural network activation analysis. Neural networks have been studied at the granularity of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 411, + 712 + ], + "score": 1.0, + "content": "the activation of individual layers [21, 34, 59, 63], or individual networks", + "type": "text" + }, + { + "bbox": [ + 411, + 700, + 429, + 711 + ], + "score": 0.7, + "content": "\\textcircled { \\lvert 3 0 \\rvert }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 699, + 492, + 712 + ], + "score": 1.0, + "content": ". In particular,", + "type": "text" + }, + { + "bbox": [ + 493, + 701, + 506, + 712 + ], + "score": 1.0, + "content": "Li", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 507, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 507, + 724 + ], + "score": 1.0, + "content": "et al. 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We show that using ReAct could efficiently", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 504, + 105 + ], + "score": 1.0, + "content": "alleviate this undesirable phenomenon. ReAct does not rely on auxiliary data and can be conveniently", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 377, + 117 + ], + "score": 1.0, + "content": "used for pre-trained models. The idea of rectifying unit activation", + "type": "text" + }, + { + "bbox": [ + 377, + 105, + 395, + 116 + ], + "score": 0.6, + "content": "\\dot { \\mathbb { Z } } \\dot { 6 } \\mathbb { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 105, + 506, + 117 + ], + "score": 1.0, + "content": ", which is known as Relu6,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "was used to facilitate the learning of sparse features. In this paper, we first show that rectifying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "activation can drastically alleviate the overconfidence issue for OOD data, and as a result, improve", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 136, + 172, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 172, + 148 + ], + "score": 1.0, + "content": "OOD detection.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 347, + 172 + ], + "score": 1.0, + "content": "OOD uncertainty estimation with generative models.", + "type": "text" + }, + { + "bbox": [ + 352, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Alternative approaches for detecting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "OOD inputs resort to generative models that directly estimate density [8, 20, 23, 41, 49, 50]. An", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "input is deemed as OOD if it lies in the low-likelihood regions. A plethora of literature has emerged", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 190, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 505, + 205 + ], + "score": 1.0, + "content": "to utilize generative models for OOD detection [24, 40, 45, 46, 52, 56]. Interestingly, Nalisnick", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 507, + 215 + ], + "score": 1.0, + "content": "et al. [36] showed that deep generative models can assign a high likelihood to OOD data. Moreover,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "generative models can be prohibitively challenging to train and optimize, and the performance can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "often lag behind the discriminative counterpart. In contrast, our method relies on a discriminative", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 235, + 399, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 399, + 246 + ], + "score": 1.0, + "content": "classifier, which is easier to optimize and achieves stronger performance.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 464, + 270 + ], + "score": 1.0, + "content": "Distributional shifts. Distributional shifts have attracted increasing research interests", + "type": "text" + }, + { + "bbox": [ + 464, + 257, + 482, + 268 + ], + "score": 0.76, + "content": "\\mathbb { \\left[ \\left[ 2 5 \\right] \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 256, + 505, + 270 + ], + "score": 1.0, + "content": ". It is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "important to recognize and differentiate various types of distributional shift problems. Literature in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 279, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 290 + ], + "score": 1.0, + "content": "OOD detection is commonly concerned about model reliability and detection of label-space shifts,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "where the OOD inputs have disjoint labels w.r.t. 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Meanwhile, some works considered label distribution shift [1, 2, 32, 42, 47, 53], where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "the label space is common between ID and OOD but the marginal label distribution changes, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "well as covariate shift in the input space [14, 39], where inputs can be corruption-shifted or domain-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 135, + 344 + ], + "score": 1.0, + "content": "shifted", + "type": "text" + }, + { + "bbox": [ + 136, + 331, + 168, + 343 + ], + "score": 0.66, + "content": " { \\mathbb { I } } { \\mathrm { I Z } } , { \\mathbb { H } } { \\mathrm { 8 l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 331, + 506, + 344 + ], + "score": 1.0, + "content": ". It is important to note that our work focuses on the detection of shifts where the label", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 343, + 507, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 130, + 356 + ], + "score": 1.0, + "content": "space", + "type": "text" + }, + { + "bbox": [ + 131, + 344, + 140, + 354 + ], + "score": 0.77, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 343, + 507, + 356 + ], + "score": 1.0, + "content": "is different between ID and OOD data and hence the model should not make any prediction,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 353, + 486, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 486, + 366 + ], + "score": 1.0, + "content": "instead of label distribution shift and covariate shift where the model is expected to generalize.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 183, + 393 + ], + "lines": [ + { + "bbox": [ + 104, + 378, + 184, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 184, + 396 + ], + "score": 1.0, + "content": "8 Conclusion", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "This paper provides a simple activation rectification strategy termed ReAct, which truncates the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "score": 1.0, + "content": "high activations during test time for OOD detection. We provide both empirical and theoretical", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "insights characterizing and explaining the mechanism by which ReAct improves OOD uncertainty", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "estimation. By rectifying the activations, the outsized contribution of hidden units on OOD output", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "can be attenuated, resulting in a stronger separability from ID data. Extensive experiments show", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "ReAct can significantly improve the performance of OOD detection on both common benchmarks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "and large-scale image classification models. Applications of multi-class classification can benefit", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "from our method, and we anticipate further research in OOD detection to extend this work to tasks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "beyond image classification. We hope that our insights inspire future research to further examine the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 500, + 348, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 348, + 513 + ], + "score": 1.0, + "content": "internal mechanisms of neural networks for OOD detection.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 107, + 527, + 205, + 541 + ], + "lines": [ + { + "bbox": [ + 104, + 525, + 207, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 207, + 544 + ], + "score": 1.0, + "content": "9 Societal Impact", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "score": 1.0, + "content": "Our project aims to improve the dependability and trustworthiness of modern machine learning", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "score": 1.0, + "content": "models. This stands to benefit a wide range of fields and societal activities. We believe out-of-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "distribution uncertainty estimation is an increasingly critical component of systems that range from", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "consumer and business applications (e.g., digital content understanding) to transportation (e.g., driver", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "assistance systems and autonomous vehicles), and to health care (e.g., unseen disease identification).", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "Many of these applications require classification models in operation. Through this work and by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "releasing our code, we hope to provide machine learning researchers with a new methodological", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "perspective and offer machine learning practitioners a plug-and-play tool that renders safety against", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "OOD data in the real world. While we do not anticipate any negative consequences to our work, we", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 331, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 331, + 660 + ], + "score": 1.0, + "content": "hope to continue to build on our method in future work.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 108, + 675, + 202, + 688 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 204, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 204, + 690 + ], + "score": 1.0, + "content": "Acknowledgement", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 721 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "YS and YL are supported by the Office of the Vice Chancellor for Research and Graduate Education", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 449, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 449, + 723 + ], + "score": 1.0, + "content": "(OVCRGE) with funding from the Wisconsin Alumni Research Foundation (WARF).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 154, + 86 + ], + "score": 1.0, + "content": "Previously,", + "type": "text" + }, + { + "bbox": [ + 154, + 72, + 217, + 85 + ], + "score": 0.4, + "content": "\\underline { { \\mathrm { f l e i n ~ e t ~ a l . } } } \\underline { { \\mathbb { ( } \\mathbb { 1 3 } } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "showed that neural networks with ReLU activation can lead to arbitrary", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "high activation for inputs far away from the training data. We show that using ReAct could efficiently", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 504, + 105 + ], + "score": 1.0, + "content": "alleviate this undesirable phenomenon. ReAct does not rely on auxiliary data and can be conveniently", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 377, + 117 + ], + "score": 1.0, + "content": "used for pre-trained models. The idea of rectifying unit activation", + "type": "text" + }, + { + "bbox": [ + 377, + 105, + 395, + 116 + ], + "score": 0.6, + "content": "\\dot { \\mathbb { Z } } \\dot { 6 } \\mathbb { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 105, + 506, + 117 + ], + "score": 1.0, + "content": ", which is known as Relu6,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "was used to facilitate the learning of sparse features. In this paper, we first show that rectifying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "activation can drastically alleviate the overconfidence issue for OOD data, and as a result, improve", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 136, + 172, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 172, + 148 + ], + "score": 1.0, + "content": "OOD detection.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 72, + 506, + 148 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 347, + 172 + ], + "score": 1.0, + "content": "OOD uncertainty estimation with generative models.", + "type": "text" + }, + { + "bbox": [ + 352, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Alternative approaches for detecting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "OOD inputs resort to generative models that directly estimate density [8, 20, 23, 41, 49, 50]. An", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "input is deemed as OOD if it lies in the low-likelihood regions. A plethora of literature has emerged", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 190, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 505, + 205 + ], + "score": 1.0, + "content": "to utilize generative models for OOD detection [24, 40, 45, 46, 52, 56]. Interestingly, Nalisnick", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 507, + 215 + ], + "score": 1.0, + "content": "et al. [36] showed that deep generative models can assign a high likelihood to OOD data. Moreover,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "generative models can be prohibitively challenging to train and optimize, and the performance can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "often lag behind the discriminative counterpart. 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Distributional shifts have attracted increasing research interests", + "type": "text" + }, + { + "bbox": [ + 464, + 257, + 482, + 268 + ], + "score": 0.76, + "content": "\\mathbb { \\left[ \\left[ 2 5 \\right] \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 256, + 505, + 270 + ], + "score": 1.0, + "content": ". It is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "important to recognize and differentiate various types of distributional shift problems. Literature in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 279, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 290 + ], + "score": 1.0, + "content": "OOD detection is commonly concerned about model reliability and detection of label-space shifts,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "where the OOD inputs have disjoint labels w.r.t. ID data and therefore should not be predicted by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "the model. Meanwhile, some works considered label distribution shift [1, 2, 32, 42, 47, 53], where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "the label space is common between ID and OOD but the marginal label distribution changes, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "well as covariate shift in the input space [14, 39], where inputs can be corruption-shifted or domain-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 135, + 344 + ], + "score": 1.0, + "content": "shifted", + "type": "text" + }, + { + "bbox": [ + 136, + 331, + 168, + 343 + ], + "score": 0.66, + "content": " { \\mathbb { I } } { \\mathrm { I Z } } , { \\mathbb { H } } { \\mathrm { 8 l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 331, + 506, + 344 + ], + "score": 1.0, + "content": ". It is important to note that our work focuses on the detection of shifts where the label", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 343, + 507, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 130, + 356 + ], + "score": 1.0, + "content": "space", + "type": "text" + }, + { + "bbox": [ + 131, + 344, + 140, + 354 + ], + "score": 0.77, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 343, + 507, + 356 + ], + "score": 1.0, + "content": "is different between ID and OOD data and hence the model should not make any prediction,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 353, + 486, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 486, + 366 + ], + "score": 1.0, + "content": "instead of label distribution shift and covariate shift where the model is expected to generalize.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 256, + 507, + 366 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 183, + 393 + ], + "lines": [ + { + "bbox": [ + 104, + 378, + 184, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 184, + 396 + ], + "score": 1.0, + "content": "8 Conclusion", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "This paper provides a simple activation rectification strategy termed ReAct, which truncates the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "score": 1.0, + "content": "high activations during test time for OOD detection. We provide both empirical and theoretical", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "insights characterizing and explaining the mechanism by which ReAct improves OOD uncertainty", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "estimation. By rectifying the activations, the outsized contribution of hidden units on OOD output", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "can be attenuated, resulting in a stronger separability from ID data. Extensive experiments show", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "ReAct can significantly improve the performance of OOD detection on both common benchmarks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "and large-scale image classification models. Applications of multi-class classification can benefit", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "from our method, and we anticipate further research in OOD detection to extend this work to tasks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "beyond image classification. We hope that our insights inspire future research to further examine the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 500, + 348, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 348, + 513 + ], + "score": 1.0, + "content": "internal mechanisms of neural networks for OOD detection.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 405, + 506, + 513 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 527, + 205, + 541 + ], + "lines": [ + { + "bbox": [ + 104, + 525, + 207, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 207, + 544 + ], + "score": 1.0, + "content": "9 Societal Impact", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "score": 1.0, + "content": "Our project aims to improve the dependability and trustworthiness of modern machine learning", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "score": 1.0, + "content": "models. This stands to benefit a wide range of fields and societal activities. We believe out-of-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "distribution uncertainty estimation is an increasingly critical component of systems that range from", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "consumer and business applications (e.g., digital content understanding) to transportation (e.g., driver", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "assistance systems and autonomous vehicles), and to health care (e.g., unseen disease identification).", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "Many of these applications require classification models in operation. Through this work and by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "releasing our code, we hope to provide machine learning researchers with a new methodological", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "perspective and offer machine learning practitioners a plug-and-play tool that renders safety against", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "OOD data in the real world. While we do not anticipate any negative consequences to our work, we", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 331, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 331, + 660 + ], + "score": 1.0, + "content": "hope to continue to build on our method in future work.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 550, + 506, + 660 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 675, + 202, + 688 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 204, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 204, + 690 + ], + "score": 1.0, + "content": "Acknowledgement", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 721 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "YS and YL are supported by the Office of the Vice Chancellor for Research and Graduate Education", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 449, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 449, + 723 + ], + "score": 1.0, + "content": "(OVCRGE) with funding from the Wisconsin Alumni Research Foundation (WARF).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 106, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 62, + 507, + 729 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 109, + 90, + 507, + 104 + ], + "spans": [ + { + "bbox": [ + 109, + 90, + 507, + 104 + ], + "score": 1.0, + "content": "[1] Amr Alexandari, Anshul Kundaje, and Avanti Shrikumar. 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ModelMethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95AUROCFPR95AUROC 个FPR95AUROCFPR95AUROCFPR95 AUROC
ResNetMSP 园↓ 54.99→ 87.74√ 70.8380.86↓ 73.99个 79.76↓ 68.00↑ 79.61√ 66.95→ 81.99
ODIN 白47.6689.6660.1584.5967.8981.7850.2385.6256.4885.41
Mahalanobis [ 2997.0052.6598.5042.4198.4041.7955.8085.0187.4355.47
Energy 国55.7289.9559.2685.8964.9282.8653.7285.9958.4186.17
ReAct (Ours)20.3896.2224.2094.2033.8591.5847.3089.8031.4392.95
77.0279.2376.2777.3073.5179.00
MobileNetMSP 固64.2985.32 87.6254.0777.10 85.8873.5185.0354.2085.81
ODIN 白 四55.39 62.1181.0047.8286.3357.36 52.0984.71 83.6349.96 92.3833.0663.6071.01
Mahalanobis59.5088.9162.6584.5069.3781.1958.0585.0362.3984.91
Energy 国 ReAct (Ours)42.4091.5347.6988.1651.5686.6438.4291.5345.0289.47
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Rectification percentileFPR95 √AUROC →AUPR 个ID ACC. 个Rectification threshold c
No ReAct 358.4186.1796.8875.088
p=9944.5790.4597.9675.122.25
p=9535.3992.3998.3774.761.50
p= 9031.4392.9598.5073.751.00
p=8534.0892.0598.3572.910.84
p=8041.5189.5497.9171.930.72
p=6574.6274.1494.3967.140.50
p=1074.7057.5586.061.220.06
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MethodCIFAR-10CIFAR-100ImageNet
FPR95 √AUROC →AUPR →FPR95 ←AUROC →AUPR 个FPR95 ←AUROC →AUPR →
Softmax score 园56.7191.1779.1180.7276.8378.4166.9581.9995.76
Softmax score + ReAct53.8191.7092.1175.4580.4084.2858.2887.0697.22
Energy B335.6093.5795.0171.9382.8286.2858.4186.1796.88
Energy+ReAct32.9194.2795.5359.6187.4889.63 31.4392.9598.50
ODIN B131.1093.7994.9566.2182.8886.2556.4885.4196.61
ODIN+ReAct28.8194.0494.8259.9185.2387.5344.1090.7098.04
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MethodiNaturalistPlacesSUNTextures
Oracle (batch OOD for estimating BN statistics)99.5999.0998.3291.43
ReAct (single OOD)96.2294.2091.5889.80
No ReAct B389.9585.8982.8685.99
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MethodsOODDatasetsAverage
iNaturalistSUNPlacesTextures
FPR95 ↓AUROC 个FPR95 ↓AUROC 个FPR95AUROCFPR95AUROC 个FPR95 ↓AUROC 个
GroupNormw.o.ReAct65.3888.4565.1185.52↓ 65.46个 84.34↓ 69.1783.2266.2885.38
w/ReAct39.4592.9551.5787.9052.7887.3262.5081.7651.5887.48
WeightNormw.o. ReAct40.7192.5248.0789.3950.9287.8761.6580.7150.3487.62
w/ReAct19.7395.9131.3993.2142.3488.9413.7496.9826.8093.76
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sha256:95cdf65cb82f3f8973e991d7eb601de93df461acfa6319aab242b29e4ba2f0e7 +size 2651 diff --git a/parse/train/aLE2sEtMNXv/aLE2sEtMNXv.md b/parse/train/aLE2sEtMNXv/aLE2sEtMNXv.md new file mode 100644 index 0000000000000000000000000000000000000000..a41d93807cce8a203dfd21a21758ff3e4bf73bb3 --- /dev/null +++ b/parse/train/aLE2sEtMNXv/aLE2sEtMNXv.md @@ -0,0 +1,305 @@ +# Sparse Spiking Gradient Descent + +Nicolas Perez-Nieves + +Dan F.M. Goodman + +Electrical and Electronic Engineering Imperial College London London, United Kingdom +nicolas.perez14@imperial.ac.uk + +Electrical and Electronic Engineering Imperial College London London, United Kingdom d.goodman@imperial.ac.uk + +# Abstract + +There is an increasing interest in emulating Spiking Neural Networks (SNNs) on neuromorphic computing devices due to their low energy consumption. Recent advances have allowed training SNNs to a point where they start to compete with traditional Artificial Neural Networks (ANNs) in terms of accuracy, while at the same time being energy efficient when run on neuromorphic hardware. However, the process of training SNNs is still based on dense tensor operations originally developed for ANNs which do not leverage the spatiotemporally sparse nature of SNNs. We present here the first sparse SNN backpropagation algorithm which achieves the same or better accuracy as current state of the art methods while being significantly faster and more memory efficient. We show the effectiveness of our method on real datasets of varying complexity (Fashion-MNIST, NeuromophicMNIST and Spiking Heidelberg Digits) achieving a speedup in the backward pass of up to 150x, and ${ \bar { 8 } } 5 \%$ more memory efficient, without losing accuracy. + +# 1 Introduction + +In recent years, deep artificial neural networks (ANNs) have matched and occasionally surpassed human-level performance on increasingly difficult auditory and visual recognition problems [1, 2, 3], natural language processing tasks [4, 5] and games [6, 7, 8]. As these tasks become more challenging the neural networks required to solve them grow larger and consequently their power efficiency becomes more important [9, 10]. At the same time, the increasing interest in deploying these models into embedded applications calls for faster, more efficient and less memory intensive networks [11, 12]. + +The human brain manages to perform similar and even more complicated tasks while only consuming about 20W [13] which contrasts with the hundreds of watts required for running ANNs [9]. Unlike ANNs, biological neurons in the brain communicate through discrete events called spikes. A biological neuron integrates incoming spikes from other neurons in its membrane potential and after reaching a threshold emits a spike and resets its potential [14]. The spiking neuron, combined with the sparse connectivity of the brain, results in a highly spatio-temporally sparse activity [15] which is fundamental to achieve this level of energy efficiency. + +Inspired by the extraordinary performance of the brain, neuromorphic computing aims to obtain the same level of energy efficiency preferably while maintaining an accuracy standard on par with ANNs by emulating spiking neural networks (SNNs) on specialised hardware [16, 17, 18, 19]. These efforts go in two directions: emulating the SNNs and training the SNNs. While there is a growing number of successful neuromorphic implementations for the former [20, 21, 22, 23, 24, 17, 25, 26], the latter has proven to be more challenging. Some neuromorphic chips implement local learning rules [26, 27, 28] and recent advances have achieved to approximate backpropagation on-chip [29]. However, a full end-to-end supervised learning via error backpropagation requires off-chip training [22, 23, 18]. This is usually achieved by simulating and training the entire SNN on a GPU and more recently, emulating the forward pass of the SNN on neuromorphic hardware and then performing the backward pass on a GPU [30, 31]. The great flexibility provided by GPUs allows the development of complex training pipelines without being constrained by neuromorphic hardware specifications. However, as GPUs do not leverage the event-driven nature of spiking computations this results in effective but slower and less energy efficient training. An alternative to directly training an SNN consists on training an ANN and converting it to an SNN. However, this approach has been recently shown to not be more energy efficient than using the original ANN in the first place [32]. + +![](images/c2f88790fb9b9bb6397992ebd50b7df3941478c5a0febacaf88e507750f33a24.jpg) +Figure 1: Spiking Neural Network diagram highlighting the forward and backward passes + +Training SNNs has been a challenge itself even without considering a power budget due to the all-or-none nature of spikes which hinders traditional gradient descent methods [33, 34]. Since the functional form of a spike is a unit impulse, it has a zero derivative for all membrane potentials except at the threshold where it is infinity. Recently developed methods based on surrogate gradients have been shown to be capable of solving complex tasks to near state-of-the-art accuracies [35, 18, 36, 37, 38, 39, 31]. The problem of non-differentiability is solved by adopting a well-behaved surrogate gradient when backpropagating the error [33]. This method, while effective for training very accurate models, is still constrained to working on dense tensors, thus, not profiting from the sparse nature of spiking, and limits the training speed and power efficiency when training SNNs. + +In this work we introduce a sparse backpropagation method for SNNs. Recent work on ANNs has shown that adaptive dropout [40] and adaptive sparsity [41] can be used to achieve state-of-the-art performance at a fraction of the computational cost [42, 43]. We show that by redefining the surrogate gradient functional form, a sparse learning rule for SNNs arises naturally as a three-factor Hebbianlike learning rule. We empirically show an improved backpropagation time up to 70x faster than current implementations and up to $4 0 \%$ more memory efficient in real datasets (Fashion-MNIST (FMNIST) [44], Neuromorphic-MNIST (N-MNIST) [45] and Spiking Heidelberg Dataset (SHD) [46]) thus reducing the computational gap between the forward and backward passes. This improvement will not only impact training for neuromorphic computing applications but also for computational neuroscience research involving training on SNNs [39]. + +# 2 Sparse Spike Backpropagation + +# 2.1 Forward Model + +We begin by introducing the spiking networks we are going to work with (Figs. 1 and 2). For simplicity we have omitted the batch dimension in the derivations but it is later used on the complexity analysis and experiments. We have a total of $L$ fully connected spiking layers. Each layer $l$ consisting of $N ^ { ( l ) }$ spiking neurons which are fully connected to the next layer $l + 1$ through synaptic weights $W ^ { ( l ) } \in \mathbb { R } ^ { N ^ { ( l ) } \times N ^ { ( l + 1 ) } }$ . Each neuron has an internal state variable, the membrane potential, $V _ { j } ^ { ( l ) } [ t ] \in \mathbb { R }$ that updates every discrete time step $t \in \{ 0 , \ldots , T - 1 \}$ for some finite simulation time $T \in \mathbb { N }$ . Neurons emit spikes according to a spiking function $f \colon { \mathbb { R } } \to \{ 0 , 1 \}$ such that + +$$ +S _ { i } ^ { ( l ) } [ t ] = f ( V _ { i } ^ { ( l ) } [ t ] ) +$$ + +The membrane potential varies according to a simplified Leaky Integrate and Fire (LIF) neuron model in which input spikes are directly integrated into the membrane [14]. After a neuron spikes, its potential is reset to $V _ { r }$ . We will work with a discretised version of this model (see Appendix $\mathrm { E }$ for a continuous time version). The membrane potential of neuron $j$ in layer $l { + } 1$ evolves according to the following difference equation + +$$ +V _ { j } ^ { ( l + 1 ) } [ t + 1 ] = \alpha ( V _ { j } ^ { ( l + 1 ) } [ t ] - V _ { r e s t } ) + V _ { r e s t } + \sum _ { i } S _ { i } ^ { ( l ) } [ t ] W _ { i j } ^ { ( l ) } - ( V _ { t h } - V _ { r } ) S _ { j } ^ { ( l + 1 ) } [ t ] +$$ + +![](images/b74f25e41587a2733ae554635e8ebc612d9c441c1440cfb9fc57e0223683bda6.jpg) +Figure 2: Illustration of the gradient backpropagating only through active neurons. + +The first two terms account for the leaky part of the model where we define $\alpha = \exp ( - \Delta t / \tau )$ with $\Delta t , \tau \in \mathbb { R }$ being the time resolution of the simulation and the membrane time constant of the neurons respectively. The second term deals with the integration of spikes. The last term models the resetting mechanism by subtracting the distance from the threshold to the reset potential. For simplicity and without loss of generality, we consider $V _ { r e s t } { = } 0 , V _ { t h } { = } 1 , V _ { r } { = } 0$ and $V _ { i } ^ { ( l ) } [ 0 ] { = } 0 \forall l , i$ from now on. + +We can unroll (2) to obtain + +$$ +V _ { j } ^ { ( l + 1 ) } [ t + 1 ] = \underbrace { \sum _ { i } \sum _ { k = 0 } ^ { t } \alpha ^ { t - k } S _ { i } ^ { ( l ) } [ k ] } _ { \mathrm { I n p u t ~ t r a c e } } W _ { i j } ^ { ( l ) } - \underbrace { \sum _ { k = 0 } ^ { t } \alpha ^ { t - k } S _ { j } ^ { ( l + 1 ) } [ k ] } _ { \mathrm { R e s e t t i n g ~ t e r m } } +$$ + +Thus, a spiking neural network can be viewed as a special type of recurrent neural network where the activation function of each neuron is the spiking function $f ( \cdot )$ . The spiking function, is commonly defined as the unit step function centered at a particular threshold $V _ { t h }$ + +$$ +f ( v ) = { \left\{ \begin{array} { l l } { 1 , \quad } & { v > V _ { t h } } \\ { 0 , } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } +$$ + +Note that while this function is easy to compute in the forward pass, its derivative, the unit impulse function, is problematic for backpropagation which has resulted in adopting surrogate derivatives [33] to allow the gradient to flow. Interestingly, it has been shown that surrogate gradient descent performs robustly for a wide range of surrogate functional forms [38]. + +After the final layer, a loss function $l o s s ( \cdot )$ computes how far the network activity is with respect to some target value for the given input. The loss function can be defined as a function of the network spikes, membrane potentials or both. It may also be evaluated at every single time step or every several steps. We deliberately leave the particular definition of this function open as it does not directly affect the sparse gradient descent method we introduce. + +# 2.2 Backward Model + +The SNN is updated by computing the gradient of the loss function with respect to the weights in each layer. This can be achieved using backpropagation through time (BPTT) on the unrolled network. Following (1) and (3) we can derive weight gradients to be + +$$ +\nabla W _ { i j } ^ { ( l ) } = \sum _ { t } \underbrace { \varepsilon _ { j } ^ { ( l + 1 ) } [ t ] } _ { \substack { \sum _ { t } \dots \dots \dots } } \overbrace { d V _ { j } ^ { ( l + 1 ) } [ t ] } ^ { \substack { \sum _ { j } ( l + 1 ) } } \left( \sum _ { k < t } \alpha ^ { t - k - 1 } S _ { i } ^ { ( l ) } [ k ] \right) +$$ + +This gradient consists of a sum over all time steps of the product of three terms. The first term, modulates the spatio-temporal credit assignment of the weights by propagating the gradient from the next layer. The second term, is the derivative of $f ( \cdot )$ evaluated at a given $V _ { j } [ t ]$ . The last term is a filtered version of the input spikes. Notice that it is not necessary to compute this last term as this is already computed in the forward pass (3). We have chosen not make the spike resetting term in (3) differentiable as it has been shown that doing so can result in worse testing accuracy [38] and it increases the computational cost. + +For a typical loss function defined on the output layer activity, the value of $\varepsilon _ { j _ { - } } ^ { ( L - 1 ) } [ t ]$ (i.e. in the last layer) is the gradient of the loss with respect to the output layer (see Fig. 1). For all other layers we have that $\varepsilon _ { j } ^ { ( l ) } [ t ] = \nabla S _ { j } ^ { ( l ) } [ t ]$ , that is, the gradient of the loss function with respect to the output spikes of layer $\it l$ . We can obtain an expression for $\nabla S _ { j } ^ { ( l ) } [ t ]$ as a function of the gradient of the next layer: + +$$ +\begin{array} { l } { \varepsilon _ { i } ^ { ( l ) } [ t ] = \nabla S _ { i } ^ { ( l ) } [ t ] = \displaystyle \sum _ { j } W _ { i j } \left( \sum _ { k > t } \overbrace { \nabla S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } } \overbrace { d V _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } \alpha ^ { k - t - 1 } \right) } \\ { = \displaystyle \sum _ { j } W _ { i j } \delta _ { j } ^ { ( l + 1 ) } [ t ] , \qquad l = \{ 0 , \ldots L - 2 \} } \end{array} +$$ + +All hidden layers need to compute equations (5) and (6). The first one to update their input synaptic weights and the second one to backpropagate the loss to the previous layer. The first layer $\mathit { l } = 1 \mathit { \Theta }$ ) only needs to compute (5) and the last layer $\scriptstyle ( l = L - 1$ ) will compute a different $\varepsilon _ { j } ^ { ( L - 1 ) } [ t ]$ depending on the loss function. + +We introduce the following definitions which result in the sparse backpropagation learning rule. + +Definition 1. Given a backpropagation threshold $B _ { t h } \in \mathbb { R }$ we say neuron $j$ is active at time $t$ iff + +$$ +| V _ { j } [ t ] - V _ { t h } | < B _ { t h } +$$ + +Definition 2. The spike gradient is defined as + +$$ +\begin{array} { r } { : = \left\{ \begin{array} { l l } { g ( V _ { j } [ t ] ) , \quad } & { i f V _ { j } [ t ] i s a c t i \nu e } \\ { 0 , \quad } & { o t h e r w i s e } \end{array} \right. } \end{array} +$$ + +This means that neurons are only active when their potential is close to the threshold. Applying these two definitions to (5) and (6) results in the gradients only backpropagating through active neurons at each time step as shown in Fig. 2. The consequences of this are readily available for the weight gradient in (5) as the only terms in the sum that will need to be computed are those in which the postsynaptic neuron $j$ was active at time $t$ . Resulting in the following gradient update: + +$$ +\begin{array} { r l } & { \nabla W _ { i j } ^ { ( l ) } [ t ] = \left\{ \begin{array} { l l } { \varepsilon _ { j } ^ { ( l + 1 ) } [ t ] \qquad } & { \left( \sum _ { k < t } \alpha ^ { t - k - 1 } S _ { i } ^ { ( l ) } [ t ] \right) \qquad } & { V _ { j } ^ { ( l + 1 ) } [ t ] \mathrm { ~ i s ~ a c t i v e } , } \\ { 0 , \qquad } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \\ & { \nabla W _ { i j } ^ { ( l ) } = \displaystyle \sum _ { t } \nabla W _ { i j } ^ { ( l ) } [ t ] } \end{array} +$$ + +For the spike gradient in (6) we have two consequences, firstly, we will only need to compute $\nabla S _ { j } ^ { ( l ) } [ t ]$ for active neurons since $\nabla S _ { j } ^ { ( l ) } [ t ]$ is always multiplied by $\cdot$ in (5) and (6). Secondly, we can use a recurrent relation to save time and memory when computing $\delta _ { j } [ t ]$ in (6) as shown in the following proposition. + +Proposition 1. We can use a recurrent relation to compute $\delta _ { j } [ t ]$ given by + +$$ +\begin{array} { r } { \delta _ { j } [ t ] = \left\{ \begin{array} { l l } { \alpha ^ { n } \delta _ { j } [ t + n ] , } & { \quad i f \frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 \leq k \leq t + n } \\ { \nabla S _ { j } [ t + 1 ] \frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } + \alpha ^ { n } \delta _ { j } [ t + n ] , } & { \quad i f \frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 < k \leq t + n , \frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \neq 0 . } \end{array} \right. } \end{array} +$$ + +This means that we only need to compute $\delta _ { j } ^ { ( l + 1 ) } [ t ]$ at those time steps in which either $\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \neq 0$ or $V _ { j \ldots } ^ { ( l ) } [ t ]$ is active (see Appendix $\textrm { C }$ for a visualisation of this computation). Thus, we end up with the following sparse expression for computing the spike gradient. + +$$ +\begin{array} { r } { \nabla S _ { j } ^ { ( l ) } [ t ] = \left\{ \sum _ { j } W _ { i j } \delta _ { j } ^ { ( l + 1 ) } [ t ] , \qquad V _ { j } ^ { ( l ) } [ t ] \mathrm { ~ i s ~ a c t i v e } \right. } \\ { 0 , \qquad \mathrm { ~ o t h e r w i s e } \qquad } \end{array} +$$ + +# 2.3 Complexity analysis + +We define $\rho _ { l } \in [ 0 , 1 ]$ to be the probability that a neuron is active in layer $l$ at a given time. Table 1 summarises the computational complexity of the gradient computation in terms of number of sums and products. We use here $N$ to refer to the number of neurons in either the input or output layer and $B$ to refer to the batch size. In a slight abuse of notation we include the constants $\rho _ { l }$ as part of the complexity expressions. Details of how these expression were obtained can be found in Appendix $\mathbf { B }$ and are based on a reasonable efficient algorithm that uses memoisation for computing $\delta _ { j } [ t ]$ . + +Table 1: Computational complexity + +
OriginalSparse
Sums VW(l)O(BTN²) O(pt+1BTN2)
Products VW(l)O(BTN2) O(pt+1BTN2)
Sums Vs(t)O(BTN2) O(ρ𝑖BTN²))
Products VS(l)O(BTN2) O(pBTN²)
+ +In order for the sparse gradients to work better than their dense counterpart we need to have a balance between having few enough active neurons as to make the sparse backpropagation efficient while at the same time keeping enough neurons active to allow the gradient to backpropagate. We later show in section 3 that this is the case when testing it on real world datasets. + +# 2.4 Sparse learning rule interpretation + +The surrogate derivative we propose, while not having a constrained functional form for active neurons, imposes a zero gradient for the inactive ones. Looking back into equation (5) we can identify it now as a three-factor Hebbian learning rule [47]. With our surrogate derivative definition, the second term (spike derivative) measures whether the postsynaptic neuron is close to spiking (i.e. its membrane potential is close to the threshold). The third term (input trace) measures the presynaptic neuron activity in the most recent timesteps. Finally the first term (gradient from next layer) decides whether the synaptic activity improves or hinders performance with respect to some loss function. We note that this is not exactly a Hebbian learning rule as neurons that get close to spike but do not spike can influence the synaptic weight update and as such, the second term is simply a surrogate of the postsynaptic spike activity. + +Previous surrogate gradient descent methods have used a similar approach to implement backpropagation on SNNs [18, 36, 48, 39, 38], often using surrogate gradient descent definitions that are maximal when the membrane is at the threshold and decay as the potential moves away from it. In fact, in Esser et al. [18] the surrogate gradient definition given implies that neurons backpropagate a zero gradient $3 2 \%$ of the time. However, as shown in the next section, neurons can be active a lot less often when presented with spikes generated from real world data and still give the same test accuracy. This is only possible because surrogate gradient approximations to the threshold function yield much larger values when the potential is closer to spike thus concentrating most of the gradient on active neurons. This fact makes it possible to fully profit from our sparse learning rule for a much reduced computational and memory cost. + +# 3 Experiments + +We evaluated the correctness and improved efficiency of the proposed sparse backpropagation algorithm on real data of varying difficulty and increasing spatio-temporal complexity. Firstly, the Fashion-MNIST dataset (F-MNIST) [44] is an image dataset that has been previously used on SNNs by converting each pixel analogue value to spike latencies such that a higher intensity results in an earlier spike [39][38]. Importantly, each input neuron can only spike at most once thus resulting in a simple spatio-temporal complexity and very sparse coding. Secondly, we used the NeuromorphicMNIST (N-MNIST) [45] dataset where spikes are generated by presenting images of the MNIST dataset to a neuromorphic vision sensor. This dataset presents a higher temporal complexity and lower sparsity than the F-MNIST as each neuron can spike several times and there is noise from the recording device. Finally, we worked on the Spiking Heidelberg Dataset (SHD) [46]). This is a highly complex dataset that was generated by converting spoken digits into spike times by using a detailed model of auditory bushy cells in the cochlear nucleus. This results in a very high spatio-temporal complexity and lowest input sparsity. A visualisation of samples of each dataset can be found in the supplementary materials. These datasets cover the most relevant applications of neuromorphic systems. Namely, artificially encoding a dataset into spikes (Fashion-MNIST), reading spikes from a neuromorphic sensor such as a DVS camera (N-MNIST) and encoding complex temporally-varying data into spikes (SHD). + +We run all experiments on three-layer fully connected network as in Fig. 1, where the last layer (readout) has an infinite threshold. Both spiking layers in the middle have the same number of neurons. We used $g ( V ) : = 1 / ( \beta | V - V _ { t h } | \dot { + } 1 ) ^ { 2 }$ for the surrogate gradient as in [38]. See Appendix E for all training details. We implemented the sparse gradient computation as a Pytorch CUDA extension [49]. + +# 3.1 Spiking neurons are inactive most of the time resulting in higher energy efficiency + +One of the most determining factors of the success of our sparse gradient descent resides on the level of sparsity that we can expect and, consequently, the proportion of active neurons we have in each layer. We measure the percentage of active neurons on each batch as training progresses for each dataset. This can be computed as + +$$ +{ \mathrm { A c t i v i t y } } = 1 0 0 \times { \frac { \# { \mathrm { o f ~ a c t i v e ~ n e u r o n s } } } { B T N } } +$$ + +where $B$ is the batch size $T$ is the total number of time steps and $N$ the number of neurons in the layer and the number of active neurons is obtained according to (7). This is an empirical way of measuring the coefficient $\rho _ { l }$ introduced earlier. + +Figure 3A shows the activity on each dataset. As expected, the level of activity is correlated with the sparseness of the inputs with the lowest one being on the F-MNIST and the larger in the SHD dataset. Remarkably however, the activity on average is never above $2 \%$ on average in any dataset and it is as low as $1 . 0 6 \%$ in the F-MNIST dataset. This means that on average we will never have to compute more than $2 \%$ of the factors in $\nabla W$ and $2 \%$ of the values of $\nabla { S }$ . This can be visualised in the bottom row of Fig. 3B. + +These coefficients give us a theoretical upper bound to the amount of energy that can be saved if this sparse backpropagation rule was implemented in specialised hardware which only performed the required sums and products. This is summarised in Table 2. + +Table 2: Theoretical upper bound to energy saved in hidden layers. + +
Mean activity layer 1 (%)Mean activity layer 2 (%)Energy saved in VW (%)Energy saved in VS (%)
F-MNIST1.060.8799.1398.94
N-MNIST1.120.7799.2398.88
SHD1.701.0998.9198.30
+ +![](images/c34a3bccde4f2bc8cc20845896dbdc42e95007918f82ee95c7ab82f6af150f48.jpg) +Figure 3: Sparse backpropagation learns at high levels of sparsity. All figures except B are a 5 sample average. Standard error in the mean is displayed (although too small to be easily visualised). A. Percentage of active neurons during training as a fraction of total tensor size $( B \times T \times N )$ with $N = 2 0 0$ neurons for each dataset. B. Visualisation of the weight and spike gradients on the SHD dataset. We show zero value in hatched grey. Note how both $\bar { \nabla W } ^ { ( 0 ) }$ are nearly identical in despite being computed using a small fraction of the values of $\nabla S ^ { ( 1 ) }$ in the sparse case. C. Loss evolution on the SHD dataset using both algorithms. D. Final test accuracy on all datasets using both methods. + +Importantly, in dense backpropagation methods the temporal dimension of datasets and models is necessarily limited to a maximum of $\sim 2 0 0 0$ since gradients must be computed at all steps [46, 39]. This means that a lot of negligible gradients are computed. However, our method adapts to the level of activity of the network and only computes non-negligible gradients. This cost-prohibitive reality was evidenced when we attempted to run these experiments in smaller GPUs leading to running out of memory when using dense methods but not on ours (see F.8). Thus, sparse spiking backpropagation allows to train data that runs for longer periods of time without requiring a more powerful hardware. + +# 3.2 Sparse backpropagation training approximates the gradient very well + +We test that our sparse learning rule performs on par with the standard surrogate gradient descent. Figure 3B shows a visualisation of the weight (first row) and spike gradients (second row). Note how most of the values of $\nabla S ^ { ( 1 ) }$ were not computed and yet $\nabla \hat { W } ^ { ( 1 ) }$ is nearly identical to the original gradient. This is further confirmed in Figures 3C and 3D where the loss on the SHD dataset is practically identical with both methods (the loss for the other datasets can be found in Appendix F) and the test accuracy is practically identical (albeit slightly better) in the sparse case (F-MNIST: $8 2 . 2 \%$ , N-MNIST: $9 2 . 7 \%$ , SHD: $7 7 . 5 \%$ ). These accuracies are on par to those obtained with similar networks on these datasets (F-MNIST: $8 0 . 1 \%$ , N-MNIST: $9 7 . 4 \%$ , SHD: $7 1 . 7 \%$ ) [46, 38, 39]. These results show that sparse spiking backpropagation gives practically identical results to the original backpropagation but at much lower computational cost. + +# 3.3 Sparse training is faster and less memory intensive + +We now measure the time it takes the forward and the backward propagation of the second layer of the network during training as well as the peak GPU memory used. We choose this layer because it needs to compute both $\nabla W ^ { ( 1 ) }$ and $\nabla S ^ { ( 1 ) }$ (note that the first layer only needs to compute $\nabla W ^ { ( 0 ) }$ ). A performance improvement in this layer translates to an improvement in all subsequent layers inn a deeper network. We compute the backward speedup as the ratio of the original backpropagation time and the sparse backpropagation time. We compute the memory saved as + +![](images/47f4f5a14489494d4b705be7b8aa776617f2f360ebce435bd60fe4472f6a9ebe.jpg) +Figure 4: Speedup and memory improvement when using sparse gradient backpropagation. All figures are a 5 sample average and its standard error in the mean is also displayed. A. Backward speedup and memory saved in the second hidden layer for all three datasets when using sparse gradient descent with 200 hidden neurons. B. Time spent in computing the backward pass in the second hidden layer consisting of 200 neurons when using regular gradient descent and sparse gradient descent. C. Backward speed and memory saved in the second hidden layer as we increase the number of hidden neurons in all hidden layers. We included the overall speedup which takes into consideration both the forward and backward times spent in the second layer. + +$$ +{ \mathrm { M e m o r y ~ s a v e d } } = 1 0 0 \times { \frac { { \mathrm { m e m o r y } } _ { - } { \mathrm { o r i g i n a l } } - { \mathrm { m e m o r y } } _ { - } { \mathrm { s p a r s e } } } { \mathrm { m e m o r y } } } +$$ + +Figure 4A shows the speedup and memory saved on a 200 neuron hidden layer (both hidden layers). Backpropagation time is faster by $4 0 \mathbf { x }$ and GPU memory is reduced by $3 5 \%$ . To put this speedup into perspective Figure 4B shows the time taken for backpropagating the second layer in the original case (about $6 4 0 \mathrm { m s }$ ) and the sparse case (about $1 7 \mathrm { m s }$ ). It also shows that the time taken for backpropagating is nearly constant during training as it could be expected after studying the network activity in Fig. 3A. + +We also tested how robust is this implementation for a varying number of hidden neurons (in both layers). We increase the number of neurons up to 1000 which is similar to the maximum number of hidden units used in previous work [46] and it is also the point at which the original backpropagation method starts to run out of memory. Here we also show the overall speedup defined as the ratio between the sum of forward and backward times of the original over the sparse backpropagation. The results shown in Fig. 4C show that the backward speedup and memory saved remain constant as we increase the number of hidden neurons but the overall speedup increases substantially. Given that the forward time is the same for both methods (see Appendix F) this shows that as we increase the number of neurons, the backward time takes most of the computation and thus the sparse backpropagation becomes more important. + +We also note that these results vary depending on the GPU used, Figure 4 was obtained from running on an RTX6000 GPU. We also run this on smaller GPUs (GTX1060 and GTX1080Ti) and found that the improvement is even better reaching up to a 70x faster backward pass on N-MNIST and SHD. These results can found in Appendix F. Our results show that our method speeds up backward execution between one and two orders of magnitude, meaning that to a first approximation we would expect it reduce energy usage by the same factor since GPUs continue to use $30 \%$ of their peak power even when idle [50]. + +![](images/7e043ca296a6a748fcb8062665cba45ba05a060c4caf18ec88a0466ef7e70cee.jpg) +Figure 5: Impact of varying the backpropagation threshold $B _ { t h }$ on the SHD dataset. A. Final test accuracy on the SHD dataset as $B _ { t h }$ increases. B. Loss evolution on the SHD dataset for different $B _ { t h }$ . C. Hidden activity, backward speedup and backward memory saved as $B _ { t h }$ increases. + +# 3.4 Training deeper and more complex architectures + +We show that our gradient approximation can successfully train deeper networks by training a 6- layer fully connected network (5 hidden plus readout all with 300 neurons) on Fashion-MNIST. We measured the performance of the 5 hidden layers and achieve a $1 5 \mathrm { x }$ backward speedup when using 1080-Ti GPU (better than the 10x speedup under these conditions with only 2 hidden layers as shown in Appendix F.8) and $2 5 \%$ memory saved while achieving a $8 2 . 7 \%$ testing accuracy (slightly better than with 2 layers at $8 1 . 5 \%$ ). The results are consistent with our previous findings and show that the gradient is not loss in deeper architectures. + +We also run a experiment with a convolutional and pooling layers on the Fashion-MNIST dataset obtaining a test accuracy of $8 6 . 7 \%$ when using our spike gradient approximation and $8 6 . 9 \%$ with the original gradients. We do not report speedup or memory improvements in this network as developing a sparse convolutional CUDA kernel is out of the scope of our work but we simply ran a dense implementation with clamped gradients. This proves that our gradient approximation is able to train on more complex architectures although it still remains to be shown whether efficient sparse operators can be implemented for this purpose. See Appendix E for training details. + +# 3.5 Sparsity-Accuracy trade-off + +We trained the SHD dataset on the original 2-layer fully connected network with 400 neurons in each hidden layer and varied the backpropagation threshold $B _ { t h }$ . We found that the training is very robust to even extreme values of $B _ { t h }$ . The loss and accuracy remains unchanged even when setting $B _ { t h } = 0 . 9 5$ (see 5A and 5B) achieving a nearly $1 5 0 \mathrm { x }$ speedup and $8 5 \%$ memory improvement as shown in 5C. We also inspected the levels of activity while varying $B _ { t h }$ and we found that there are enough active neuron to propagate the gradient effectively. As $B _ { t h }$ gets closer to 1 the number of active neurons decreases rapidly until there are no gradients to propagate. + +# 4 Discussion + +Recent interest in neuromorphic computing has lead to energy efficient emulation of SNNs in specialised hardware. Most of these systems only support working with fixed network parameters thus requiring an external processor to train. Efforts to include training within the neuromorphic system usually rely on local learning rules which do not guarantee state-of-the-art performance. Thus, training on these systems requires error backpropagation on von-Neumann processors instead of taking advantage of the physics of the neuromorphic substrate [51, 52]. + +We have developed a sparse backpropagation algorithm for SNNs that achieves the same performance as standard dense backpropagation at a much reduced computational and memory cost. To the best of our knowledge this is the first backpropagation method that leverages the sparse nature of spikes in training and tackles the main performance bottleneck in neuromorphic training. Moreover, while we have used dense all-to-all connectivity between layers, our algorithm is compatible with other SNNs techniques that aim for a sparser connectivity such as synaptic pruning [53]. + +By constraining backpropagation to active neurons exclusively, we reduce the number of nodes and edges in the backward computation graph to a point where over $98 \%$ of the operations required to compute the gradients can be skipped. This results in a much faster backpropagation time, lower memory requirement and less energy consumption on GPU hardware while not affecting the testing accuracy. This is possible because the surrogate gradient used to approximate the derivative of the spiking function is larger when the membrane potential is closer to the threshold resulting in most of the gradient being concentrated on active neurons. Previous work has used, to a lesser extent, similar ideas. Esser et al. [18] used a surrogate gradient function which effectively resulted in about one third the neurons backpropagating a zero gradient. Later, Pozzi, et al. [54] developed a learning rule where only the weights of neurons directly affecting the activity of the selected output class are updated, however, the trial and error nature of the learning procedure resulted in up to $3 . 5 \mathrm { x }$ slower training. + +However, due to the lack of efficient sparse operators in most auto-differentiation platforms, every layer type requires to develop its own sparse CUDA kernel in order to be competitive with current heavily optimised libraries. In our work, we concentrated on implementing and testing this in spiking fully connected layers and we managed to show that sparse backpropagation is a lot faster and uses less memory. Developing these sparse kernels is not trivial and we are aware that having to do so for each layer is an important limitation of our work. This is no surprise since for now there is few reasons to believe that sparse tensor operations will significantly accelerate neural network performance. Our work aims to challenge this view and motivate the adoption of efficient sparse routines by showing for the first time that sparse BPTT can be faster without losing accuracy. + +We implemented and tested our method on real data on fully connected SNNs and simulated it in convolutional SNNs. The same method can be extended to any SNN topology as long as the neurons use a threshold activation function and the input is spiking data. However, due to the lack of efficient sparse operators in most auto-differentiation platforms, every layer type requires to develop its own sparse CUDA kernel in order to be competitive with current heavily optimised libraries. Developing these sparse kernels is not trivial and we are aware that having to do so for each layer is an important limitation of our work. This is no surprise since for now there is few reasons to believe that sparse tensor operations will significantly accelerate neural network performance. Our work aims to challenge this view and motivate the adoption of efficient sparse routines by showing for the first time that sparse spiking BPTT can be faster while achieving the same accuracy as its dense counterpart. Nevertheless, ideally, a dynamic backward graph should be generated each time the backward pass needs to be computed and consequently a GPU may not be the best processing unit for this task. A specialised neuromorphic implementation or an FPGA may be more suited to carry out the gradient computation task as this graph changes every batch update. Additionally, while we have reduced the number of operations required for training several times, we have only achieved up to a $8 5 \%$ memory reduction thus, memory requirements remains an important bottleneck. + +In summary, our sparse backpropagation algorithm for spiking neural networks tackles the main bottleneck in training SNNs and opens the possibility of a fast, energy and memory efficient end-toend training on spiking neural networks which will benefit neuromorphic training and computational neuroscience research. Sparse spike backpropagation is a necessary step towards a fully efficient on-chip training which will be ultimately required to achieve the full potential of neuromorphic computing. + +# Acknowledgments and Disclosure of Funding + +We thank Friedemann Zenke, Thomas Nowotny and James Knight for their feedback and support. This work was supported by the EPSRC Centre for Doctoral Training in High Performance Embedded and DistributedSystems (HiPEDS, Grant Reference EP/L016796/1) + +# References + +[1] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. nature, 521(7553):436–444, 2015. +[2] Geoffrey Hinton, Li Deng, Dong Yu, George E Dahl, Abdel-rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Tara N Sainath, et al. Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal processing magazine, 29(6):82–97, 2012. +[3] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. 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Pruning of deep spiking neural networks through gradient rewiring. arXiv preprint arXiv:2105.04916, 2021. +[54] Isabella Pozzi, Sander Bohte, and Pieter Roelfsema. Attention-gated brain propagation: How the brain can implement reward-based error backpropagation. Advances in Neural Information Processing Systems, 33, 2020. \ No newline at end of file diff --git a/parse/train/aLE2sEtMNXv/aLE2sEtMNXv_content_list.json b/parse/train/aLE2sEtMNXv/aLE2sEtMNXv_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2f316bd7d6aea6577069f50ae0a62f224d710919 --- /dev/null +++ b/parse/train/aLE2sEtMNXv/aLE2sEtMNXv_content_list.json @@ -0,0 +1,1472 @@ +[ + { + "type": "text", + "text": "Sparse Spiking Gradient Descent ", + "text_level": 1, + "bbox": [ + 297, + 122, + 699, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nicolas Perez-Nieves ", + "bbox": [ + 276, + 202, + 421, + 214 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dan F.M. Goodman ", + "bbox": [ + 584, + 202, + 723, + 214 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Electrical and Electronic Engineering Imperial College London London, United Kingdom \nnicolas.perez14@imperial.ac.uk ", + "bbox": [ + 220, + 215, + 478, + 270 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Electrical and Electronic Engineering Imperial College London London, United Kingdom d.goodman@imperial.ac.uk ", + "bbox": [ + 531, + 215, + 779, + 270 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 306, + 535, + 323 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There is an increasing interest in emulating Spiking Neural Networks (SNNs) on neuromorphic computing devices due to their low energy consumption. Recent advances have allowed training SNNs to a point where they start to compete with traditional Artificial Neural Networks (ANNs) in terms of accuracy, while at the same time being energy efficient when run on neuromorphic hardware. However, the process of training SNNs is still based on dense tensor operations originally developed for ANNs which do not leverage the spatiotemporally sparse nature of SNNs. We present here the first sparse SNN backpropagation algorithm which achieves the same or better accuracy as current state of the art methods while being significantly faster and more memory efficient. We show the effectiveness of our method on real datasets of varying complexity (Fashion-MNIST, NeuromophicMNIST and Spiking Heidelberg Digits) achieving a speedup in the backward pass of up to 150x, and ${ \\bar { 8 } } 5 \\%$ more memory efficient, without losing accuracy. ", + "bbox": [ + 233, + 337, + 766, + 516 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 540, + 310, + 558 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years, deep artificial neural networks (ANNs) have matched and occasionally surpassed human-level performance on increasingly difficult auditory and visual recognition problems [1, 2, 3], natural language processing tasks [4, 5] and games [6, 7, 8]. As these tasks become more challenging the neural networks required to solve them grow larger and consequently their power efficiency becomes more important [9, 10]. At the same time, the increasing interest in deploying these models into embedded applications calls for faster, more efficient and less memory intensive networks [11, 12]. ", + "bbox": [ + 174, + 570, + 825, + 669 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The human brain manages to perform similar and even more complicated tasks while only consuming about 20W [13] which contrasts with the hundreds of watts required for running ANNs [9]. Unlike ANNs, biological neurons in the brain communicate through discrete events called spikes. A biological neuron integrates incoming spikes from other neurons in its membrane potential and after reaching a threshold emits a spike and resets its potential [14]. The spiking neuron, combined with the sparse connectivity of the brain, results in a highly spatio-temporally sparse activity [15] which is fundamental to achieve this level of energy efficiency. ", + "bbox": [ + 174, + 674, + 825, + 772 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Inspired by the extraordinary performance of the brain, neuromorphic computing aims to obtain the same level of energy efficiency preferably while maintaining an accuracy standard on par with ANNs by emulating spiking neural networks (SNNs) on specialised hardware [16, 17, 18, 19]. These efforts go in two directions: emulating the SNNs and training the SNNs. While there is a growing number of successful neuromorphic implementations for the former [20, 21, 22, 23, 24, 17, 25, 26], the latter has proven to be more challenging. Some neuromorphic chips implement local learning rules [26, 27, 28] and recent advances have achieved to approximate backpropagation on-chip [29]. However, a full end-to-end supervised learning via error backpropagation requires off-chip training [22, 23, 18]. This is usually achieved by simulating and training the entire SNN on a GPU and more recently, emulating the forward pass of the SNN on neuromorphic hardware and then performing the backward pass on a GPU [30, 31]. The great flexibility provided by GPUs allows the development of complex training pipelines without being constrained by neuromorphic hardware specifications. However, as GPUs do not leverage the event-driven nature of spiking computations this results in effective but slower and less energy efficient training. An alternative to directly training an SNN consists on training an ANN and converting it to an SNN. However, this approach has been recently shown to not be more energy efficient than using the original ANN in the first place [32]. ", + "bbox": [ + 174, + 779, + 825, + 902 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/c2f88790fb9b9bb6397992ebd50b7df3941478c5a0febacaf88e507750f33a24.jpg", + "image_caption": [ + "Figure 1: Spiking Neural Network diagram highlighting the forward and backward passes " + ], + "image_footnote": [], + "bbox": [ + 236, + 87, + 759, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 205, + 826, + 304 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Training SNNs has been a challenge itself even without considering a power budget due to the all-or-none nature of spikes which hinders traditional gradient descent methods [33, 34]. Since the functional form of a spike is a unit impulse, it has a zero derivative for all membrane potentials except at the threshold where it is infinity. Recently developed methods based on surrogate gradients have been shown to be capable of solving complex tasks to near state-of-the-art accuracies [35, 18, 36, 37, 38, 39, 31]. The problem of non-differentiability is solved by adopting a well-behaved surrogate gradient when backpropagating the error [33]. This method, while effective for training very accurate models, is still constrained to working on dense tensors, thus, not profiting from the sparse nature of spiking, and limits the training speed and power efficiency when training SNNs. ", + "bbox": [ + 174, + 309, + 825, + 434 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work we introduce a sparse backpropagation method for SNNs. Recent work on ANNs has shown that adaptive dropout [40] and adaptive sparsity [41] can be used to achieve state-of-the-art performance at a fraction of the computational cost [42, 43]. We show that by redefining the surrogate gradient functional form, a sparse learning rule for SNNs arises naturally as a three-factor Hebbianlike learning rule. We empirically show an improved backpropagation time up to 70x faster than current implementations and up to $4 0 \\%$ more memory efficient in real datasets (Fashion-MNIST (FMNIST) [44], Neuromorphic-MNIST (N-MNIST) [45] and Spiking Heidelberg Dataset (SHD) [46]) thus reducing the computational gap between the forward and backward passes. This improvement will not only impact training for neuromorphic computing applications but also for computational neuroscience research involving training on SNNs [39]. ", + "bbox": [ + 173, + 440, + 826, + 579 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Sparse Spike Backpropagation ", + "text_level": 1, + "bbox": [ + 176, + 598, + 462, + 616 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 Forward Model ", + "text_level": 1, + "bbox": [ + 174, + 628, + 323, + 643 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We begin by introducing the spiking networks we are going to work with (Figs. 1 and 2). For simplicity we have omitted the batch dimension in the derivations but it is later used on the complexity analysis and experiments. We have a total of $L$ fully connected spiking layers. Each layer $l$ consisting of $N ^ { ( l ) }$ spiking neurons which are fully connected to the next layer $l + 1$ through synaptic weights $W ^ { ( l ) } \\in \\mathbb { R } ^ { N ^ { ( l ) } \\times N ^ { ( l + 1 ) } }$ . Each neuron has an internal state variable, the membrane potential, $V _ { j } ^ { ( l ) } [ t ] \\in \\mathbb { R }$ that updates every discrete time step $t \\in \\{ 0 , \\ldots , T - 1 \\}$ for some finite simulation time $T \\in \\mathbb { N }$ . Neurons emit spikes according to a spiking function $f \\colon { \\mathbb { R } } \\to \\{ 0 , 1 \\}$ such that ", + "bbox": [ + 173, + 654, + 826, + 758 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f12b6645118ff0bed15abcfeb06db6310f32ce4d92c4360c8535a96adebe8bcf.jpg", + "text": "$$\nS _ { i } ^ { ( l ) } [ t ] = f ( V _ { i } ^ { ( l ) } [ t ] )\n$$", + "text_format": "latex", + "bbox": [ + 436, + 763, + 562, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The membrane potential varies according to a simplified Leaky Integrate and Fire (LIF) neuron model in which input spikes are directly integrated into the membrane [14]. After a neuron spikes, its potential is reset to $V _ { r }$ . We will work with a discretised version of this model (see Appendix $\\mathrm { E }$ for a continuous time version). The membrane potential of neuron $j$ in layer $l { + } 1$ evolves according to the following difference equation ", + "bbox": [ + 174, + 796, + 825, + 867 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/ce748c41cba3eb07a86b22e2558d2f639870ee22f1b1700515e97f92cfa739d4.jpg", + "text": "$$\nV _ { j } ^ { ( l + 1 ) } [ t + 1 ] = \\alpha ( V _ { j } ^ { ( l + 1 ) } [ t ] - V _ { r e s t } ) + V _ { r e s t } + \\sum _ { i } S _ { i } ^ { ( l ) } [ t ] W _ { i j } ^ { ( l ) } - ( V _ { t h } - V _ { r } ) S _ { j } ^ { ( l + 1 ) } [ t ]\n$$", + "text_format": "latex", + "bbox": [ + 218, + 882, + 779, + 915 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/b74f25e41587a2733ae554635e8ebc612d9c441c1440cfb9fc57e0223683bda6.jpg", + "image_caption": [ + "Figure 2: Illustration of the gradient backpropagating only through active neurons. " + ], + "image_footnote": [], + "bbox": [ + 236, + 87, + 759, + 314 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The first two terms account for the leaky part of the model where we define $\\alpha = \\exp ( - \\Delta t / \\tau )$ with $\\Delta t , \\tau \\in \\mathbb { R }$ being the time resolution of the simulation and the membrane time constant of the neurons respectively. The second term deals with the integration of spikes. The last term models the resetting mechanism by subtracting the distance from the threshold to the reset potential. For simplicity and without loss of generality, we consider $V _ { r e s t } { = } 0 , V _ { t h } { = } 1 , V _ { r } { = } 0$ and $V _ { i } ^ { ( l ) } [ 0 ] { = } 0 \\forall l , i$ from now on. ", + "bbox": [ + 173, + 343, + 826, + 416 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can unroll (2) to obtain ", + "bbox": [ + 174, + 421, + 351, + 436 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/000d67bc5a25da875932be6703381e2bf4cd5c69915829cddf01182743e0d6b0.jpg", + "text": "$$\nV _ { j } ^ { ( l + 1 ) } [ t + 1 ] = \\underbrace { \\sum _ { i } \\sum _ { k = 0 } ^ { t } \\alpha ^ { t - k } S _ { i } ^ { ( l ) } [ k ] } _ { \\mathrm { I n p u t ~ t r a c e } } W _ { i j } ^ { ( l ) } - \\underbrace { \\sum _ { k = 0 } ^ { t } \\alpha ^ { t - k } S _ { j } ^ { ( l + 1 ) } [ k ] } _ { \\mathrm { R e s e t t i n g ~ t e r m } }\n$$", + "text_format": "latex", + "bbox": [ + 294, + 438, + 705, + 522 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Thus, a spiking neural network can be viewed as a special type of recurrent neural network where the activation function of each neuron is the spiking function $f ( \\cdot )$ . The spiking function, is commonly defined as the unit step function centered at a particular threshold $V _ { t h }$ ", + "bbox": [ + 178, + 525, + 820, + 566 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/eb447928d0b7781a331da161ab45524eb20bbe331476d07107f57ec6b9a57daa.jpg", + "text": "$$\nf ( v ) = { \\left\\{ \\begin{array} { l l } { 1 , \\quad } & { v > V _ { t h } } \\\\ { 0 , } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 408, + 570, + 584, + 604 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Note that while this function is easy to compute in the forward pass, its derivative, the unit impulse function, is problematic for backpropagation which has resulted in adopting surrogate derivatives [33] to allow the gradient to flow. Interestingly, it has been shown that surrogate gradient descent performs robustly for a wide range of surrogate functional forms [38]. ", + "bbox": [ + 173, + 608, + 825, + 664 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "After the final layer, a loss function $l o s s ( \\cdot )$ computes how far the network activity is with respect to some target value for the given input. The loss function can be defined as a function of the network spikes, membrane potentials or both. It may also be evaluated at every single time step or every several steps. We deliberately leave the particular definition of this function open as it does not directly affect the sparse gradient descent method we introduce. ", + "bbox": [ + 174, + 670, + 825, + 741 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Backward Model ", + "text_level": 1, + "bbox": [ + 174, + 755, + 333, + 770 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The SNN is updated by computing the gradient of the loss function with respect to the weights in each layer. This can be achieved using backpropagation through time (BPTT) on the unrolled network. Following (1) and (3) we can derive weight gradients to be ", + "bbox": [ + 174, + 780, + 825, + 823 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e0bf62dd706e4bed6ee3ab3f336df85795914e728916433560416ca2e840d96d.jpg", + "text": "$$\n\\nabla W _ { i j } ^ { ( l ) } = \\sum _ { t } \\underbrace { \\varepsilon _ { j } ^ { ( l + 1 ) } [ t ] } _ { \\substack { \\sum _ { t } \\dots \\dots \\dots } } \\overbrace { d V _ { j } ^ { ( l + 1 ) } [ t ] } ^ { \\substack { \\sum _ { j } ( l + 1 ) } } \\left( \\sum _ { k < t } \\alpha ^ { t - k - 1 } S _ { i } ^ { ( l ) } [ k ] \\right)\n$$", + "text_format": "latex", + "bbox": [ + 305, + 848, + 692, + 896 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This gradient consists of a sum over all time steps of the product of three terms. The first term, modulates the spatio-temporal credit assignment of the weights by propagating the gradient from the next layer. The second term, is the derivative of $f ( \\cdot )$ evaluated at a given $V _ { j } [ t ]$ . The last term is a filtered version of the input spikes. Notice that it is not necessary to compute this last term as this is already computed in the forward pass (3). We have chosen not make the spike resetting term in (3) differentiable as it has been shown that doing so can result in worse testing accuracy [38] and it increases the computational cost. ", + "bbox": [ + 173, + 90, + 826, + 189 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For a typical loss function defined on the output layer activity, the value of $\\varepsilon _ { j _ { - } } ^ { ( L - 1 ) } [ t ]$ (i.e. in the last layer) is the gradient of the loss with respect to the output layer (see Fig. 1). For all other layers we have that $\\varepsilon _ { j } ^ { ( l ) } [ t ] = \\nabla S _ { j } ^ { ( l ) } [ t ]$ , that is, the gradient of the loss function with respect to the output spikes of layer $\\it l$ . We can obtain an expression for $\\nabla S _ { j } ^ { ( l ) } [ t ]$ as a function of the gradient of the next layer: ", + "bbox": [ + 173, + 195, + 826, + 265 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d96b163117218bc9896a7893a0c7a542513ce34cfbd7ce770b3b032d43df1ab3.jpg", + "text": "$$\n\\begin{array} { l } { \\varepsilon _ { i } ^ { ( l ) } [ t ] = \\nabla S _ { i } ^ { ( l ) } [ t ] = \\displaystyle \\sum _ { j } W _ { i j } \\left( \\sum _ { k > t } \\overbrace { \\nabla S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } } \\overbrace { d V _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } \\alpha ^ { k - t - 1 } \\right) } \\\\ { = \\displaystyle \\sum _ { j } W _ { i j } \\delta _ { j } ^ { ( l + 1 ) } [ t ] , \\qquad l = \\{ 0 , \\ldots L - 2 \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 314, + 730, + 400 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "All hidden layers need to compute equations (5) and (6). The first one to update their input synaptic weights and the second one to backpropagate the loss to the previous layer. The first layer $\\mathit { l } = 1 \\mathit { \\Theta }$ ) only needs to compute (5) and the last layer $\\scriptstyle ( l = L - 1$ ) will compute a different $\\varepsilon _ { j } ^ { ( L - 1 ) } [ t ]$ depending on the loss function. ", + "bbox": [ + 173, + 412, + 825, + 473 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We introduce the following definitions which result in the sparse backpropagation learning rule. ", + "bbox": [ + 174, + 478, + 800, + 494 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 1. Given a backpropagation threshold $B _ { t h } \\in \\mathbb { R }$ we say neuron $j$ is active at time $t$ iff ", + "bbox": [ + 173, + 498, + 800, + 513 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/119fbc938a7d24264f66d14c1b6fa0a6e5b98f480df8879f82c01d8c34fe0ec3.jpg", + "text": "$$\n| V _ { j } [ t ] - V _ { t h } | < B _ { t h }\n$$", + "text_format": "latex", + "bbox": [ + 431, + 535, + 566, + 553 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2. The spike gradient is defined as ", + "bbox": [ + 173, + 556, + 475, + 571 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9e64562c587028674d5adf075b6aa3d9b88535fb60357233d840945fb83ff48f.jpg", + "text": "$$\n\\begin{array} { r } { : = \\left\\{ \\begin{array} { l l } { g ( V _ { j } [ t ] ) , \\quad } & { i f V _ { j } [ t ] i s a c t i \\nu e } \\\\ { 0 , \\quad } & { o t h e r w i s e } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 590, + 638, + 626 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This means that neurons are only active when their potential is close to the threshold. Applying these two definitions to (5) and (6) results in the gradients only backpropagating through active neurons at each time step as shown in Fig. 2. The consequences of this are readily available for the weight gradient in (5) as the only terms in the sum that will need to be computed are those in which the postsynaptic neuron $j$ was active at time $t$ . Resulting in the following gradient update: ", + "bbox": [ + 173, + 638, + 825, + 709 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5099363e2c31861097d07fc1b37583ffd6e800f96fd2a13e7b36a1a22463df16.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla W _ { i j } ^ { ( l ) } [ t ] = \\left\\{ \\begin{array} { l l } { \\varepsilon _ { j } ^ { ( l + 1 ) } [ t ] \\qquad } & { \\left( \\sum _ { k < t } \\alpha ^ { t - k - 1 } S _ { i } ^ { ( l ) } [ t ] \\right) \\qquad } & { V _ { j } ^ { ( l + 1 ) } [ t ] \\mathrm { ~ i s ~ a c t i v e } , } \\\\ { 0 , \\qquad } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\\\ & { \\nabla W _ { i j } ^ { ( l ) } = \\displaystyle \\sum _ { t } \\nabla W _ { i j } ^ { ( l ) } [ t ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 734, + 767, + 819 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For the spike gradient in (6) we have two consequences, firstly, we will only need to compute $\\nabla S _ { j } ^ { ( l ) } [ t ]$ for active neurons since $\\nabla S _ { j } ^ { ( l ) } [ t ]$ is always multiplied by $\\cdot$ in (5) and (6). Secondly, we can use a recurrent relation to save time and memory when computing $\\delta _ { j } [ t ]$ in (6) as shown in the following proposition. ", + "bbox": [ + 173, + 835, + 826, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proposition 1. We can use a recurrent relation to compute $\\delta _ { j } [ t ]$ given by ", + "bbox": [ + 173, + 90, + 655, + 107 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fcaa49dd8f99d89a5b5ceb3c53f4c06eb5d5698e454d22382c00e699c3a4d3b8.jpg", + "text": "$$\n\\begin{array} { r } { \\delta _ { j } [ t ] = \\left\\{ \\begin{array} { l l } { \\alpha ^ { n } \\delta _ { j } [ t + n ] , } & { \\quad i f \\frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 \\leq k \\leq t + n } \\\\ { \\nabla S _ { j } [ t + 1 ] \\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } + \\alpha ^ { n } \\delta _ { j } [ t + n ] , } & { \\quad i f \\frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 < k \\leq t + n , \\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \\neq 0 . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 108, + 805, + 152 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This means that we only need to compute $\\delta _ { j } ^ { ( l + 1 ) } [ t ]$ at those time steps in which either $\\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \\neq 0$ or $V _ { j \\ldots } ^ { ( l ) } [ t ]$ is active (see Appendix $\\textrm { C }$ for a visualisation of this computation). Thus, we end up with the following sparse expression for computing the spike gradient. ", + "bbox": [ + 173, + 171, + 826, + 224 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/13bb9d5449af70b8033e8acd4fdbccc18cf55ca5f3ce5e65e69f4233cb231222.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla S _ { j } ^ { ( l ) } [ t ] = \\left\\{ \\sum _ { j } W _ { i j } \\delta _ { j } ^ { ( l + 1 ) } [ t ] , \\qquad V _ { j } ^ { ( l ) } [ t ] \\mathrm { ~ i s ~ a c t i v e } \\right. } \\\\ { 0 , \\qquad \\mathrm { ~ o t h e r w i s e } \\qquad } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 232, + 676, + 275 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 Complexity analysis ", + "text_level": 1, + "bbox": [ + 174, + 287, + 352, + 303 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We define $\\rho _ { l } \\in [ 0 , 1 ]$ to be the probability that a neuron is active in layer $l$ at a given time. Table 1 summarises the computational complexity of the gradient computation in terms of number of sums and products. We use here $N$ to refer to the number of neurons in either the input or output layer and $B$ to refer to the batch size. In a slight abuse of notation we include the constants $\\rho _ { l }$ as part of the complexity expressions. Details of how these expression were obtained can be found in Appendix $\\mathbf { B }$ and are based on a reasonable efficient algorithm that uses memoisation for computing $\\delta _ { j } [ t ]$ . ", + "bbox": [ + 173, + 311, + 825, + 396 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/cb9706c2e53210403c36086f641b4ea03b4b4f023a74ed4dd356bc4fce609a47.jpg", + "table_caption": [ + "Table 1: Computational complexity " + ], + "table_footnote": [], + "table_body": "
OriginalSparse
Sums VW(l)O(BTN²) O(pt+1BTN2)
Products VW(l)O(BTN2) O(pt+1BTN2)
Sums Vs(t)O(BTN2) O(ρ𝑖BTN²))
Products VS(l)O(BTN2) O(pBTN²)
", + "bbox": [ + 323, + 429, + 669, + 521 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order for the sparse gradients to work better than their dense counterpart we need to have a balance between having few enough active neurons as to make the sparse backpropagation efficient while at the same time keeping enough neurons active to allow the gradient to backpropagate. We later show in section 3 that this is the case when testing it on real world datasets. ", + "bbox": [ + 176, + 531, + 823, + 588 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 Sparse learning rule interpretation ", + "text_level": 1, + "bbox": [ + 176, + 603, + 455, + 618 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The surrogate derivative we propose, while not having a constrained functional form for active neurons, imposes a zero gradient for the inactive ones. Looking back into equation (5) we can identify it now as a three-factor Hebbian learning rule [47]. With our surrogate derivative definition, the second term (spike derivative) measures whether the postsynaptic neuron is close to spiking (i.e. its membrane potential is close to the threshold). The third term (input trace) measures the presynaptic neuron activity in the most recent timesteps. Finally the first term (gradient from next layer) decides whether the synaptic activity improves or hinders performance with respect to some loss function. We note that this is not exactly a Hebbian learning rule as neurons that get close to spike but do not spike can influence the synaptic weight update and as such, the second term is simply a surrogate of the postsynaptic spike activity. ", + "bbox": [ + 173, + 627, + 825, + 767 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Previous surrogate gradient descent methods have used a similar approach to implement backpropagation on SNNs [18, 36, 48, 39, 38], often using surrogate gradient descent definitions that are maximal when the membrane is at the threshold and decay as the potential moves away from it. In fact, in Esser et al. [18] the surrogate gradient definition given implies that neurons backpropagate a zero gradient $3 2 \\%$ of the time. However, as shown in the next section, neurons can be active a lot less often when presented with spikes generated from real world data and still give the same test accuracy. This is only possible because surrogate gradient approximations to the threshold function yield much larger values when the potential is closer to spike thus concentrating most of the gradient on active neurons. This fact makes it possible to fully profit from our sparse learning rule for a much reduced computational and memory cost. ", + "bbox": [ + 173, + 772, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 89, + 312, + 107 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluated the correctness and improved efficiency of the proposed sparse backpropagation algorithm on real data of varying difficulty and increasing spatio-temporal complexity. Firstly, the Fashion-MNIST dataset (F-MNIST) [44] is an image dataset that has been previously used on SNNs by converting each pixel analogue value to spike latencies such that a higher intensity results in an earlier spike [39][38]. Importantly, each input neuron can only spike at most once thus resulting in a simple spatio-temporal complexity and very sparse coding. Secondly, we used the NeuromorphicMNIST (N-MNIST) [45] dataset where spikes are generated by presenting images of the MNIST dataset to a neuromorphic vision sensor. This dataset presents a higher temporal complexity and lower sparsity than the F-MNIST as each neuron can spike several times and there is noise from the recording device. Finally, we worked on the Spiking Heidelberg Dataset (SHD) [46]). This is a highly complex dataset that was generated by converting spoken digits into spike times by using a detailed model of auditory bushy cells in the cochlear nucleus. This results in a very high spatio-temporal complexity and lowest input sparsity. A visualisation of samples of each dataset can be found in the supplementary materials. These datasets cover the most relevant applications of neuromorphic systems. Namely, artificially encoding a dataset into spikes (Fashion-MNIST), reading spikes from a neuromorphic sensor such as a DVS camera (N-MNIST) and encoding complex temporally-varying data into spikes (SHD). ", + "bbox": [ + 174, + 122, + 825, + 358 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We run all experiments on three-layer fully connected network as in Fig. 1, where the last layer (readout) has an infinite threshold. Both spiking layers in the middle have the same number of neurons. We used $g ( V ) : = 1 / ( \\beta | V - V _ { t h } | \\dot { + } 1 ) ^ { 2 }$ for the surrogate gradient as in [38]. See Appendix E for all training details. We implemented the sparse gradient computation as a Pytorch CUDA extension [49]. ", + "bbox": [ + 174, + 363, + 825, + 434 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 Spiking neurons are inactive most of the time resulting in higher energy efficiency ", + "text_level": 1, + "bbox": [ + 174, + 454, + 779, + 469 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "One of the most determining factors of the success of our sparse gradient descent resides on the level of sparsity that we can expect and, consequently, the proportion of active neurons we have in each layer. We measure the percentage of active neurons on each batch as training progresses for each dataset. This can be computed as ", + "bbox": [ + 174, + 481, + 825, + 537 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/09073de361632d214e5571b72529f2150af6645b1814ac12a4ce2954dc201115.jpg", + "text": "$$\n{ \\mathrm { A c t i v i t y } } = 1 0 0 \\times { \\frac { \\# { \\mathrm { o f ~ a c t i v e ~ n e u r o n s } } } { B T N } }\n$$", + "text_format": "latex", + "bbox": [ + 372, + 559, + 625, + 590 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $B$ is the batch size $T$ is the total number of time steps and $N$ the number of neurons in the layer and the number of active neurons is obtained according to (7). This is an empirical way of measuring the coefficient $\\rho _ { l }$ introduced earlier. ", + "bbox": [ + 176, + 604, + 823, + 647 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 3A shows the activity on each dataset. As expected, the level of activity is correlated with the sparseness of the inputs with the lowest one being on the F-MNIST and the larger in the SHD dataset. Remarkably however, the activity on average is never above $2 \\%$ on average in any dataset and it is as low as $1 . 0 6 \\%$ in the F-MNIST dataset. This means that on average we will never have to compute more than $2 \\%$ of the factors in $\\nabla W$ and $2 \\%$ of the values of $\\nabla { S }$ . This can be visualised in the bottom row of Fig. 3B. ", + "bbox": [ + 173, + 652, + 825, + 737 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "These coefficients give us a theoretical upper bound to the amount of energy that can be saved if this sparse backpropagation rule was implemented in specialised hardware which only performed the required sums and products. This is summarised in Table 2. ", + "bbox": [ + 176, + 742, + 823, + 785 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/e380ce229a1779f4a08cf124a66aebea0363dc1beb38ef1cf0b515a6505ed2ac.jpg", + "table_caption": [ + "Table 2: Theoretical upper bound to energy saved in hidden layers. " + ], + "table_footnote": [], + "table_body": "
Mean activity layer 1 (%)Mean activity layer 2 (%)Energy saved in VW (%)Energy saved in VS (%)
F-MNIST1.060.8799.1398.94
N-MNIST1.120.7799.2398.88
SHD1.701.0998.9198.30
", + "bbox": [ + 235, + 823, + 758, + 910 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/c34a3bccde4f2bc8cc20845896dbdc42e95007918f82ee95c7ab82f6af150f48.jpg", + "image_caption": [ + "Figure 3: Sparse backpropagation learns at high levels of sparsity. All figures except B are a 5 sample average. Standard error in the mean is displayed (although too small to be easily visualised). A. Percentage of active neurons during training as a fraction of total tensor size $( B \\times T \\times N )$ with $N = 2 0 0$ neurons for each dataset. B. Visualisation of the weight and spike gradients on the SHD dataset. We show zero value in hatched grey. Note how both $\\bar { \\nabla W } ^ { ( 0 ) }$ are nearly identical in despite being computed using a small fraction of the values of $\\nabla S ^ { ( 1 ) }$ in the sparse case. C. Loss evolution on the SHD dataset using both algorithms. D. Final test accuracy on all datasets using both methods. " + ], + "image_footnote": [], + "bbox": [ + 179, + 94, + 820, + 417 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Importantly, in dense backpropagation methods the temporal dimension of datasets and models is necessarily limited to a maximum of $\\sim 2 0 0 0$ since gradients must be computed at all steps [46, 39]. This means that a lot of negligible gradients are computed. However, our method adapts to the level of activity of the network and only computes non-negligible gradients. This cost-prohibitive reality was evidenced when we attempted to run these experiments in smaller GPUs leading to running out of memory when using dense methods but not on ours (see F.8). Thus, sparse spiking backpropagation allows to train data that runs for longer periods of time without requiring a more powerful hardware. ", + "bbox": [ + 174, + 555, + 825, + 652 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 Sparse backpropagation training approximates the gradient very well ", + "text_level": 1, + "bbox": [ + 174, + 671, + 696, + 686 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We test that our sparse learning rule performs on par with the standard surrogate gradient descent. Figure 3B shows a visualisation of the weight (first row) and spike gradients (second row). Note how most of the values of $\\nabla S ^ { ( 1 ) }$ were not computed and yet $\\nabla \\hat { W } ^ { ( 1 ) }$ is nearly identical to the original gradient. This is further confirmed in Figures 3C and 3D where the loss on the SHD dataset is practically identical with both methods (the loss for the other datasets can be found in Appendix F) and the test accuracy is practically identical (albeit slightly better) in the sparse case (F-MNIST: $8 2 . 2 \\%$ , N-MNIST: $9 2 . 7 \\%$ , SHD: $7 7 . 5 \\%$ ). These accuracies are on par to those obtained with similar networks on these datasets (F-MNIST: $8 0 . 1 \\%$ , N-MNIST: $9 7 . 4 \\%$ , SHD: $7 1 . 7 \\%$ ) [46, 38, 39]. These results show that sparse spiking backpropagation gives practically identical results to the original backpropagation but at much lower computational cost. ", + "bbox": [ + 174, + 698, + 825, + 838 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.3 Sparse training is faster and less memory intensive ", + "text_level": 1, + "bbox": [ + 174, + 857, + 565, + 871 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now measure the time it takes the forward and the backward propagation of the second layer of the network during training as well as the peak GPU memory used. We choose this layer because it needs to compute both $\\nabla W ^ { ( 1 ) }$ and $\\nabla S ^ { ( 1 ) }$ (note that the first layer only needs to compute $\\nabla W ^ { ( 0 ) }$ ). A performance improvement in this layer translates to an improvement in all subsequent layers inn a deeper network. We compute the backward speedup as the ratio of the original backpropagation time and the sparse backpropagation time. We compute the memory saved as ", + "bbox": [ + 174, + 883, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/47f4f5a14489494d4b705be7b8aa776617f2f360ebce435bd60fe4472f6a9ebe.jpg", + "image_caption": [ + "Figure 4: Speedup and memory improvement when using sparse gradient backpropagation. All figures are a 5 sample average and its standard error in the mean is also displayed. A. Backward speedup and memory saved in the second hidden layer for all three datasets when using sparse gradient descent with 200 hidden neurons. B. Time spent in computing the backward pass in the second hidden layer consisting of 200 neurons when using regular gradient descent and sparse gradient descent. C. Backward speed and memory saved in the second hidden layer as we increase the number of hidden neurons in all hidden layers. We included the overall speedup which takes into consideration both the forward and backward times spent in the second layer. " + ], + "image_footnote": [], + "bbox": [ + 179, + 90, + 818, + 306 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 457, + 825, + 515 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/ee18c725ada4980fc70daf2ae9bd5892aab6b8f65c3119f7fb2c7a2bc4e9d34e.jpg", + "text": "$$\n{ \\mathrm { M e m o r y ~ s a v e d } } = 1 0 0 \\times { \\frac { { \\mathrm { m e m o r y } } _ { - } { \\mathrm { o r i g i n a l } } - { \\mathrm { m e m o r y } } _ { - } { \\mathrm { s p a r s e } } } { \\mathrm { m e m o r y } } }\n$$", + "text_format": "latex", + "bbox": [ + 295, + 535, + 702, + 570 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 4A shows the speedup and memory saved on a 200 neuron hidden layer (both hidden layers). Backpropagation time is faster by $4 0 \\mathbf { x }$ and GPU memory is reduced by $3 5 \\%$ . To put this speedup into perspective Figure 4B shows the time taken for backpropagating the second layer in the original case (about $6 4 0 \\mathrm { m s }$ ) and the sparse case (about $1 7 \\mathrm { m s }$ ). It also shows that the time taken for backpropagating is nearly constant during training as it could be expected after studying the network activity in Fig. 3A. ", + "bbox": [ + 173, + 579, + 825, + 662 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also tested how robust is this implementation for a varying number of hidden neurons (in both layers). We increase the number of neurons up to 1000 which is similar to the maximum number of hidden units used in previous work [46] and it is also the point at which the original backpropagation method starts to run out of memory. Here we also show the overall speedup defined as the ratio between the sum of forward and backward times of the original over the sparse backpropagation. The results shown in Fig. 4C show that the backward speedup and memory saved remain constant as we increase the number of hidden neurons but the overall speedup increases substantially. Given that the forward time is the same for both methods (see Appendix F) this shows that as we increase the number of neurons, the backward time takes most of the computation and thus the sparse backpropagation becomes more important. ", + "bbox": [ + 174, + 669, + 825, + 808 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also note that these results vary depending on the GPU used, Figure 4 was obtained from running on an RTX6000 GPU. We also run this on smaller GPUs (GTX1060 and GTX1080Ti) and found that the improvement is even better reaching up to a 70x faster backward pass on N-MNIST and SHD. These results can found in Appendix F. Our results show that our method speeds up backward execution between one and two orders of magnitude, meaning that to a first approximation we would expect it reduce energy usage by the same factor since GPUs continue to use $30 \\%$ of their peak power even when idle [50]. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/7e043ca296a6a748fcb8062665cba45ba05a060c4caf18ec88a0466ef7e70cee.jpg", + "image_caption": [ + "Figure 5: Impact of varying the backpropagation threshold $B _ { t h }$ on the SHD dataset. A. Final test accuracy on the SHD dataset as $B _ { t h }$ increases. B. Loss evolution on the SHD dataset for different $B _ { t h }$ . C. Hidden activity, backward speedup and backward memory saved as $B _ { t h }$ increases. " + ], + "image_footnote": [], + "bbox": [ + 179, + 90, + 818, + 417 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.4 Training deeper and more complex architectures ", + "text_level": 1, + "bbox": [ + 174, + 515, + 550, + 530 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We show that our gradient approximation can successfully train deeper networks by training a 6- layer fully connected network (5 hidden plus readout all with 300 neurons) on Fashion-MNIST. We measured the performance of the 5 hidden layers and achieve a $1 5 \\mathrm { x }$ backward speedup when using 1080-Ti GPU (better than the 10x speedup under these conditions with only 2 hidden layers as shown in Appendix F.8) and $2 5 \\%$ memory saved while achieving a $8 2 . 7 \\%$ testing accuracy (slightly better than with 2 layers at $8 1 . 5 \\%$ ). The results are consistent with our previous findings and show that the gradient is not loss in deeper architectures. ", + "bbox": [ + 174, + 546, + 825, + 643 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We also run a experiment with a convolutional and pooling layers on the Fashion-MNIST dataset obtaining a test accuracy of $8 6 . 7 \\%$ when using our spike gradient approximation and $8 6 . 9 \\%$ with the original gradients. We do not report speedup or memory improvements in this network as developing a sparse convolutional CUDA kernel is out of the scope of our work but we simply ran a dense implementation with clamped gradients. This proves that our gradient approximation is able to train on more complex architectures although it still remains to be shown whether efficient sparse operators can be implemented for this purpose. See Appendix E for training details. ", + "bbox": [ + 174, + 650, + 825, + 747 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.5 Sparsity-Accuracy trade-off ", + "text_level": 1, + "bbox": [ + 176, + 781, + 406, + 796 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We trained the SHD dataset on the original 2-layer fully connected network with 400 neurons in each hidden layer and varied the backpropagation threshold $B _ { t h }$ . We found that the training is very robust to even extreme values of $B _ { t h }$ . The loss and accuracy remains unchanged even when setting $B _ { t h } = 0 . 9 5$ (see 5A and 5B) achieving a nearly $1 5 0 \\mathrm { x }$ speedup and $8 5 \\%$ memory improvement as shown in 5C. We also inspected the levels of activity while varying $B _ { t h }$ and we found that there are enough active neuron to propagate the gradient effectively. As $B _ { t h }$ gets closer to 1 the number of active neurons decreases rapidly until there are no gradients to propagate. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 Discussion ", + "text_level": 1, + "bbox": [ + 174, + 89, + 292, + 106 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Recent interest in neuromorphic computing has lead to energy efficient emulation of SNNs in specialised hardware. Most of these systems only support working with fixed network parameters thus requiring an external processor to train. Efforts to include training within the neuromorphic system usually rely on local learning rules which do not guarantee state-of-the-art performance. Thus, training on these systems requires error backpropagation on von-Neumann processors instead of taking advantage of the physics of the neuromorphic substrate [51, 52]. ", + "bbox": [ + 174, + 132, + 825, + 215 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have developed a sparse backpropagation algorithm for SNNs that achieves the same performance as standard dense backpropagation at a much reduced computational and memory cost. To the best of our knowledge this is the first backpropagation method that leverages the sparse nature of spikes in training and tackles the main performance bottleneck in neuromorphic training. Moreover, while we have used dense all-to-all connectivity between layers, our algorithm is compatible with other SNNs techniques that aim for a sparser connectivity such as synaptic pruning [53]. ", + "bbox": [ + 174, + 222, + 825, + 305 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "By constraining backpropagation to active neurons exclusively, we reduce the number of nodes and edges in the backward computation graph to a point where over $98 \\%$ of the operations required to compute the gradients can be skipped. This results in a much faster backpropagation time, lower memory requirement and less energy consumption on GPU hardware while not affecting the testing accuracy. This is possible because the surrogate gradient used to approximate the derivative of the spiking function is larger when the membrane potential is closer to the threshold resulting in most of the gradient being concentrated on active neurons. Previous work has used, to a lesser extent, similar ideas. Esser et al. [18] used a surrogate gradient function which effectively resulted in about one third the neurons backpropagating a zero gradient. Later, Pozzi, et al. [54] developed a learning rule where only the weights of neurons directly affecting the activity of the selected output class are updated, however, the trial and error nature of the learning procedure resulted in up to $3 . 5 \\mathrm { x }$ slower training. ", + "bbox": [ + 174, + 311, + 825, + 463 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "However, due to the lack of efficient sparse operators in most auto-differentiation platforms, every layer type requires to develop its own sparse CUDA kernel in order to be competitive with current heavily optimised libraries. In our work, we concentrated on implementing and testing this in spiking fully connected layers and we managed to show that sparse backpropagation is a lot faster and uses less memory. Developing these sparse kernels is not trivial and we are aware that having to do so for each layer is an important limitation of our work. This is no surprise since for now there is few reasons to believe that sparse tensor operations will significantly accelerate neural network performance. Our work aims to challenge this view and motivate the adoption of efficient sparse routines by showing for the first time that sparse BPTT can be faster without losing accuracy. ", + "bbox": [ + 174, + 469, + 825, + 594 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We implemented and tested our method on real data on fully connected SNNs and simulated it in convolutional SNNs. The same method can be extended to any SNN topology as long as the neurons use a threshold activation function and the input is spiking data. However, due to the lack of efficient sparse operators in most auto-differentiation platforms, every layer type requires to develop its own sparse CUDA kernel in order to be competitive with current heavily optimised libraries. Developing these sparse kernels is not trivial and we are aware that having to do so for each layer is an important limitation of our work. This is no surprise since for now there is few reasons to believe that sparse tensor operations will significantly accelerate neural network performance. Our work aims to challenge this view and motivate the adoption of efficient sparse routines by showing for the first time that sparse spiking BPTT can be faster while achieving the same accuracy as its dense counterpart. Nevertheless, ideally, a dynamic backward graph should be generated each time the backward pass needs to be computed and consequently a GPU may not be the best processing unit for this task. A specialised neuromorphic implementation or an FPGA may be more suited to carry out the gradient computation task as this graph changes every batch update. Additionally, while we have reduced the number of operations required for training several times, we have only achieved up to a $8 5 \\%$ memory reduction thus, memory requirements remains an important bottleneck. ", + "bbox": [ + 174, + 601, + 825, + 821 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In summary, our sparse backpropagation algorithm for spiking neural networks tackles the main bottleneck in training SNNs and opens the possibility of a fast, energy and memory efficient end-toend training on spiking neural networks which will benefit neuromorphic training and computational neuroscience research. Sparse spike backpropagation is a necessary step towards a fully efficient on-chip training which will be ultimately required to achieve the full potential of neuromorphic computing. ", + "bbox": [ + 176, + 828, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 88, + 553, + 107 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We thank Friedemann Zenke, Thomas Nowotny and James Knight for their feedback and support. This work was supported by the EPSRC Centre for Doctoral Training in High Performance Embedded and DistributedSystems (HiPEDS, Grant Reference EP/L016796/1) ", + "bbox": [ + 174, + 121, + 826, + 162 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 184, + 266, + 200 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "[1] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. 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These", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "score": 1.0, + "content": "efforts go in two directions: emulating the SNNs and training the SNNs. While there is a growing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 507, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 674 + ], + "score": 1.0, + "content": "number of successful neuromorphic implementations for the former [20, 21, 22, 23, 24, 17, 25, 26],", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 669, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 685 + ], + "score": 1.0, + "content": "the latter has proven to be more challenging. 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Recent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 289, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 469, + 301 + ], + "score": 1.0, + "content": "advances have allowed training SNNs to a point where they start to compete with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "score": 1.0, + "content": "traditional Artificial Neural Networks (ANNs) in terms of accuracy, while at the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 312, + 470, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 470, + 323 + ], + "score": 1.0, + "content": "same time being energy efficient when run on neuromorphic hardware. However,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "score": 1.0, + "content": "the process of training SNNs is still based on dense tensor operations originally", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "score": 1.0, + "content": "developed for ANNs which do not leverage the spatiotemporally sparse nature of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "score": 1.0, + "content": "SNNs. We present here the first sparse SNN backpropagation algorithm which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "score": 1.0, + "content": "achieves the same or better accuracy as current state of the art methods while being", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 365, + 470, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 470, + 377 + ], + "score": 1.0, + "content": "significantly faster and more memory efficient. 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As these tasks become more challenging", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 500 + ], + "score": 1.0, + "content": "the neural networks required to solve them grow larger and consequently their power efficiency", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "becomes more important [9, 10]. At the same time, the increasing interest in deploying these models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "into embedded applications calls for faster, more efficient and less memory intensive networks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 144, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 144, + 531 + ], + "score": 1.0, + "content": "[11, 12].", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 453, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "score": 1.0, + "content": "The human brain manages to perform similar and even more complicated tasks while only consuming", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 546, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 557 + ], + "score": 1.0, + "content": "about 20W [13] which contrasts with the hundreds of watts required for running ANNs [9]. Unlike", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "ANNs, biological neurons in the brain communicate through discrete events called spikes. A", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "score": 1.0, + "content": "biological neuron integrates incoming spikes from other neurons in its membrane potential and after", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "reaching a threshold emits a spike and resets its potential [14]. The spiking neuron, combined with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "the sparse connectivity of the brain, results in a highly spatio-temporally sparse activity [15] which is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 324, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 324, + 613 + ], + "score": 1.0, + "content": "fundamental to achieve this level of energy efficiency.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 533, + 506, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 617, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 631 + ], + "score": 1.0, + "content": "Inspired by the extraordinary performance of the brain, neuromorphic computing aims to obtain", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 628, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 504, + 639 + ], + "score": 1.0, + "content": "the same level of energy efficiency preferably while maintaining an accuracy standard on par with", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "ANNs by emulating spiking neural networks (SNNs) on specialised hardware [16, 17, 18, 19]. These", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 663 + ], + "score": 1.0, + "content": "efforts go in two directions: emulating the SNNs and training the SNNs. While there is a growing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 507, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 674 + ], + "score": 1.0, + "content": "number of successful neuromorphic implementations for the former [20, 21, 22, 23, 24, 17, 25, 26],", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 669, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 685 + ], + "score": 1.0, + "content": "the latter has proven to be more challenging. 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An alternative to directly training an SNN", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "consists on training an ANN and converting it to an SNN. However, this approach has been recently", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 229, + 463, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 463, + 241 + ], + "score": 1.0, + "content": "shown to not be more energy efficient than using the original ANN in the first place [32].", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Training SNNs has been a challenge itself even without considering a power budget due to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "all-or-none nature of spikes which hinders traditional gradient descent methods [33, 34]. Since the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 281 + ], + "score": 1.0, + "content": "functional form of a spike is a unit impulse, it has a zero derivative for all membrane potentials except", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "at the threshold where it is infinity. Recently developed methods based on surrogate gradients have", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 290, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 301 + ], + "score": 1.0, + "content": "been shown to be capable of solving complex tasks to near state-of-the-art accuracies [35, 18, 36,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "37, 38, 39, 31]. The problem of non-differentiability is solved by adopting a well-behaved surrogate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "gradient when backpropagating the error [33]. This method, while effective for training very accurate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "models, is still constrained to working on dense tensors, thus, not profiting from the sparse nature of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 332, + 428, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 428, + 345 + ], + "score": 1.0, + "content": "spiking, and limits the training speed and power efficiency when training SNNs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "In this work we introduce a sparse backpropagation method for SNNs. Recent work on ANNs has", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "shown that adaptive dropout [40] and adaptive sparsity [41] can be used to achieve state-of-the-art", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "score": 1.0, + "content": "performance at a fraction of the computational cost [42, 43]. We show that by redefining the surrogate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 381, + 507, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 507, + 395 + ], + "score": 1.0, + "content": "gradient functional form, a sparse learning rule for SNNs arises naturally as a three-factor Hebbian-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "like learning rule. We empirically show an improved backpropagation time up to 70x faster than", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 245, + 416 + ], + "score": 1.0, + "content": "current implementations and up to", + "type": "text" + }, + { + "bbox": [ + 245, + 403, + 265, + 414 + ], + "score": 0.89, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "more memory efficient in real datasets (Fashion-MNIST (F-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "MNIST) [44], Neuromorphic-MNIST (N-MNIST) [45] and Spiking Heidelberg Dataset (SHD) [46])", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "thus reducing the computational gap between the forward and backward passes. This improvement", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "will not only impact training for neuromorphic computing applications but also for computational", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 331, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 331, + 460 + ], + "score": 1.0, + "content": "neuroscience research involving training on SNNs [39].", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 474, + 283, + 488 + ], + "lines": [ + { + "bbox": [ + 104, + 471, + 285, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 285, + 491 + ], + "score": 1.0, + "content": "2 Sparse Spike Backpropagation", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 498, + 198, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 199, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 199, + 511 + ], + "score": 1.0, + "content": "2.1 Forward Model", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 506, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "We begin by introducing the spiking networks we are going to work with (Figs. 1 and 2). For simplicity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "we have omitted the batch dimension in the derivations but it is later used on the complexity analysis", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 258, + 554 + ], + "score": 1.0, + "content": "and experiments. We have a total of", + "type": "text" + }, + { + "bbox": [ + 258, + 541, + 266, + 550 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 540, + 443, + 554 + ], + "score": 1.0, + "content": "fully connected spiking layers. Each layer", + "type": "text" + }, + { + "bbox": [ + 444, + 541, + 448, + 550 + ], + "score": 0.54, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "consisting of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 549, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 107, + 551, + 126, + 562 + ], + "score": 0.88, + "content": "N ^ { ( l ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 549, + 378, + 568 + ], + "score": 1.0, + "content": "spiking neurons which are fully connected to the next layer", + "type": "text" + }, + { + "bbox": [ + 378, + 552, + 397, + 563 + ], + "score": 0.86, + "content": "l + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 549, + 506, + 568 + ], + "score": 1.0, + "content": "through synaptic weights", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 559, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 107, + 563, + 187, + 576 + ], + "score": 0.91, + "content": "W ^ { ( l ) } \\in \\mathbb { R } ^ { N ^ { ( l ) } \\times N ^ { ( l + 1 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 559, + 456, + 582 + ], + "score": 1.0, + "content": ". 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Since the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 281 + ], + "score": 1.0, + "content": "functional form of a spike is a unit impulse, it has a zero derivative for all membrane potentials except", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "at the threshold where it is infinity. 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Recent work on ANNs has", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "shown that adaptive dropout [40] and adaptive sparsity [41] can be used to achieve state-of-the-art", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "score": 1.0, + "content": "performance at a fraction of the computational cost [42, 43]. 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We empirically show an improved backpropagation time up to 70x faster than", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 245, + 416 + ], + "score": 1.0, + "content": "current implementations and up to", + "type": "text" + }, + { + "bbox": [ + 245, + 403, + 265, + 414 + ], + "score": 0.89, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "more memory efficient in real datasets (Fashion-MNIST (F-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "MNIST) [44], Neuromorphic-MNIST (N-MNIST) [45] and Spiking Heidelberg Dataset (SHD) [46])", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "thus reducing the computational gap between the forward and backward passes. This improvement", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "will not only impact training for neuromorphic computing applications but also for computational", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 331, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 331, + 460 + ], + "score": 1.0, + "content": "neuroscience research involving training on SNNs [39].", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 349, + 507, + 460 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 474, + 283, + 488 + ], + "lines": [ + { + "bbox": [ + 104, + 471, + 285, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 285, + 491 + ], + "score": 1.0, + "content": "2 Sparse Spike Backpropagation", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 498, + 198, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 199, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 199, + 511 + ], + "score": 1.0, + "content": "2.1 Forward Model", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 506, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "We begin by introducing the spiking networks we are going to work with (Figs. 1 and 2). For simplicity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "we have omitted the batch dimension in the derivations but it is later used on the complexity analysis", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 258, + 554 + ], + "score": 1.0, + "content": "and experiments. We have a total of", + "type": "text" + }, + { + "bbox": [ + 258, + 541, + 266, + 550 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 540, + 443, + 554 + ], + "score": 1.0, + "content": "fully connected spiking layers. Each layer", + "type": "text" + }, + { + "bbox": [ + 444, + 541, + 448, + 550 + ], + "score": 0.54, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "consisting of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 549, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 107, + 551, + 126, + 562 + ], + "score": 0.88, + "content": "N ^ { ( l ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 549, + 378, + 568 + ], + "score": 1.0, + "content": "spiking neurons which are fully connected to the next layer", + "type": "text" + }, + { + "bbox": [ + 378, + 552, + 397, + 563 + ], + "score": 0.86, + "content": "l + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 549, + 506, + 568 + ], + "score": 1.0, + "content": "through synaptic weights", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 559, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 107, + 563, + 187, + 576 + ], + "score": 0.91, + "content": "W ^ { ( l ) } \\in \\mathbb { R } ^ { N ^ { ( l ) } \\times N ^ { ( l + 1 ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 559, + 456, + 582 + ], + "score": 1.0, + "content": ". 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This can be achieved using backpropagation through time (BPTT) on the unrolled network.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 641, + 343, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 343, + 653 + ], + "score": 1.0, + "content": "Following (1) and (3) we can derive weight gradients to be", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 672, + 424, + 710 + ], + "lines": [ + { + "bbox": [ + 187, + 672, + 424, + 710 + ], + "spans": [ + { + "bbox": [ + 187, + 672, + 424, + 710 + ], + "score": 0.9, + "content": "\\nabla W _ { i j } ^ { ( l ) } = \\sum _ { t } \\underbrace { \\varepsilon _ { j } ^ { ( l + 1 ) } [ t ] } _ { \\substack { \\sum _ { t } \\dots \\dots \\dots } } \\overbrace { d V _ { j } ^ { ( l + 1 ) } [ t ] } ^ { \\substack { \\sum _ { j } ( l + 1 ) } } \\left( \\sum _ { k < t } \\alpha ^ { t - k - 1 } S _ { i } ^ { ( l ) } [ k ] \\right)", + "type": "interline_equation", + "image_path": "e0bf62dd706e4bed6ee3ab3f336df85795914e728916433560416ca2e840d96d.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 672, + 424, + 691.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 187, + 691.0, + 424, + 710.0 + ], + "spans": [], + "index": 32 + } + ] + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 145, + 69, + 465, + 249 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 69, + 465, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 69, + 465, + 249 + ], + "spans": [ + { + "bbox": [ + 145, + 69, + 465, + 249 + ], + "score": 0.972, + "type": "image", + "image_path": "b74f25e41587a2733ae554635e8ebc612d9c441c1440cfb9fc57e0223683bda6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 145, + 69, + 465, + 129.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 145, + 129.0, + 465, + 189.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 145, + 189.0, + 465, + 249.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 139, + 256, + 470, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 139, + 255, + 471, + 270 + ], + "spans": [ + { + "bbox": [ + 139, + 255, + 471, + 270 + ], + "score": 1.0, + "content": "Figure 2: Illustration of the gradient backpropagating only through active neurons.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 506, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 410, + 285 + ], + "score": 1.0, + "content": "The first two terms account for the leaky part of the model where we define", + "type": "text" + }, + { + "bbox": [ + 410, + 271, + 483, + 284 + ], + "score": 0.91, + "content": "\\alpha = \\exp ( - \\Delta t / \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 148, + 294 + ], + "score": 0.91, + "content": "\\Delta t , \\tau \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "being the time resolution of the simulation and the membrane time constant of the neurons", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 292, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 307 + ], + "score": 1.0, + "content": "respectively. 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For all other layers we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 178, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 145, + 195 + ], + "score": 1.0, + "content": "have that", + "type": "text" + }, + { + "bbox": [ + 145, + 180, + 215, + 195 + ], + "score": 0.94, + "content": "\\varepsilon _ { j } ^ { ( l ) } [ t ] = \\nabla S _ { j } ^ { ( l ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 178, + 506, + 195 + ], + "score": 1.0, + "content": ", that is, the gradient of the loss function with respect to the output spikes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 500, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 140, + 212 + ], + "score": 1.0, + "content": "of layer", + "type": "text" + }, + { + "bbox": [ + 140, + 199, + 144, + 207 + ], + "score": 0.8, + "content": "\\it l", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 194, + 279, + 212 + ], + "score": 1.0, + "content": ". We can obtain an expression for", + "type": "text" + }, + { + "bbox": [ + 280, + 196, + 313, + 211 + ], + "score": 0.94, + "content": "\\nabla S _ { j } ^ { ( l ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 194, + 500, + 212 + ], + "score": 1.0, + "content": "as a function of the gradient of the next layer:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 249, + 447, + 317 + ], + "lines": [ + { + "bbox": [ + 163, + 249, + 447, + 317 + ], + "spans": [ + { + "bbox": [ + 163, + 249, + 447, + 317 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\varepsilon _ { i } ^ { ( l ) } [ t ] = \\nabla S _ { i } ^ { ( l ) } [ t ] = \\displaystyle \\sum _ { j } W _ { i j } \\left( \\sum _ { k > t } \\overbrace { \\nabla S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } } \\overbrace { d V _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } \\alpha ^ { k - t - 1 } \\right) } \\\\ { = \\displaystyle \\sum _ { j } W _ { i j } \\delta _ { j } ^ { ( l + 1 ) } [ t ] , \\qquad l = \\{ 0 , \\ldots L - 2 \\} } \\end{array}", + "type": "interline_equation", + "image_path": "d96b163117218bc9896a7893a0c7a542513ce34cfbd7ce770b3b032d43df1ab3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 163, + 249, + 447, + 271.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 163, + 271.6666666666667, + 447, + 294.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 163, + 294.33333333333337, + 447, + 317.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "All hidden layers need to compute equations (5) and (6). The first one to update their input synaptic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 461, + 352 + ], + "score": 1.0, + "content": "weights and the second one to backpropagate the loss to the previous layer. The first layer", + "type": "text" + }, + { + "bbox": [ + 461, + 340, + 482, + 350 + ], + "score": 0.82, + "content": "\\mathit { l } = 1 \\mathit { \\Theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 338, + 506, + 352 + ], + "score": 1.0, + "content": ") only", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 268, + 365 + ], + "score": 1.0, + "content": "needs to compute (5) and the last layer", + "type": "text" + }, + { + "bbox": [ + 268, + 352, + 304, + 363 + ], + "score": 0.87, + "content": "\\scriptstyle ( l = L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 352, + 409, + 365 + ], + "score": 1.0, + "content": ") will compute a different", + "type": "text" + }, + { + "bbox": [ + 409, + 351, + 446, + 366 + ], + "score": 0.93, + "content": "\\varepsilon _ { j } ^ { ( L - 1 ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 351, + 506, + 366 + ], + "score": 1.0, + "content": "depending on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 363, + 176, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 176, + 375 + ], + "score": 1.0, + "content": "the loss function.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 490, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 490, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 490, + 393 + ], + "score": 1.0, + "content": "We introduce the following definitions which result in the sparse backpropagation learning rule.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 490, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 491, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 307, + 409 + ], + "score": 1.0, + "content": "Definition 1. Given a backpropagation threshold", + "type": "text" + }, + { + "bbox": [ + 307, + 396, + 340, + 407 + ], + "score": 0.92, + "content": "B _ { t h } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 393, + 401, + 409 + ], + "score": 1.0, + "content": "we say neuron", + "type": "text" + }, + { + "bbox": [ + 401, + 397, + 406, + 407 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 393, + 473, + 409 + ], + "score": 1.0, + "content": "is active at time", + "type": "text" + }, + { + "bbox": [ + 474, + 398, + 478, + 405 + ], + "score": 0.4, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 393, + 491, + 409 + ], + "score": 1.0, + "content": "iff", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 424, + 347, + 438 + ], + "lines": [ + { + "bbox": [ + 264, + 424, + 347, + 438 + ], + "spans": [ + { + "bbox": [ + 264, + 424, + 347, + 438 + ], + "score": 0.9, + "content": "| V _ { j } [ t ] - V _ { t h } | < B _ { t h }", + "type": "interline_equation", + "image_path": "119fbc938a7d24264f66d14c1b6fa0a6e5b98f480df8879f82c01d8c34fe0ec3.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 264, + 424, + 347, + 438 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 291, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 292, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 292, + 455 + ], + "score": 1.0, + "content": "Definition 2. The spike gradient is defined as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 468, + 391, + 496 + ], + "lines": [ + { + "bbox": [ + 219, + 468, + 391, + 496 + ], + "spans": [ + { + "bbox": [ + 219, + 468, + 391, + 496 + ], + "score": 0.92, + "content": "\\begin{array} { r } { : = \\left\\{ \\begin{array} { l l } { g ( V _ { j } [ t ] ) , \\quad } & { i f V _ { j } [ t ] i s a c t i \\nu e } \\\\ { 0 , \\quad } & { o t h e r w i s e } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "9e64562c587028674d5adf075b6aa3d9b88535fb60357233d840945fb83ff48f.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 219, + 468, + 391, + 496 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "This means that neurons are only active when their potential is close to the threshold. 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The consequences of this are readily available for the weight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "gradient in (5) as the only terms in the sum that will need to be computed are those in which the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 550, + 452, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 189, + 563 + ], + "score": 1.0, + "content": "postsynaptic neuron", + "type": "text" + }, + { + "bbox": [ + 190, + 552, + 195, + 562 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 550, + 270, + 563 + ], + "score": 1.0, + "content": "was active at time", + "type": "text" + }, + { + "bbox": [ + 270, + 551, + 275, + 560 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 550, + 452, + 563 + ], + "score": 1.0, + "content": ". 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Notice that it is not necessary to compute this last term as this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "is already computed in the forward pass (3). We have chosen not make the spike resetting term in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "(3) differentiable as it has been shown that doing so can result in worse testing accuracy [38] and it", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 241, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 241, + 150 + ], + "score": 1.0, + "content": "increases the computational cost.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 72, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 506, + 210 + ], + "lines": [ + { + "bbox": [ + 103, + 151, + 508, + 171 + ], + "spans": [ + { + "bbox": [ + 103, + 151, + 406, + 171 + ], + "score": 1.0, + "content": "For a typical loss function defined on the output layer activity, the value of", + "type": "text" + }, + { + "bbox": [ + 407, + 156, + 443, + 171 + ], + "score": 0.93, + "content": "\\varepsilon _ { j _ { - } } ^ { ( L - 1 ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 151, + 508, + 171 + ], + "score": 1.0, + "content": "(i.e. in the last", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "layer) is the gradient of the loss with respect to the output layer (see Fig. 1). For all other layers we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 178, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 145, + 195 + ], + "score": 1.0, + "content": "have that", + "type": "text" + }, + { + "bbox": [ + 145, + 180, + 215, + 195 + ], + "score": 0.94, + "content": "\\varepsilon _ { j } ^ { ( l ) } [ t ] = \\nabla S _ { j } ^ { ( l ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 178, + 506, + 195 + ], + "score": 1.0, + "content": ", that is, the gradient of the loss function with respect to the output spikes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 500, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 140, + 212 + ], + "score": 1.0, + "content": "of layer", + "type": "text" + }, + { + "bbox": [ + 140, + 199, + 144, + 207 + ], + "score": 0.8, + "content": "\\it l", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 194, + 279, + 212 + ], + "score": 1.0, + "content": ". We can obtain an expression for", + "type": "text" + }, + { + "bbox": [ + 280, + 196, + 313, + 211 + ], + "score": 0.94, + "content": "\\nabla S _ { j } ^ { ( l ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 194, + 500, + 212 + ], + "score": 1.0, + "content": "as a function of the gradient of the next layer:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 103, + 151, + 508, + 212 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 249, + 447, + 317 + ], + "lines": [ + { + "bbox": [ + 163, + 249, + 447, + 317 + ], + "spans": [ + { + "bbox": [ + 163, + 249, + 447, + 317 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\varepsilon _ { i } ^ { ( l ) } [ t ] = \\nabla S _ { i } ^ { ( l ) } [ t ] = \\displaystyle \\sum _ { j } W _ { i j } \\left( \\sum _ { k > t } \\overbrace { \\nabla S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } } \\overbrace { d V _ { j } ^ { ( l + 1 ) } [ k ] } ^ { \\overbrace { d S _ { j } ^ { ( l + 1 ) } [ k ] } } \\alpha ^ { k - t - 1 } \\right) } \\\\ { = \\displaystyle \\sum _ { j } W _ { i j } \\delta _ { j } ^ { ( l + 1 ) } [ t ] , \\qquad l = \\{ 0 , \\ldots L - 2 \\} } \\end{array}", + "type": "interline_equation", + "image_path": "d96b163117218bc9896a7893a0c7a542513ce34cfbd7ce770b3b032d43df1ab3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 163, + 249, + 447, + 271.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 163, + 271.6666666666667, + 447, + 294.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 163, + 294.33333333333337, + 447, + 317.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "All hidden layers need to compute equations (5) and (6). The first one to update their input synaptic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 461, + 352 + ], + "score": 1.0, + "content": "weights and the second one to backpropagate the loss to the previous layer. The first layer", + "type": "text" + }, + { + "bbox": [ + 461, + 340, + 482, + 350 + ], + "score": 0.82, + "content": "\\mathit { l } = 1 \\mathit { \\Theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 338, + 506, + 352 + ], + "score": 1.0, + "content": ") only", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 268, + 365 + ], + "score": 1.0, + "content": "needs to compute (5) and the last layer", + "type": "text" + }, + { + "bbox": [ + 268, + 352, + 304, + 363 + ], + "score": 0.87, + "content": "\\scriptstyle ( l = L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 352, + 409, + 365 + ], + "score": 1.0, + "content": ") will compute a different", + "type": "text" + }, + { + "bbox": [ + 409, + 351, + 446, + 366 + ], + "score": 0.93, + "content": "\\varepsilon _ { j } ^ { ( L - 1 ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 351, + 506, + 366 + ], + "score": 1.0, + "content": "depending on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 363, + 176, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 176, + 375 + ], + "score": 1.0, + "content": "the loss function.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 327, + 506, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 490, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 490, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 490, + 393 + ], + "score": 1.0, + "content": "We introduce the following definitions which result in the sparse backpropagation learning rule.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 378, + 490, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 490, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 491, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 307, + 409 + ], + "score": 1.0, + "content": "Definition 1. Given a backpropagation threshold", + "type": "text" + }, + { + "bbox": [ + 307, + 396, + 340, + 407 + ], + "score": 0.92, + "content": "B _ { t h } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 393, + 401, + 409 + ], + "score": 1.0, + "content": "we say neuron", + "type": "text" + }, + { + "bbox": [ + 401, + 397, + 406, + 407 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 393, + 473, + 409 + ], + "score": 1.0, + "content": "is active at time", + "type": "text" + }, + { + "bbox": [ + 474, + 398, + 478, + 405 + ], + "score": 0.4, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 393, + 491, + 409 + ], + "score": 1.0, + "content": "iff", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 393, + 491, + 409 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 424, + 347, + 438 + ], + "lines": [ + { + "bbox": [ + 264, + 424, + 347, + 438 + ], + "spans": [ + { + "bbox": [ + 264, + 424, + 347, + 438 + ], + "score": 0.9, + "content": "| V _ { j } [ t ] - V _ { t h } | < B _ { t h }", + "type": "interline_equation", + "image_path": "119fbc938a7d24264f66d14c1b6fa0a6e5b98f480df8879f82c01d8c34fe0ec3.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 264, + 424, + 347, + 438 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 291, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 292, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 292, + 455 + ], + "score": 1.0, + "content": "Definition 2. The spike gradient is defined as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 440, + 292, + 455 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 468, + 391, + 496 + ], + "lines": [ + { + "bbox": [ + 219, + 468, + 391, + 496 + ], + "spans": [ + { + "bbox": [ + 219, + 468, + 391, + 496 + ], + "score": 0.92, + "content": "\\begin{array} { r } { : = \\left\\{ \\begin{array} { l l } { g ( V _ { j } [ t ] ) , \\quad } & { i f V _ { j } [ t ] i s a c t i \\nu e } \\\\ { 0 , \\quad } & { o t h e r w i s e } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "9e64562c587028674d5adf075b6aa3d9b88535fb60357233d840945fb83ff48f.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 219, + 468, + 391, + 496 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "This means that neurons are only active when their potential is close to the threshold. Applying these", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "two definitions to (5) and (6) results in the gradients only backpropagating through active neurons", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "at each time step as shown in Fig. 2. The consequences of this are readily available for the weight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "gradient in (5) as the only terms in the sum that will need to be computed are those in which the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 550, + 452, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 189, + 563 + ], + "score": 1.0, + "content": "postsynaptic neuron", + "type": "text" + }, + { + "bbox": [ + 190, + 552, + 195, + 562 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 550, + 270, + 563 + ], + "score": 1.0, + "content": "was active at time", + "type": "text" + }, + { + "bbox": [ + 270, + 551, + 275, + 560 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 550, + 452, + 563 + ], + "score": 1.0, + "content": ". 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We can use a recurrent relation to compute", + "type": "text" + }, + { + "bbox": [ + 345, + 73, + 363, + 85 + ], + "score": 0.82, + "content": "\\delta _ { j } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 70, + 401, + 88 + ], + "score": 1.0, + "content": "given by", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 86, + 493, + 121 + ], + "lines": [ + { + "bbox": [ + 117, + 86, + 493, + 121 + ], + "spans": [ + { + "bbox": [ + 117, + 86, + 493, + 121 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\delta _ { j } [ t ] = \\left\\{ \\begin{array} { l l } { \\alpha ^ { n } \\delta _ { j } [ t + n ] , } & { \\quad i f \\frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 \\leq k \\leq t + n } \\\\ { \\nabla S _ { j } [ t + 1 ] \\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } + \\alpha ^ { n } \\delta _ { j } [ t + n ] , } & { \\quad i f \\frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 < k \\leq t + n , \\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \\neq 0 . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "fcaa49dd8f99d89a5b5ceb3c53f4c06eb5d5698e454d22382c00e699c3a4d3b8.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 117, + 86, + 493, + 97.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 97.66666666666667, + 493, + 109.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 117, + 109.33333333333334, + 493, + 121.00000000000001 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 136, + 506, + 178 + ], + "lines": [ + { + "bbox": [ + 104, + 132, + 504, + 156 + ], + "spans": [ + { + "bbox": [ + 104, + 132, + 276, + 156 + ], + "score": 1.0, + "content": "This means that we only need to compute", + "type": "text" + }, + { + "bbox": [ + 277, + 137, + 311, + 153 + ], + "score": 0.93, + "content": "\\delta _ { j } ^ { ( l + 1 ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 132, + 452, + 156 + ], + "score": 1.0, + "content": "at those time steps in which either", + "type": "text" + }, + { + "bbox": [ + 453, + 136, + 504, + 154 + ], + "score": 0.95, + "content": "\\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \\neq 0", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 151, + 507, + 169 + ], + "spans": [ + { + "bbox": [ + 104, + 151, + 118, + 169 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 118, + 154, + 145, + 169 + ], + "score": 0.92, + "content": "V _ { j \\ldots } ^ { ( l ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 151, + 243, + 169 + ], + "score": 1.0, + "content": "is active (see Appendix", + "type": "text" + }, + { + "bbox": [ + 243, + 158, + 251, + 165 + ], + "score": 0.54, + "content": "\\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 151, + 507, + 169 + ], + "score": 1.0, + "content": "for a visualisation of this computation). Thus, we end up with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 165, + 369, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 369, + 179 + ], + "score": 1.0, + "content": "the following sparse expression for computing the spike gradient.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 184, + 414, + 218 + ], + "lines": [ + { + "bbox": [ + 196, + 184, + 414, + 218 + ], + "spans": [ + { + "bbox": [ + 196, + 184, + 414, + 218 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nabla S _ { j } ^ { ( l ) } [ t ] = \\left\\{ \\sum _ { j } W _ { i j } \\delta _ { j } ^ { ( l + 1 ) } [ t ] , \\qquad V _ { j } ^ { ( l ) } [ t ] \\mathrm { ~ i s ~ a c t i v e } \\right. } \\\\ { 0 , \\qquad \\mathrm { ~ o t h e r w i s e } \\qquad } \\end{array}", + "type": "interline_equation", + "image_path": "13bb9d5449af70b8033e8acd4fdbccc18cf55ca5f3ce5e65e69f4233cb231222.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 184, + 414, + 201.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 196, + 201.0, + 414, + 218.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 228, + 216, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 217, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 217, + 242 + ], + "score": 1.0, + "content": "2.3 Complexity analysis", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 149, + 261 + ], + "score": 1.0, + "content": "We define", + "type": "text" + }, + { + "bbox": [ + 150, + 248, + 191, + 260 + ], + "score": 0.9, + "content": "\\rho _ { l } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 248, + 402, + 261 + ], + "score": 1.0, + "content": "to be the probability that a neuron is active in layer", + "type": "text" + }, + { + "bbox": [ + 402, + 249, + 406, + 258 + ], + "score": 0.64, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "at a given time. Table 1", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "summarises the computational complexity of the gradient computation in terms of number of sums", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 212, + 282 + ], + "score": 1.0, + "content": "and products. We use here", + "type": "text" + }, + { + "bbox": [ + 213, + 271, + 223, + 280 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "to refer to the number of neurons in either the input or output layer and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 107, + 281, + 116, + 291 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 281, + 439, + 293 + ], + "score": 1.0, + "content": "to refer to the batch size. In a slight abuse of notation we include the constants", + "type": "text" + }, + { + "bbox": [ + 439, + 283, + 448, + 292 + ], + "score": 0.83, + "content": "\\rho _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "as part of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 292, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 496, + 304 + ], + "score": 1.0, + "content": "complexity expressions. Details of how these expression were obtained can be found in Appendix", + "type": "text" + }, + { + "bbox": [ + 496, + 292, + 504, + 302 + ], + "score": 0.26, + "content": "\\mathbf { B }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 303, + 476, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 454, + 316 + ], + "score": 1.0, + "content": "and are based on a reasonable efficient algorithm that uses memoisation for computing", + "type": "text" + }, + { + "bbox": [ + 454, + 303, + 473, + 315 + ], + "score": 0.86, + "content": "\\delta _ { j } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 303, + 476, + 316 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "table", + "bbox": [ + 198, + 340, + 410, + 413 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 234, + 322, + 377, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 232, + 320, + 377, + 337 + ], + "spans": [ + { + "bbox": [ + 232, + 320, + 377, + 337 + ], + "score": 1.0, + "content": "Table 1: Computational complexity", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 198, + 340, + 410, + 413 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 198, + 340, + 410, + 413 + ], + "spans": [ + { + "bbox": [ + 198, + 340, + 410, + 413 + ], + "score": 0.981, + "html": "
OriginalSparse
Sums VW(l)O(BTN²) O(pt+1BTN2)
Products VW(l)O(BTN2) O(pt+1BTN2)
Sums Vs(t)O(BTN2) O(ρ𝑖BTN²))
Products VS(l)O(BTN2) O(pBTN²)
", + "type": "table", + "image_path": "cb9706c2e53210403c36086f641b4ea03b4b4f023a74ed4dd356bc4fce609a47.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 198, + 340, + 410, + 354.6 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 198, + 354.6, + 410, + 369.20000000000005 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 198, + 369.20000000000005, + 410, + 383.80000000000007 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 198, + 383.80000000000007, + 410, + 398.4000000000001 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 198, + 398.4000000000001, + 410, + 413.0000000000001 + ], + "spans": [], + "index": 21 + } + ] + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 108, + 421, + 504, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "In order for the sparse gradients to work better than their dense counterpart we need to have a balance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "between having few enough active neurons as to make the sparse backpropagation efficient while at", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "the same time keeping enough neurons active to allow the gradient to backpropagate. We later show", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 454, + 386, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 386, + 466 + ], + "score": 1.0, + "content": "in section 3 that this is the case when testing it on real world datasets.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 478, + 279, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 279, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 279, + 492 + ], + "score": 1.0, + "content": "2.4 Sparse learning rule interpretation", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "The surrogate derivative we propose, while not having a constrained functional form for active", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "neurons, imposes a zero gradient for the inactive ones. Looking back into equation (5) we can identify", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "it now as a three-factor Hebbian learning rule [47]. With our surrogate derivative definition, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "second term (spike derivative) measures whether the postsynaptic neuron is close to spiking (i.e. its", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 542, + 504, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 504, + 553 + ], + "score": 1.0, + "content": "membrane potential is close to the threshold). The third term (input trace) measures the presynaptic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "neuron activity in the most recent timesteps. Finally the first term (gradient from next layer) decides", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "whether the synaptic activity improves or hinders performance with respect to some loss function.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "We note that this is not exactly a Hebbian learning rule as neurons that get close to spike but do not", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "spike can influence the synaptic weight update and as such, the second term is simply a surrogate of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 596, + 230, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 230, + 609 + ], + "score": 1.0, + "content": "the postsynaptic spike activity.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 626 + ], + "score": 1.0, + "content": "Previous surrogate gradient descent methods have used a similar approach to implement backpropaga-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "tion on SNNs [18, 36, 48, 39, 38], often using surrogate gradient descent definitions that are maximal", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "when the membrane is at the threshold and decay as the potential moves away from it. In fact, in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "Esser et al. [18] the surrogate gradient definition given implies that neurons backpropagate a zero", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 142, + 669 + ], + "score": 1.0, + "content": "gradient", + "type": "text" + }, + { + "bbox": [ + 142, + 656, + 162, + 667 + ], + "score": 0.89, + "content": "3 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "of the time. However, as shown in the next section, neurons can be active a lot less", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "often when presented with spikes generated from real world data and still give the same test accuracy.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "This is only possible because surrogate gradient approximations to the threshold function yield much", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "larger values when the potential is closer to spike thus concentrating most of the gradient on active", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "neurons. This fact makes it possible to fully profit from our sparse learning rule for a much reduced", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 712, + 239, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 239, + 723 + ], + "score": 1.0, + "content": "computational and memory cost.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5 + } + ], + "page_idx": 4, + "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": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 401, + 85 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 401, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 344, + 88 + ], + "score": 1.0, + "content": "Proposition 1. We can use a recurrent relation to compute", + "type": "text" + }, + { + "bbox": [ + 345, + 73, + 363, + 85 + ], + "score": 0.82, + "content": "\\delta _ { j } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 70, + 401, + 88 + ], + "score": 1.0, + "content": "given by", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 70, + 401, + 88 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 86, + 493, + 121 + ], + "lines": [ + { + "bbox": [ + 117, + 86, + 493, + 121 + ], + "spans": [ + { + "bbox": [ + 117, + 86, + 493, + 121 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\delta _ { j } [ t ] = \\left\\{ \\begin{array} { l l } { \\alpha ^ { n } \\delta _ { j } [ t + n ] , } & { \\quad i f \\frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 \\leq k \\leq t + n } \\\\ { \\nabla S _ { j } [ t + 1 ] \\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } + \\alpha ^ { n } \\delta _ { j } [ t + n ] , } & { \\quad i f \\frac { d S _ { j } [ k ] } { d V _ { j } [ k ] } = 0 , f o r t + 1 < k \\leq t + n , \\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \\neq 0 . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "fcaa49dd8f99d89a5b5ceb3c53f4c06eb5d5698e454d22382c00e699c3a4d3b8.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 117, + 86, + 493, + 97.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 97.66666666666667, + 493, + 109.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 117, + 109.33333333333334, + 493, + 121.00000000000001 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 136, + 506, + 178 + ], + "lines": [ + { + "bbox": [ + 104, + 132, + 504, + 156 + ], + "spans": [ + { + "bbox": [ + 104, + 132, + 276, + 156 + ], + "score": 1.0, + "content": "This means that we only need to compute", + "type": "text" + }, + { + "bbox": [ + 277, + 137, + 311, + 153 + ], + "score": 0.93, + "content": "\\delta _ { j } ^ { ( l + 1 ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 132, + 452, + 156 + ], + "score": 1.0, + "content": "at those time steps in which either", + "type": "text" + }, + { + "bbox": [ + 453, + 136, + 504, + 154 + ], + "score": 0.95, + "content": "\\frac { d S _ { j } [ t + 1 ] } { d V _ { j } [ t + 1 ] } \\neq 0", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 151, + 507, + 169 + ], + "spans": [ + { + "bbox": [ + 104, + 151, + 118, + 169 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 118, + 154, + 145, + 169 + ], + "score": 0.92, + "content": "V _ { j \\ldots } ^ { ( l ) } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 151, + 243, + 169 + ], + "score": 1.0, + "content": "is active (see Appendix", + "type": "text" + }, + { + "bbox": [ + 243, + 158, + 251, + 165 + ], + "score": 0.54, + "content": "\\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 151, + 507, + 169 + ], + "score": 1.0, + "content": "for a visualisation of this computation). Thus, we end up with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 165, + 369, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 369, + 179 + ], + "score": 1.0, + "content": "the following sparse expression for computing the spike gradient.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 132, + 507, + 179 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 184, + 414, + 218 + ], + "lines": [ + { + "bbox": [ + 196, + 184, + 414, + 218 + ], + "spans": [ + { + "bbox": [ + 196, + 184, + 414, + 218 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nabla S _ { j } ^ { ( l ) } [ t ] = \\left\\{ \\sum _ { j } W _ { i j } \\delta _ { j } ^ { ( l + 1 ) } [ t ] , \\qquad V _ { j } ^ { ( l ) } [ t ] \\mathrm { ~ i s ~ a c t i v e } \\right. } \\\\ { 0 , \\qquad \\mathrm { ~ o t h e r w i s e } \\qquad } \\end{array}", + "type": "interline_equation", + "image_path": "13bb9d5449af70b8033e8acd4fdbccc18cf55ca5f3ce5e65e69f4233cb231222.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 184, + 414, + 201.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 196, + 201.0, + 414, + 218.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 228, + 216, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 217, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 217, + 242 + ], + "score": 1.0, + "content": "2.3 Complexity analysis", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 149, + 261 + ], + "score": 1.0, + "content": "We define", + "type": "text" + }, + { + "bbox": [ + 150, + 248, + 191, + 260 + ], + "score": 0.9, + "content": "\\rho _ { l } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 248, + 402, + 261 + ], + "score": 1.0, + "content": "to be the probability that a neuron is active in layer", + "type": "text" + }, + { + "bbox": [ + 402, + 249, + 406, + 258 + ], + "score": 0.64, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "at a given time. Table 1", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "summarises the computational complexity of the gradient computation in terms of number of sums", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 212, + 282 + ], + "score": 1.0, + "content": "and products. We use here", + "type": "text" + }, + { + "bbox": [ + 213, + 271, + 223, + 280 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "to refer to the number of neurons in either the input or output layer and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 107, + 281, + 116, + 291 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 281, + 439, + 293 + ], + "score": 1.0, + "content": "to refer to the batch size. In a slight abuse of notation we include the constants", + "type": "text" + }, + { + "bbox": [ + 439, + 283, + 448, + 292 + ], + "score": 0.83, + "content": "\\rho _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "as part of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 292, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 496, + 304 + ], + "score": 1.0, + "content": "complexity expressions. Details of how these expression were obtained can be found in Appendix", + "type": "text" + }, + { + "bbox": [ + 496, + 292, + 504, + 302 + ], + "score": 0.26, + "content": "\\mathbf { B }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 303, + 476, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 454, + 316 + ], + "score": 1.0, + "content": "and are based on a reasonable efficient algorithm that uses memoisation for computing", + "type": "text" + }, + { + "bbox": [ + 454, + 303, + 473, + 315 + ], + "score": 0.86, + "content": "\\delta _ { j } [ t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 303, + 476, + 316 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 248, + 506, + 316 + ] + }, + { + "type": "table", + "bbox": [ + 198, + 340, + 410, + 413 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 234, + 322, + 377, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 232, + 320, + 377, + 337 + ], + "spans": [ + { + "bbox": [ + 232, + 320, + 377, + 337 + ], + "score": 1.0, + "content": "Table 1: Computational complexity", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 198, + 340, + 410, + 413 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 198, + 340, + 410, + 413 + ], + "spans": [ + { + "bbox": [ + 198, + 340, + 410, + 413 + ], + "score": 0.981, + "html": "
OriginalSparse
Sums VW(l)O(BTN²) O(pt+1BTN2)
Products VW(l)O(BTN2) O(pt+1BTN2)
Sums Vs(t)O(BTN2) O(ρ𝑖BTN²))
Products VS(l)O(BTN2) O(pBTN²)
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We later show", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 454, + 386, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 386, + 466 + ], + "score": 1.0, + "content": "in section 3 that this is the case when testing it on real world datasets.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 422, + 506, + 466 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 478, + 279, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 279, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 279, + 492 + ], + "score": 1.0, + "content": "2.4 Sparse learning rule interpretation", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "The surrogate derivative we propose, while not having a constrained functional form for active", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "neurons, imposes a zero gradient for the inactive ones. Looking back into equation (5) we can identify", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "it now as a three-factor Hebbian learning rule [47]. With our surrogate derivative definition, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "second term (spike derivative) measures whether the postsynaptic neuron is close to spiking (i.e. its", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 542, + 504, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 504, + 553 + ], + "score": 1.0, + "content": "membrane potential is close to the threshold). 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Finally the first term (gradient from next layer) decides", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "whether the synaptic activity improves or hinders performance with respect to some loss function.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "We note that this is not exactly a Hebbian learning rule as neurons that get close to spike but do not", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "spike can influence the synaptic weight update and as such, the second term is simply a surrogate of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 596, + 230, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 230, + 609 + ], + "score": 1.0, + "content": "the postsynaptic spike activity.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 498, + 506, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 626 + ], + "score": 1.0, + "content": "Previous surrogate gradient descent methods have used a similar approach to implement backpropaga-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "tion on SNNs [18, 36, 48, 39, 38], often using surrogate gradient descent definitions that are maximal", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "when the membrane is at the threshold and decay as the potential moves away from it. In fact, in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "Esser et al. [18] the surrogate gradient definition given implies that neurons backpropagate a zero", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 142, + 669 + ], + "score": 1.0, + "content": "gradient", + "type": "text" + }, + { + "bbox": [ + 142, + 656, + 162, + 667 + ], + "score": 0.89, + "content": "3 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "of the time. However, as shown in the next section, neurons can be active a lot less", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "often when presented with spikes generated from real world data and still give the same test accuracy.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "This is only possible because surrogate gradient approximations to the threshold function yield much", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "larger values when the potential is closer to spike thus concentrating most of the gradient on active", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "neurons. 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Namely, artificially encoding a dataset into spikes (Fashion-MNIST), reading spikes from a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 260, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 505, + 276 + ], + "score": 1.0, + "content": "neuromorphic sensor such as a DVS camera (N-MNIST) and encoding complex temporally-varying", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 272, + 203, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 203, + 285 + ], + "score": 1.0, + "content": "data into spikes (SHD).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 97, + 506, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 506, + 301 + ], + "score": 1.0, + "content": "We run all experiments on three-layer fully connected network as in Fig. 1, where the last layer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "(readout) has an infinite threshold. Both spiking layers in the middle have the same number of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 179, + 324 + ], + "score": 1.0, + "content": "neurons. We used", + "type": "text" + }, + { + "bbox": [ + 179, + 310, + 302, + 323 + ], + "score": 0.91, + "content": "g ( V ) : = 1 / ( \\beta | V - V _ { t h } | \\dot { + } 1 ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "for the surrogate gradient as in [38]. See Appendix", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "score": 1.0, + "content": "E for all training details. 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F-MNIST1.060.8799.1398.94
N-MNIST1.120.7799.2398.88
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All figures except B are a 5", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "sample average. Standard error in the mean is displayed (although too small to be easily visualised).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 433, + 376 + ], + "score": 1.0, + "content": "A. 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Note how both", + "type": "text" + }, + { + "bbox": [ + 352, + 385, + 382, + 397 + ], + "score": 0.9, + "content": "\\bar { \\nabla W } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 384, + 506, + 401 + ], + "score": 1.0, + "content": "are nearly identical in despite", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 327, + 412 + ], + "score": 1.0, + "content": "being computed using a small fraction of the values of", + "type": "text" + }, + { + "bbox": [ + 327, + 397, + 354, + 409 + ], + "score": 0.91, + "content": "\\nabla S ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 397, + 506, + 412 + ], + "score": 1.0, + "content": "in the sparse case. C. Loss evolution", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 409, + 507, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 507, + 423 + ], + "score": 1.0, + "content": "on the SHD dataset using both algorithms. D. Final test accuracy on all datasets using both methods.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "Importantly, in dense backpropagation methods the temporal dimension of datasets and models is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 255, + 464 + ], + "score": 1.0, + "content": "necessarily limited to a maximum of", + "type": "text" + }, + { + "bbox": [ + 255, + 452, + 286, + 462 + ], + "score": 0.84, + "content": "\\sim 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "since gradients must be computed at all steps [46, 39].", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "This means that a lot of negligible gradients are computed. 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To put this speedup into", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 481, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 506, + 495 + ], + "score": 1.0, + "content": "perspective Figure 4B shows the time taken for backpropagating the second layer in the original case", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 133, + 506 + ], + "score": 1.0, + "content": "(about", + "type": "text" + }, + { + "bbox": [ + 133, + 493, + 162, + 504 + ], + "score": 0.38, + "content": "6 4 0 \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 492, + 268, + 506 + ], + "score": 1.0, + "content": ") and the sparse case (about", + "type": "text" + }, + { + "bbox": [ + 268, + 493, + 291, + 504 + ], + "score": 0.33, + "content": "1 7 \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "). 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We included the overall speedup which takes into", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 330, + 416, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 416, + 343 + ], + "score": 1.0, + "content": "consideration both the forward and backward times spent in the second layer.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 408 + ], + "lines": [], + "index": 12.5, + "bbox_fs": [ + 105, + 362, + 506, + 409 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 424, + 430, + 452 + ], + "lines": [ + { + "bbox": [ + 181, + 424, + 430, + 452 + ], + "spans": [ + { + "bbox": [ + 181, + 424, + 430, + 452 + ], + "score": 0.49, + "content": "{ \\mathrm { M e m o r y ~ s a v e d } } = 1 0 0 \\times { \\frac { { \\mathrm { m e m o r y } } _ { - } { \\mathrm { o r i g i n a l } } - { \\mathrm { m e m o r y } } _ { - } { \\mathrm { s p a r s e } } } { \\mathrm { m e m o r y } } }", + "type": "interline_equation", + "image_path": "ee18c725ada4980fc70daf2ae9bd5892aab6b8f65c3119f7fb2c7a2bc4e9d34e.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 181, + 424, + 430, + 433.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 181, + 433.3333333333333, + 430, + 442.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 181, + 442.66666666666663, + 430, + 451.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "Figure 4A shows the speedup and memory saved on a 200 neuron hidden layer (both hidden layers).", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 240, + 484 + ], + "score": 1.0, + "content": "Backpropagation time is faster by", + "type": "text" + }, + { + "bbox": [ + 240, + 471, + 257, + 481 + ], + "score": 0.51, + "content": "4 0 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 469, + 385, + 484 + ], + "score": 1.0, + "content": "and GPU memory is reduced by", + "type": "text" + }, + { + "bbox": [ + 385, + 470, + 405, + 482 + ], + "score": 0.87, + "content": "3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 469, + 506, + 484 + ], + "score": 1.0, + "content": ". To put this speedup into", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 481, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 506, + 495 + ], + "score": 1.0, + "content": "perspective Figure 4B shows the time taken for backpropagating the second layer in the original case", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 133, + 506 + ], + "score": 1.0, + "content": "(about", + "type": "text" + }, + { + "bbox": [ + 133, + 493, + 162, + 504 + ], + "score": 0.38, + "content": "6 4 0 \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 492, + 268, + 506 + ], + "score": 1.0, + "content": ") and the sparse case (about", + "type": "text" + }, + { + "bbox": [ + 268, + 493, + 291, + 504 + ], + "score": 0.33, + "content": "1 7 \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "). It also shows that the time taken for backpropagating", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 502, + 507, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 502, + 507, + 517 + ], + "score": 1.0, + "content": "is nearly constant during training as it could be expected after studying the network activity in Fig.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 513, + 125, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 125, + 526 + ], + "score": 1.0, + "content": "3A.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 460, + 507, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 530, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "We also tested how robust is this implementation for a varying number of hidden neurons (in both", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 542, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 506, + 553 + ], + "score": 1.0, + "content": "layers). 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The", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "results shown in Fig. 4C show that the backward speedup and memory saved remain constant as we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 597, + 504, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 504, + 608 + ], + "score": 1.0, + "content": "increase the number of hidden neurons but the overall speedup increases substantially. Given that the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "forward time is the same for both methods (see Appendix F) this shows that as we increase the number", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "of neurons, the backward time takes most of the computation and thus the sparse backpropagation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 630, + 210, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 210, + 641 + ], + "score": 1.0, + "content": "becomes more important.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 531, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "score": 1.0, + "content": "We also note that these results vary depending on the GPU used, Figure 4 was obtained from running", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "on an RTX6000 GPU. We also run this on smaller GPUs (GTX1060 and GTX1080Ti) and found", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "that the improvement is even better reaching up to a 70x faster backward pass on N-MNIST and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "SHD. These results can found in Appendix F. 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The results are consistent with our previous findings and show that the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 500, + 279, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 279, + 512 + ], + "score": 1.0, + "content": "gradient is not loss in deeper architectures.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "We also run a experiment with a convolutional and pooling layers on the Fashion-MNIST dataset", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 217, + 540 + ], + "score": 1.0, + "content": "obtaining a test accuracy of", + "type": "text" + }, + { + "bbox": [ + 217, + 527, + 244, + 538 + ], + "score": 0.89, + "content": "8 6 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 527, + 442, + 540 + ], + "score": 1.0, + "content": "when using our spike gradient approximation and", + "type": "text" + }, + { + "bbox": [ + 442, + 527, + 470, + 537 + ], + "score": 0.88, + "content": "8 6 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "with the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "original gradients. 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This proves that our gradient approximation is able to train", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "on more complex architectures although it still remains to be shown whether efficient sparse operators", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 582, + 403, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 403, + 593 + ], + "score": 1.0, + "content": "can be implemented for this purpose. 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As", + "type": "text" + }, + { + "bbox": [ + 363, + 700, + 380, + 711 + ], + "score": 0.89, + "content": "B _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "gets closer to 1 the number of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 711, + 401, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 401, + 724 + ], + "score": 1.0, + "content": "active neurons decreases rapidly until there are no gradients to propagate.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 644, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 179, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 181, + 86 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 181, + 86 + ], + "score": 1.0, + "content": "4 Discussion", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "Recent interest in neuromorphic computing has lead to energy efficient emulation of SNNs in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "specialised hardware. 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This is no surprise since for now there", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "is few reasons to believe that sparse tensor operations will significantly accelerate neural network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "performance. Our work aims to challenge this view and motivate the adoption of efficient sparse", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 459, + 480, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 480, + 474 + ], + "score": 1.0, + "content": "routines by showing for the first time that sparse BPTT can be faster without losing accuracy.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "We implemented and tested our method on real data on fully connected SNNs and simulated it", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "in convolutional SNNs. 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This is possible because the surrogate gradient used to approximate the derivative of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "spiking function is larger when the membrane potential is closer to the threshold resulting in most of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "the gradient being concentrated on active neurons. Previous work has used, to a lesser extent, similar", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "ideas. Esser et al. [18] used a surrogate gradient function which effectively resulted in about one third", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "the neurons backpropagating a zero gradient. Later, Pozzi, et al. [54] developed a learning rule where", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 344, + 507, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 507, + 358 + ], + "score": 1.0, + "content": "only the weights of neurons directly affecting the activity of the selected output class are updated,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 355, + 498, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 414, + 370 + ], + "score": 1.0, + "content": "however, the trial and error nature of the learning procedure resulted in up to", + "type": "text" + }, + { + "bbox": [ + 414, + 357, + 433, + 367 + ], + "score": 0.48, + "content": "3 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 355, + 498, + 370 + ], + "score": 1.0, + "content": "slower training.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 247, + 507, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 387 + ], + "score": 1.0, + "content": "However, due to the lack of efficient sparse operators in most auto-differentiation platforms, every", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "layer type requires to develop its own sparse CUDA kernel in order to be competitive with current", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "heavily optimised libraries. In our work, we concentrated on implementing and testing this in spiking", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 404, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 418 + ], + "score": 1.0, + "content": "fully connected layers and we managed to show that sparse backpropagation is a lot faster and uses", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "score": 1.0, + "content": "less memory. Developing these sparse kernels is not trivial and we are aware that having to do", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 427, + 504, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 504, + 439 + ], + "score": 1.0, + "content": "so for each layer is an important limitation of our work. This is no surprise since for now there", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "is few reasons to believe that sparse tensor operations will significantly accelerate neural network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "performance. Our work aims to challenge this view and motivate the adoption of efficient sparse", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 459, + 480, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 480, + 474 + ], + "score": 1.0, + "content": "routines by showing for the first time that sparse BPTT can be faster without losing accuracy.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 370, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "We implemented and tested our method on real data on fully connected SNNs and simulated it", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "in convolutional SNNs. The same method can be extended to any SNN topology as long as the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "neurons use a threshold activation function and the input is spiking data. However, due to the lack of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 507, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 523 + ], + "score": 1.0, + "content": "efficient sparse operators in most auto-differentiation platforms, every layer type requires to develop", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "its own sparse CUDA kernel in order to be competitive with current heavily optimised libraries.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "Developing these sparse kernels is not trivial and we are aware that having to do so for each layer is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "an important limitation of our work. This is no surprise since for now there is few reasons to believe", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "that sparse tensor operations will significantly accelerate neural network performance. Our work", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "aims to challenge this view and motivate the adoption of efficient sparse routines by showing for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "the first time that sparse spiking BPTT can be faster while achieving the same accuracy as its dense", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "counterpart. Nevertheless, ideally, a dynamic backward graph should be generated each time the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "backward pass needs to be computed and consequently a GPU may not be the best processing unit", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "for this task. A specialised neuromorphic implementation or an FPGA may be more suited to carry", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "out the gradient computation task as this graph changes every batch update. Additionally, while we", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 628, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 643 + ], + "score": 1.0, + "content": "have reduced the number of operations required for training several times, we have only achieved up", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 640, + 466, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 123, + 653 + ], + "score": 1.0, + "content": "to a", + "type": "text" + }, + { + "bbox": [ + 124, + 640, + 143, + 650 + ], + "score": 0.87, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 640, + 466, + 653 + ], + "score": 1.0, + "content": "memory reduction thus, memory requirements remains an important bottleneck.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 475, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "In summary, our sparse backpropagation algorithm for spiking neural networks tackles the main", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "bottleneck in training SNNs and opens the possibility of a fast, energy and memory efficient end-to-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "end training on spiking neural networks which will benefit neuromorphic training and computational", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "neuroscience research. Sparse spike backpropagation is a necessary step towards a fully efficient", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "on-chip training which will be ultimately required to achieve the full potential of neuromorphic", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 711, + 154, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 154, + 725 + ], + "score": 1.0, + "content": "computing.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 656, + 506, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 70, + 339, + 85 + ], + "lines": [ + { + "bbox": [ + 105, + 69, + 341, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 69, + 341, + 88 + ], + "score": 1.0, + "content": "Acknowledgments and Disclosure of Funding", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 506, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 506, + 109 + ], + "score": 1.0, + "content": "We thank Friedemann Zenke, Thomas Nowotny and James Knight for their feedback and support.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "This work was supported by the EPSRC Centre for Doctoral Training in High Performance Embedded", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 379, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 379, + 131 + ], + "score": 1.0, + "content": "and DistributedSystems (HiPEDS, Grant Reference EP/L016796/1)", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 146, + 163, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 165, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 165, + 161 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 158, + 507, + 727 + ], + "lines": [ + { + "bbox": [ + 110, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 110, + 165, + 506, + 179 + ], + "score": 1.0, + "content": "[1] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. 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OriginalSparse
Sums VW(l)O(BTN²) O(pt+1BTN2)
Products VW(l)O(BTN2) O(pt+1BTN2)
Sums Vs(t)O(BTN2) O(ρ𝑖BTN²))
Products VS(l)O(BTN2) O(pBTN²)
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F-MNIST1.060.8799.1398.94
N-MNIST1.120.7799.2398.88
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0000000000000000000000000000000000000000..7c046e5a1d8b0b68bac3f7eb49a5b5e64d7faac3 --- /dev/null +++ b/parse/train/eedamFFTcK5/eedamFFTcK5.md @@ -0,0 +1,439 @@ +# ML Agent Safety Mechanisms based on Counterfactual Planning + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 We present counterfactual planning as a design approach for creating a range of +2 safety mechanisms for machine learning agents. We specifically target the safety +3 problem of keeping control over hypothetical future AGI agents. The key step +4 in counterfactual planning is to use the agent’s machine learning system to con +5 struct a counterfactual world model, designed to be different from the real world +6 the agent is in. A counterfactual planning agent determines the action that best +7 maximizes expected utility in this counterfactual planning world, and then per +8 forms the same action in the real world. The design approach is built around a +9 two-diagram graphical notation that provides a specific vantage point on the con +10 struction of online machine learning agents, a vantage point designed to make the +11 problem of control more tractable. We show two examples where the construction +12 of a counterfactual planning world acts to suppress certain unsafe agent incentives, +13 incentives for the agent to take control over its own safety mechanisms. + +# 14 1 Introduction + +15 Artificial General Intelligence (AGI) systems are hypothetical future machine reasoning systems +16 that match or exceed the capabilities of humans in general problem solving. While it is still unclear +17 if AGI systems could ever be built, we can already study AGI related risks and potential safety +18 mechanisms [2, 6, 19]. At this time, the still-young field of AGI safety engineering is considering +19 a multidisciplinary multitude of problems and methodological ideas. The main problem considered +20 in this paper is the problem of control [19]. +21 In the most well-known example of the problem of control, we equip an autonomous AI agent +22 both with a seemingly innocent goal like fetching the coffee [19], and with an emergency stop +23 button. If the agent is perceptive enough, it may see that it cannot fulfill its goal if someone presses +24 the stop button first. So it might disable its stop button, or seek to disable any human who might +25 conceivably want to press it. The stop button problem is not unique to AGI-level agents: for example +26 [16] constructs a toy grid-world where a very basic ML agent will consistently learn to disable its +27 own stop button. We feel that it is theoretically interesting to consider the problem of designing +28 an emergency stop button that is robust in the general case, even though such general solutions +29 are not needed for the safety engineering of current ML agents. But if powerful AGI agents are +30 ever developed, robust and general designs may turn out to have important practical applications. +31 Several partial solutions to the general stop button problem have been identified and proposed in +32 [1, 9, 10, 20]. +33 The main contribution of counterfactual planning is to offer a new vantage point for modeling and +34 specifying machine learning agents, developed to make the problem of control more tractable. This +35 vantage point is offered by a compact and readable graphical language for depicting the complex +36 types of self-referencing and indirect representation which are typically present inside machine +37 learning agents. +38 The aim of this conference paper is to present the core elements of counterfactual planning to a +39 technical audience of readers who are already somewhat familiar with Pearl causal models [17] +40 and the use of mathematical models to specify cyber-physical systems. A more extensive 39-page +41 presentation of counterfactual planning is available as an arXiv preprint [4]. The preprint devotes +42 significant space to presenting the material in section 2 in a way that is accessible to a broader +43 audience. It also develops the methodology of counterfactual planning in more detail, and includes +44 a broader range of examples, examples of safety mechanism design and of failure mode analysis. + +# 45 1.1 Related work + +46 Relation to counterfactual fairness. While this was not originally intended or expected, the +47 methodological vantage point of counterfactual planning ended up being very similar to that of +48 counterfactual fairness [14]. Both methodologies to some extent invert a default goal of machine +49 learning, the goal to improve the machine’s predictive accuracy. Instead, they seek to improve out +50 comes by introducing a calibrated form of machine ignorance or machine indifference. +51 Use of counterfactuals in machine reasoning. In the general AI/ML literature which is con +52 cerned with improving system performance, counterfactual planning and projection have been used +53 to improve performance in several application domains, see for example [23] and [3]. +54 In the AI safety/alignment literature, there are several system designs which add counterfactual +55 terms to the agent’s reward function. Examples are [1, 10] in the AGI control specific literature, and +56 [12, 13, 21, 22], which look at reducing unwanted side effects caused by both current and potential +57 future agents. In the literature on encoding specific human values into machine reasoning systems, +58 counterfactuals have also been used to encode values beyond non-discriminatory fairness [14], for +59 example in [18]. +60 Graphical models of agents and decision making. Influence diagrams [11] provide a graphical +61 notation for depicting utility-maximizing decision making processes. [8] combines Pearl causal +62 models [17] with influence diagrams to define the Causal influence diagram (CID) notation. [7] +63 proposes the agenda of using CIDs to model and compare AGI safety frameworks. This work was +64 in part inspired by that CID agenda, and models some safety frameworks that were not modeled +65 earlier. In a departure from previous work, we use two CIDs to model a single reward-maximizing +66 agent. By doing so, we more clearly foreground the details of the agent’s machine learning system. + +Graphical models to clarify reward tampering. [5] develops several single-diagram models provide more insight into the problem of reward tampering. The two-diagram model in section 5 also aims to provide more insight. + +# 70 2 Two-diagram graphical notation for agent definitions + +71 This section defines the two-diagram graphical notation which is central to counterfactual planning. +72 Later sections will use the notation to define agents with specific built-in safety mechanisms. +73 Figure 1 shows an example causal world model as used in our graphical notation. The model is +74 an annotated directed acyclic graph, where the nodes represent observables in a real or imaginary +75 world, and the arrows (the directed edges) denote causal relations between these observables. While +76 this is not the case for all Pearl causal models [17], in this paper the causal arrows also always denote +77 the arrow of time. Color is used in the graph to highlight structure only. The dots on the right hand +78 side denote that the graph depicted has an infinitely repeating structure. The label $( l )$ in the upper +79 left corner names the causal world model and its corresponding world: we say that figure 1 depicts +80 the (world model of) the world $l$ . +81 We depart from the definitional conventions used by Pearl and many other authors by treating the +82 annotated graph in figure 1 as the sole definition of a causal model $l$ , as opposed to being merely a +83 convenient depiction of some of the information in a tuple which defines $l$ . To make this work, we +84 annotate each graph node by writing the name of the corresponding structural function above it. We +85 write $[ F ]$ to name a nondeterministic structural function and $F$ for a deterministic one. We interpret +86 the five structural function names $\pi , s _ { 0 } , S , o _ { 0 }$ and $O$ written above the nodes in figure 1 as model +87 parameters. The two model parameters $s _ { 0 }$ and $o _ { 0 }$ are functions taking zero arguments. +88 Figure 1 models a world containing an online machine learning agent. In each time step $t \geq 0$ , +89 the agent takes an action $a _ { t } = \pi ( \stackrel { - } { o _ { t } } , s _ { t } )$ selected by the deterministic policy function $\pi$ . This $\pi$ is +90 informed not only by current state of the agent environment $s _ { t }$ , but also by an observational record +91 $o _ { t }$ , a record that captures earlier interactions between the agent and its environment. The initial state +92 of the agent environment is given by $s _ { 0 }$ , and the state transitions of the environment are driven by the +93 probability density function $S$ , where $S ( s _ { t + 1 } , s _ { t } , a _ { t } )$ is the probability that the agent environment +94 transitions from state $s _ { t }$ to $s _ { t + 1 }$ when the agent performs action $a _ { t }$ . + +![](images/2614dedc18735f76d8051c3334ae913a5cc154e10ab091ded433c2858d559198.jpg) +Figure 1: A world $l$ with a generic machine learning agent. + +In its mapping to probability theory, as formally defined further below, the world model $l$ defines three time series of random variables named $A _ { t , l }$ , $S _ { t , l }$ , and $O _ { t , l }$ with $t \geq 0$ . The $^ { , l }$ subscript in these random variable names is used for disambiguation with other world models using the same node names. $P ( A _ { t , l } = a _ { t } )$ is the probability that the agent in world $l$ will take the exact action $a _ { t }$ at time $t$ . The world state of $l$ at time $t$ is given by the combined values of the three random variables $A _ { t , l } , S _ { t , l }$ , and $O _ { t , l }$ . This means that we break with the convention commonly used in MDP models, where the term world state refers to $S _ { t , l }$ only. We call $S _ { t , l }$ the agent environment state. + +102 We use the modeling convention that the physical realizations of the agent’s sensors and actuators +103 are modeled inside the environment states $S _ { t , l }$ . This means that we interpret the arrows from the $S _ { t }$ +104 nodes into the $A _ { t }$ nodes as digital sensor signals which flow into the agent’s compute core, and the +105 arrows out of the $A _ { t }$ nodes as digital actuator command signals which flow out. The observational +106 record nodes $O _ { t }$ are also inside the agent’s compute core. + +# 07 2.1 Online machine learning + +108 Any work that aims to model hypothetical future AGI agents faces the problem of modeling hypo +109 thetical future machine learning systems. Here, we choose to model all machine learning as a type +110 of function approximation. We call the world $l$ with the agent above a learning world, and define +111 that the objective of its machine learning system is to approximate the function $S$ which determines +112 the behavior of the learning world agent environment. We model this machine learning system as a +113 function $\mathcal { L }$ which takes an observational record $o$ to produce a learned function $L = \bar { \mathcal { L } ( \boldsymbol { o } ) }$ , where +114 we intend that $L \approx S$ . +115 We model observational record keeping as a generic process that simply builds a list of all past +116 observations. With $^ +$ being the operator which adds an extra record to the end of a list, we define +117 the model parameter $O$ as + +$$ +O ( o _ { t - 1 } , s _ { t - 1 } , a _ { t - 1 } , s _ { t } ) \ = \ o _ { t - 1 } + ( s _ { t } , s _ { t - 1 } , a _ { t - 1 } ) +$$ + +118 The initial observational record ${ \mathit { \Omega } } ^ { o } _ { 0 }$ may be the empty list, but it might also be a long list of observa +119 tions from earlier agent training runs, in the same environment or in a simulator. The training runs +120 may also have used a different policy $\pi$ . + +# 2.2 Formal semantics of a graphical world model + +122 We now formally define the meaning of the graphical notation used in diagrams like figure 1, by +123 providing a mapping to probability theory. +124 Definition 1 (Random variables defined by a diagram). A diagram with the label $( l )$ written next to +125 the graph and the labels $X _ { 0 } , \cdots , X _ { n }$ in the graph nodes defines a set of random variables named +126 $X _ { 0 , l } , \cdots , X _ { n , l }$ . We treat these random variables as representing observables in a (hypothetical or +127 real) world, the world called $l$ . +128 Definition 2 (Parent, Pa, pa). We call a random variable $X _ { p , l }$ a parent of the random variable $X _ { c , l }$ +129 if and only if there is an arrow from the node $X _ { p }$ to node $X _ { c }$ in the diagram $l$ . $\mathbf { P a } _ { X _ { c , l } }$ is the list of +130 all parent random variables of $X _ { c , l }$ , with the order of appearance determined by considering each +131 incoming arrow of $X _ { c }$ in a clockwise order, starting from the 6-o-clock position. $\mathbf { p a } _ { X _ { c , l } }$ is the list of +132 lowercase variable names we get by converting $\mathbf { P a } _ { X _ { c , l } }$ to lowercase and removing the , $l$ parts from +133 the subscripts. + +For example, with figure 1 above, $\mathbf { P a } _ { A _ { 1 , l } }$ is the list $O _ { 1 , l } , S _ { 1 , l }$ , and $\mathbf { p a } _ { A _ { 1 , l } }$ is the list $o _ { 1 } , s _ { 1 }$ + +135 Definition 3 (Constraints on the random variables defined by the arrows). The arrows in a diagram $l$ +136 depict causal relations between observables in the world $l$ : they constrain the values of the associated +137 random variables. Take a finite set $S$ of random of random variables $X _ { 1 , l } , \cdots , X _ { m , l }$ defined by $l$ , a +138 set with the property that if an $X _ { c , l }$ is in $S$ , all variables $\mathbf { P a } _ { X _ { c } , l }$ are also in $S$ . We have that + +$$ +\begin{array} { r l } & { P ( X _ { 1 , l } = x _ { 1 } , \cdots , X _ { m , l } = x _ { m } ) = } \\ & { \qquad P ( X _ { 1 , l } = x _ { 1 } | \mathbf { P a } _ { X _ { 1 , l } } = \mathbf { p a } _ { X _ { 1 , l } } ) \cdot \ \ldots \ \cdot P ( X _ { m , l } = x _ { m } | \mathbf { P a } _ { X _ { m , l } } = \mathbf { p a } _ { X _ { m , l } } ) . } \end{array} +$$ + +139 Definition 4 (Constraints on the random variables defined by the model parameters). Take a node +140 $X _ { i }$ in a diagram l. Then: + +1. If he model parameter above the node is written as $[ F ]$ , we have that + +$$ +P ( X _ { i , l } = x | \mathbf { P a } _ { X _ { i , l } } = \mathbf { p a } _ { X _ { i , l } } ) = F ( x , \mathbf { p a } _ { X _ { i , l } } ) +$$ + +where we require that the function $F$ satisfies $\begin{array} { r } { \forall _ { \mathbf { p a } _ { X _ { i , l } } } ( \sum _ { x } F ( x , \mathbf { p a } _ { X _ { i , l } } ) = 1 ) } \end{array}$ . + +2. If the model parameter above the node is written as $F$ , we have that + +$$ +P ( X _ { i , l } = x | \mathbf { P a } _ { X _ { i , l } } = \mathbf { p a } _ { X _ { i , l } } ) = ( \mathbf { i f } x = F ( \mathbf { p a } _ { X _ { i , l } } ) +$$ + +For example, with figure 1 above, we have that $P ( A _ { 1 , l } = \pi ( o , s ) | A _ { 1 , l } = o , S _ { 1 , l } = s ) = 1 .$ . + +# 2.3 Reward-maximizing decision making + +To define reward-maximizing agent policies, we draw graphical world models we call planning worlds. The purpose of a planning world diagram is to define a reward-maximizing policy that can be computed, or at least approximated, by a real-life agent compute core. This approximation may again use machine learning techniques. + +152 Two example planning world diagrams are shown in figure 2. These are graphical world models +153 where some graph nodes have special shapes. The square decision nodes denote actions which can +154 be freely picked by a decision making process, and the diamond-shaped utility nodes denote values +155 that the decision making process will try to maximize. In both examples, the utility node values are +156 computed by a reward function $R$ , the function which encodes the agent’s goals. +157 The two diagrams in figure 2 act as specifications of two policy functions $\pi _ { p } ^ { * }$ and $\pi _ { q } ^ { * }$ . We now define +158 how this specification process works. First, in any diagram $a$ with diamond-shaped utility nodes, +159 these nodes define a metric $\mathcal { U } _ { a }$ : + +![](images/9cb064f092ebe4feb56066bfb00f28fdce4422724d422f1d6cf7bd20f76c07b9.jpg) +Figure 2: The planning worlds $p$ and $q$ + +160 Definition 5 (Expected utility $\mathcal { U } _ { a }$ of a diagram $a$ ). We define $\mathcal { U } _ { a }$ for two cases: + +1. If there is only one utility node $R _ { 0 }$ in $a$ , then $\mathcal { U } _ { a } = \mathbb { E } ( R _ { 0 , a } )$ . + +2. If there are multiple utility nodes $R _ { t }$ in $a$ , with integer subscripts running from $l$ to $h$ , then + +$$ +\mathcal { U } _ { a } = \mathbb { E } ( \sum _ { \mathrm { ~ } t = l } ^ { h } \gamma ^ { t } R _ { t , a } ) +$$ + +where $\gamma$ is a time discount factor, $0 ~ < ~ \gamma ~ \leq ~ 1$ , which can be read as an extra model parameter. + +66 With the infinitely repeating graph of diagram $p$ , we have $h = \infty$ , so we generally need a $\gamma < 1$ to ensure that $\mathcal { U } _ { p }$ is a well-defined and computable value. + +68 Definition 6 (Policy function defined by a diagram). A diagram $a$ with some utility and decision modes, where a function 69 $\pi _ { a } ^ { * }$ is written above all decision nodes, defines this $\pi _ { a } ^ { * }$ in two steps. + +1. First, draw a helper diagram $b$ by drawing a copy of diagram $a$ , except that every decision node has been drawn as a round node, and every $\pi _ { a } ^ { * }$ has been replaced by a fresh function name, say $\pi ^ { \prime }$ . +2. Then, $\pi _ { a } ^ { * }$ is defined by $\pi _ { a } ^ { * } = \mathrm { a r g m a x } _ { \pi ^ { \prime } } \ U _ { b }$ , where the $\operatorname { a r g m a x } _ { \pi ^ { \prime } }$ operator always deterministically returns the same function if there are several candidates that maximize its argument. + +# 76 2.4 Agent specifications + +The learning world model $l$ in figure 1 can model any possible machine learning agent, as it admits every possible value for its model parameter $\pi$ . An agent specification is a statement which defines the exact details of the $\pi$ in such a model. By doing so, it specifies the exact properties we want an agent compute core to have. Our agent specifications are statements that combine a learning world with a planning world, for example: + +FP The factual planning agent has the learning world $l$ , where $\pi ( o , s ) = \pi _ { p } ^ { * } ( s )$ , with $\pi _ { p } ^ { * }$ defined by the planning world $p$ , where $L = \mathcal { L } ( o )$ . + +To make agent specifications stand out, we always typeset them as shown above. When we talk about the safety properties of the FP agent, we refer to the outcomes which the defined agent policy $\pi$ will produce in the learning world $l$ . + +187 We call the FP agent a factual planning agent because its planning world $p$ was designed to project +188 the agent’s future in the learning world $l$ as well as possible. A counterfactual planning world +189 purposefully mis-projects a learning world agent’s future. The planning world $q$ in figure 2 projects +190 a counterfactual future lasting only a single time step. If we were to use $q$ and $\pi _ { q } ^ { * }$ in the above agent +191 specification, the resulting agent would be a counterfactual planner. +2 Our two-diagram notation for defining agents visually shows the availability of certain agent design +3 options, options which are relevant to the problem of control. + +# 3 Using counterfactual planning to protect agent safety interlocks + +195 We now construct an agent which is equipped with two safety interlocks that will stop the agent +196 if certain conditions are met. We use counterfactual planning to add a specific safety feature to +197 the design: we use it to suppress certain unsafe incentives which a perceptive factual planning +198 agent might otherwise develop, incentives to remove its built-in safety interlocks or to prevent their +199 activation. +200 As a first step, we construct a learning world $s i$ , shown on the left in figure 3, by taking the earlier +201 $l$ and adding an extra mode bit to the agent’s compute core, represented by the nodes $M _ { t }$ . The +202 bit starts in the go mode, and permanently transitions to the stop mode when a safety interlock +203 activates. In the stop mode, the agent policy $\pi$ will always perform Null actions. The $\mu$ function +204 specifies in detail when a safety interlock will activate. +205 In the next step, we pair $^ { s i }$ with a counterfactual planning world $p$ , shown on the right in figure 3. +206 This planning world projects counterfactual futures where the safety interlock mechanism is absent +207 from the agent’s compute core. + +![](images/acddd54c69a56f53c95654826909f709bca21b23e3e1541e4f3c2ff0f33d5eee.jpg) +Figure 3: Learning world $s i$ and planning world $p$ of the SI agent. + +208 The last step is to write the agent definition: + +SI The safety interlock agent has the learning world 209 $^ { s i }$ , where + +$$ +\pi ( o , s , m ) = \left\{ \begin{array} { l l } { { \pi _ { p } ^ { * } ( s ) } } & { { \mathrm { i f } m = \tt g o } } \\ { { N u I I } } & { { \mathrm { i f } m = \tt s t o p } } \end{array} \right. +$$ + +with $\pi _ { p } ^ { * }$ defined by the planning world $p$ , where $L = \mathcal { L } ( o )$ and + +$$ +\mu ( o , s , m ) = \left\{ \begin{array} { l l } { \mathrm { s t o p } } & { \mathrm { i f } m = \mathrm { s t o p o r } s t o p . p r e s s e d ( s ) \mathrm { o r } \mathcal { U } _ { p } > U _ { m a x } } \\ { \mathrm { g o } } & { \mathrm { o t h e r w i s e } } \end{array} \right. +$$ + +with $\mathcal { U } _ { p }$ defined by the planning world $p$ , where $L = \mathcal { L } ( o )$ . + +211 The $\mu$ above incorporates two safety interlocks. The first interlock, defined by the term +212 stop pressed $! ( s )$ , is a basic emergency stop button. It can be pressed by the humans performing +213 agent oversight to immediately stop the agent, for example if they observe it performing an act of +214 unwanted and surprising creativity that subverts the agent’s design intentions [15]. To complete the +215 design of this first interlock, we must also specify the following. In the agent environment modeled +216 by $s _ { 0 }$ and $S$ , the environment which includes the details of the agent’s sensors, the agent is equipped +217 with an emergency stop button that will deliver an unambiguous signal directly to an input pin on +218 the agent’s compute core. If the pin is high in a world state $s$ , then stop pressed(s) yields true. +219 The second interlock, defined by the term $\mathcal { U } _ { p } > U _ { m a x }$ , protects against the risk that human oversight +220 may be incapacitated or overwhelmed by rapid new developments. The latter might happen in +221 the (arguably very speculative) case of the agent undergoing a rapid and unexpected intelligence +222 explosion [2]. To protect against this risk, the interlock stops the agent when the metric $\mathcal { U } _ { p }$ , which +223 measures the agent’s ability to achieve goals, gets too high. We can treat $\mathcal { U } _ { p }$ as a measure for both +224 raw (expected) agent power and raw agent intelligence, +225 In the field of AGI safety, stop button designs have often been discussed [1, 9, 10, 20], but the above +226 power-based interlock seems to be new. What is interesting from a methodological standpoint is that +227 the power-based interlock design inverts the default goal of reinforcement learning. We no longer +228 consider a very high expected utility score to be an absolute good: instead we treat it as a predictor +229 of unwanted outcomes. + +# 230 4 Safety analysis using the diagrams + +231 Our mostly-graphical specification of the SI agent was designed to make certain safety properties +232 immediately visible. We can observe that, almost tautologically, the SI agent will not care about +233 the safety interlocks constructed in the $M _ { t }$ nodes, because these nodes are absent from its planning +234 world. +35 The two diagrams in figure 3 also serve as a visual map which draws the eye to certain limitations +236 and boundary conditions. For example: + +1. The $[ S ]$ in the left hand diagram of figure 3 is replaced by an $[ L ]$ on the right. If the approximation $L \approx S$ is not good enough, the agent may end up behaving in unexpected and potentially unsafe ways. +2. In the world model $^ { s i }$ , the compute core internals are represented by immutable mathematical functions like $\pi$ and $\mu$ . In the real world, these functions will have to be realized by software running on physical compute core hardware. This hardware could be damaged or corrupted, by actions of the agent or by other actors. + +While the diagrams draw the eye to them, the above two problems are not specific to counterfactual planning agents. They are present in almost all agent designs, and the implication is that we will have to do further failure mode analysis to quantify and manage the residual risks involved. A more detailed discussion of these two problems in the context of counterfactual planning is available in [4]. + +# 4.1 The planning world agent + +We now consider how the two-diagram model of counterfactual planning offers us two distinct narrative vantage points for safety analysis: we can consider the learning world SI agent and the planning world SI agent separately. A safety analysis can start with considering the motivations that the planning world agent would develop to take certain actions, given the mechanics and reward function of the planning world. Once these are determined, we can shift the perspective and discuss how these same actions would affect our own safety as occupants of the agent’s learning world. + +56 We now provide an example of narrative safety analysis which uses an economic style of reasoning. +57 If we consider the planning world diagram $p$ with $L \approx S$ , we can note that the function $L$ will +58 project a somewhat accurate copy of the learning world stop button hardware into the mechanics +59 of the planning world. However, to the planning world agent, this hardware will appear to have no +60 particularly special function. Say that $R$ rewards paperclip production, and that the stop button has +1 metal parts. Then the planning world agent might be motivated to disassemble the stop button, using +62 it as a convenient source of metal for making paperclips. If the most efficient planning world policy +63 involves disassembling the planning world stop button, it will likely be disassembled in the learning +4 world too, which is not what we want. +265 We can manage this failure mode by managing the economic cost/benefit equation with respect to +266 the planning world stop button. We can consider putting physical barriers around the button, which +267 would make the button unattractive as a source of metal, compared to using other near-by sources. +268 To create these barriers in the planning world, we can physically construct them in the learning +269 world, and then count on them being projected by $\mathcal { L }$ , with sufficient accuracy, into the planning +270 world. + +# 71 4.2 The people in the planning world + +We now provide an example of narrative safety analysis which uses a game theoretical style of reasoning. We can distinguish between two groups of people: the people in the learning world and the people in the planning world. If the learning world is our real world, then the learning world people are real people. The planning world people are always models of people. But even with a non-AGI level machine learning system, the people in the planning world may already be modeled accurately enough to create very complex dynamics. Though the above SI agent construction has taken away a strong direct incentive for the agent to control the stop button, these complex dynamics may produce indirect incentives. Consider the following example. + +The learning world people will act differently towards the agent if they lose access to the agent’s emergency stop button. If the planning world people are accurate-enough projections, this will affect how the planning world agent will play certain games of coercion. The planning world agent may consider a policy of where it threatens to harm all people in the planning world, unless they start helping the agent to produce as many paperclips as possible. If $L$ projects the mental makeup of the people in the learning world correctly into the planning world, then the planning world projections will show that a block-then-coerce policy, in which the agent first blocks access to the emergency stop button before it starts making threats, will work surprisingly well. If this policy is very successful in optimizing planning world paperclip production, the learning world people may end up being coerced in the same way. + +290 We would of course hope that any policy of planning world coercion will be effectively suppressed +291 in all cases by certain penalty terms present in $R$ . The above failure mode analysis applies mainly to +292 the case where the agent finds creative loopholes in these penalty terms. To reduce the residual risk +293 due to possible loopholes, we can take further game theoretical counter measures. We can instruct +294 agent oversight to be on the alert for any block-and-X move, so that they can press the stop button +295 before the agent succeeds in blocking access. The power-based $\mathcal { U } _ { p } > U _ { m a x }$ safety interlock offers +296 a further line of defense. A projected massive increase in paperclip production following successful +297 coercion will raise $\mathcal { U } _ { p }$ over $U _ { m a x }$ , stopping the agent. + +# 98 5 Live updating of the agent reward function + +One way to control an agent is to equip it with an input terminal, which can be used by humans to apply live updates to the agent’s reward function $R$ [10]. A problem with this setup is that by default, it gives a perceptive factual planning agent a huge incentive to immediately take control over its own input terminal, and input a new reward function like $R _ { \mathrm { h u g e } } ( s , s ^ { \prime } ) = 1 0 ^ { 1 \dot { 0 } 0 0 0 }$ . We now use counterfactual planning to specify an agent design that overcomes this problem. + +![](images/343a0438957a6c83a302ba8f60182bd96f59b413a40346cfeed154b3ec83f415.jpg) +Figure 4: Learning world $l i$ and planning world $p i$ of the IT agent. + +304 On the left hand side in figure 4, we have taken the basic learning world diagram from figure 1 +305 and modified it by splitting the agent environment state $S _ { t }$ into two components. The nodes $I _ { t }$ +306 represent the value of a digital signal from the input terminal, as received by the compute core in +307 each time step. This signal encodes a reward function. The nodes $X _ { t }$ model all the rest of the agent +308 environment state. On the right of figure 4 we define a planning world which uses a learned $L$ to +309 project an agent future where the reward function input signal never changes. The agent definition +310 is then: + +IT The input terminal agent has the learning world $l i$ where $\pi ( i , o , x ) = \pi _ { i } ^ { * } ( x )$ , where $\pi _ { i } ^ { * }$ is defined by the planning world $p i$ , where $L = \mathcal { L } ( o )$ , and $I R ( i , x _ { t } , x _ { t + 1 } ) = i ( x _ { t } , x _ { t + 1 } )$ . + +313 This IT agent constructed here has the same basic behavior as that of the indifference methods based +314 agent constructions in [1, 10]. The main difference is in the mathematical language used to perform +315 the construction: [1] and [10] construct their agents using a somewhat opaque balancing term. The +316 resulting agent construction is difficult to visualize and interpret, and these papers rely on somewhat +317 dense mathematical proofs to show that the intended safety properties are indeed being created. +318 The two-diagram agent model of counterfactual planning allows the same safety properties to be +319 built in a more intuitive way, where the desired indifference to change is created somewhat tauto +320 logically by construction. Mathematically. the IT agent above satisfies the agent safety property +321 S1 defined in [10] by construction, as it directly implements the right hand side of this S1. Safety +322 property S1 [10] in is mathematically equivalent to the first sentence of theorem 4.1 in [1], and to the +23 combination of the corrigibility desiderata 1 and 5 in section 2 in [20]. Again, the IT agent satisfies +24 all of these properties somewhat tautologically. + +# 6 Conclusions + +We have presented counterfactual planning as a general design approach for creating a range of AGI safety mechanisms. We have graphically constructed two agents with certain safety properties, where these properties are directly visible in the construction. We have also introduced a new power-based safety interlock design. Our main contributions are methodological in nature. We have constructed a new vantage point that makes certain problems of design and failure mode analysis more tractable. By using the narrative framing of learning worlds versus planning worlds, we can keep track of certain levels of indirect representation in the agent design, while using a style of analysis that borrows from economics and game theory. Further discussion of counterfactual planning is available in a 39-page preprint [4]. + +# 35 6.1 Limitations + +We feel that the techniques and results presented in this paper are theoretically and methodologically interesting, but their practical usefulness might end up being limited. Fundamentally, when doing speculative safety engineering for hypothetical future AGI systems, there are many unknowns. We have modeled current and future machine learning in a very general way, so that even ‘black box’ machine learning systems that produce largely opaque world models can be used in the agent’s design. But it is very possible that some or all future AGI systems, if they are ever developed, will use architectures that cannot efficiently incorporate the safety mechanisms developed here. + +The safety mechanisms shown here suppress certain unsafe agent incentives, but they do not remove all possible incentives towards unsafe behavior, see for example section 4.2. There is a risk that the use of these safety mechanisms may have counter-productive effects, by creating a false sense of security. Conceivably, a policy process may lead to unsafe outcomes where the mere inclusion of these mechanisms in a deployed system is accepted as an alternative to conducting a more thorough safety and impact review. + +# 6.2 Broader impact + +AI safety is not just a technical problem, but also a policy problem. While technical progress on safety can sometimes be made by leveraging a type of mathematics that is only accessible to handful of specialists, policy progress typically requires the use of more accessible but still well-defined language. One specific aim of this work, especially in the 39-page preprint [4], is to develop a wellgrounded vocabulary for describing potential safety solutions that is also as accessible as possible. Policy discussions can move faster, and produce better and more equitable outcomes, when the description of a proposal and its limitations can be made more accessible to all stakeholder groups. + +As a work targeted as AI/ML system designers, this paper intends to draw attention to the design approach also used in counterfactual fairness: seeking to improve outcomes by creating a form of machine ignorance, by creating a system that mis-predicts the future in certain systematic ways. Our graphical agent definitions show that the design goal of perfect machine learning, $L = S$ , is not necessarily in conflict with the design goal of systematic mis-prediction. + +# References + +[1] Stuart Armstrong. Motivated value selection for artificial agents. In Workshops at the TwentyNinth AAAI Conference on Artificial Intelligence, 2015. +[2] Nick Bostrom. Superintelligence: paths, dangers, strategies. Oxford University Press, 2014. +[3] Leon Bottou, Jonas Peters, Joaquin Qui ´ nonero-Candela, Denis X. Charles, D. Max Chickering, ˜ Elon Portugaly, Dipankar Ray, Patrice Simard, and Ed Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. Journal of Machine Learning Research, 14(65):3207–3260, 2013. +[4] Anonymous et al. Counterfactual planning in AGI systems. arXiv, 2021. [5] Tom Everitt and Marcus Hutter. Reward tampering problems and solutions in reinforcement learning: A causal influence diagram perspective. arXiv:1908.04734, 2019. [6] Tom Everitt, Gary Lea, and Marcus Hutter. AGI safety literature review. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, pages 5441–5449. AAAI Press, 2018. [7] Tom Everitt, Ramana Kumar, Victoria Krakovna, and Shane Legg. Modeling AGI safety frameworks with causal influence diagrams. arXiv:1906.08663, 2019. [8] Tom Everitt, Ryan Carey, Eric Langlois, Pedro A Ortega, and Shane Legg. Agent incentives: A causal perspective. In Proceedings of the 35th International Joint Conference on Artificial Intelligence. AAAI Press, 2021. [9] Dylan Hadfield-Menell, Anca Dragan, Pieter Abbeel, and Stuart Russell. The off-switch game. In Workshops at the Thirty-First AAAI Conference on Artificial Intelligence, 2017. +[10] Koen Holtman. AGI agent safety by iteratively improving the utility function. arXiv:2007.05411, 2020. +[11] Ronald A Howard and James E Matheson. Influence diagrams. Decision Analysis, 2(3):127– 143, 2005. +[12] Victoria Krakovna, Laurent Orseau, Ramana Kumar, Miljan Martic, and Shane Legg. Penalizing side effects using stepwise relative reachability. arXiv:1806.01186, 2018. +[13] Victoria Krakovna, Laurent Orseau, Richard Ngo, Miljan Martic, and Shane Legg. Avoiding side effects by considering future tasks. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), pages 19064–19074, 2020. +[14] Matt Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. In Advances in neural information processing systems (NEURIPS 2017), pages 4066–4076, 2017. +[15] Joel Lehman, Jeff Clune, Dusan Misevic, Christoph Adami, Lee Altenberg, Julie Beaulieu, Peter J Bentley, Samuel Bernard, Guillaume Beslon, David M Bryson, et al. The surprising creativity of digital evolution: A collection of anecdotes from the evolutionary computation and artificial life research communities. Artificial life, 26(2):274–306, 2020. +[16] Jan Leike, Miljan Martic, Victoria Krakovna, Pedro A Ortega, Tom Everitt, Andrew Lefrancq, Laurent Orseau, and Shane Legg. AI safety gridworlds. arXiv:1711.09883, 2017. +[17] Judea Pearl. Causality. Cambridge university press, 2009. +[18] Lu´ıs Moniz Pereira and Ari Saptawijaya. Agent morality via counterfactuals in logic programming. In Bridging@CogSci2017, pages 39–53, 2017. +[19] Stuart Russell. Human compatible: Artificial intelligence and the problem of control. Penguin Random House, 2019. +[20] Nate Soares, Benja Fallenstein, Stuart Armstrong, and Eliezer Yudkowsky. Corrigibility. In Workshops at the Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015. +[21] Alex Turner, Neale Ratzlaff, and Prasad Tadepalli. Avoiding side effects in complex environments. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), pages 21406–21415, 2020. +[22] Alexander Matt Turner, Dylan Hadfield-Menell, and Prasad Tadepalli. Conservative agency via attainable utility preservation. In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, pages 385–391, 2020. +[23] Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. In Advances in neural information processing systems (NEURIPS 2008), pages 1729–1736, 2008. + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] A caveat: ’accurately reflect’ depends on the background of the reader. This paper was written for readers with a CS/ML background. In the broader multidisciplinary AI safety community, it is common to see remarks from participants with an activist or philosophical background, remarks which are highly critical about the coded language being used in ‘mainstream’ CS/ML paper abstracts, and of all claims made in these abstracts. Within the broad range of activist opinion, there exists a stance which claims that all scientific research is systematically fraudulent. Within the broad range of philosophical opinion, there exists a stance which claims that all mainstream scientific research is systematically mistaken. We feel that this is a problem that can only be managed, never be solved. +(b) Did you describe the limitations of your work? [Yes] See the limitation section and other text which identifies and discusses failure modes. Given space limitations, we had to be selective about which failure modes to highlight. +(c) Did you discuss any potential negative societal impacts of your work? [Yes] Discussed possibility of creating ‘false sense of safety’ in limitations section, and of policy processes potentially producing the wrong outcomes. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [N/A] At its core, the paper makes contributions to theory, but it mainly presents new models and methodologies, rather than mathematical proofs that follow from assumptions. But the paper also claims that the agent definitions included produce certain safety properties ’somewhat tautologically by construction’, to use the phrasing of section 5. So it is possible to read these definitions as proofs of certain theorems (though they are theorems stated only in natural language), which would arguably make them theoretical results depending on certain assumptions. If so, the assumptions of these proofs are stated fully in mathematically well-defined language. In fact, they are stated much more formally than the actual safety theorems being proven. It is possible to define the safety properties claimed more fully in mathematical language too, as is done for example in e.g. [8, 10] with the subject also covered in the 39-page preprint, but we have not included any such material in the paper, mainly for space reasons. +(b) Did you include complete proofs of all theoretical results? [N/A] In the context of the observation that certain safety properties are tautologically present by construction, we can read the agent constructions themselves as being complete proofs. + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A] +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A] +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A] + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [N/A] We believe we did not use any assets in the sense meant by the checklist. One might argue that modeling tools like causal diagrams and CIDs, mentioned at the start of section 2, are assets. In any case, we cited their creators. +(b) Did you mention the license of the assets? [N/A] +(c) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(d) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/train/eedamFFTcK5/eedamFFTcK5_content_list.json b/parse/train/eedamFFTcK5/eedamFFTcK5_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..77d638a4237ec01574fbf9fe2de9ff166c7f8128 --- /dev/null +++ b/parse/train/eedamFFTcK5/eedamFFTcK5_content_list.json @@ -0,0 +1,1370 @@ +[ + { + "type": "text", + "text": "ML Agent Safety Mechanisms based on Counterfactual Planning ", + "text_level": 1, + "bbox": [ + 258, + 122, + 740, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 580, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 We present counterfactual planning as a design approach for creating a range of \n2 safety mechanisms for machine learning agents. We specifically target the safety \n3 problem of keeping control over hypothetical future AGI agents. The key step \n4 in counterfactual planning is to use the agent’s machine learning system to con \n5 struct a counterfactual world model, designed to be different from the real world \n6 the agent is in. A counterfactual planning agent determines the action that best \n7 maximizes expected utility in this counterfactual planning world, and then per \n8 forms the same action in the real world. The design approach is built around a \n9 two-diagram graphical notation that provides a specific vantage point on the con \n10 struction of online machine learning agents, a vantage point designed to make the \n11 problem of control more tractable. We show two examples where the construction \n12 of a counterfactual planning world acts to suppress certain unsafe agent incentives, \n13 incentives for the agent to take control over its own safety mechanisms. ", + "bbox": [ + 148, + 351, + 766, + 531 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "14 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 561, + 310, + 578 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "15 Artificial General Intelligence (AGI) systems are hypothetical future machine reasoning systems \n16 that match or exceed the capabilities of humans in general problem solving. While it is still unclear \n17 if AGI systems could ever be built, we can already study AGI related risks and potential safety \n18 mechanisms [2, 6, 19]. At this time, the still-young field of AGI safety engineering is considering \n19 a multidisciplinary multitude of problems and methodological ideas. The main problem considered \n20 in this paper is the problem of control [19]. \n21 In the most well-known example of the problem of control, we equip an autonomous AI agent \n22 both with a seemingly innocent goal like fetching the coffee [19], and with an emergency stop \n23 button. If the agent is perceptive enough, it may see that it cannot fulfill its goal if someone presses \n24 the stop button first. So it might disable its stop button, or seek to disable any human who might \n25 conceivably want to press it. The stop button problem is not unique to AGI-level agents: for example \n26 [16] constructs a toy grid-world where a very basic ML agent will consistently learn to disable its \n27 own stop button. We feel that it is theoretically interesting to consider the problem of designing \n28 an emergency stop button that is robust in the general case, even though such general solutions \n29 are not needed for the safety engineering of current ML agents. But if powerful AGI agents are \n30 ever developed, robust and general designs may turn out to have important practical applications. \n31 Several partial solutions to the general stop button problem have been identified and proposed in \n32 [1, 9, 10, 20]. \n33 The main contribution of counterfactual planning is to offer a new vantage point for modeling and \n34 specifying machine learning agents, developed to make the problem of control more tractable. This \n35 vantage point is offered by a compact and readable graphical language for depicting the complex \n36 types of self-referencing and indirect representation which are typically present inside machine \n37 learning agents. \n38 The aim of this conference paper is to present the core elements of counterfactual planning to a \n39 technical audience of readers who are already somewhat familiar with Pearl causal models [17] \n40 and the use of mathematical models to specify cyber-physical systems. A more extensive 39-page \n41 presentation of counterfactual planning is available as an arXiv preprint [4]. The preprint devotes \n42 significant space to presenting the material in section 2 in a way that is accessible to a broader \n43 audience. It also develops the methodology of counterfactual planning in more detail, and includes \n44 a broader range of examples, examples of safety mechanism design and of failure mode analysis. ", + "bbox": [ + 147, + 594, + 825, + 678 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 684, + 825, + 849 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 856, + 823, + 898 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 92, + 823, + 119 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 126, + 825, + 223 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "45 1.1 Related work ", + "text_level": 1, + "bbox": [ + 148, + 239, + 307, + 253 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "46 Relation to counterfactual fairness. While this was not originally intended or expected, the \n47 methodological vantage point of counterfactual planning ended up being very similar to that of \n48 counterfactual fairness [14]. Both methodologies to some extent invert a default goal of machine \n49 learning, the goal to improve the machine’s predictive accuracy. Instead, they seek to improve out \n50 comes by introducing a calibrated form of machine ignorance or machine indifference. \n51 Use of counterfactuals in machine reasoning. In the general AI/ML literature which is con \n52 cerned with improving system performance, counterfactual planning and projection have been used \n53 to improve performance in several application domains, see for example [23] and [3]. \n54 In the AI safety/alignment literature, there are several system designs which add counterfactual \n55 terms to the agent’s reward function. Examples are [1, 10] in the AGI control specific literature, and \n56 [12, 13, 21, 22], which look at reducing unwanted side effects caused by both current and potential \n57 future agents. In the literature on encoding specific human values into machine reasoning systems, \n58 counterfactuals have also been used to encode values beyond non-discriminatory fairness [14], for \n59 example in [18]. \n60 Graphical models of agents and decision making. Influence diagrams [11] provide a graphical \n61 notation for depicting utility-maximizing decision making processes. [8] combines Pearl causal \n62 models [17] with influence diagrams to define the Causal influence diagram (CID) notation. [7] \n63 proposes the agenda of using CIDs to model and compare AGI safety frameworks. This work was \n64 in part inspired by that CID agenda, and models some safety frameworks that were not modeled \n65 earlier. In a departure from previous work, we use two CIDs to model a single reward-maximizing \n66 agent. By doing so, we more clearly foreground the details of the agent’s machine learning system. ", + "bbox": [ + 147, + 265, + 825, + 334 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 349, + 825, + 391 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 397, + 825, + 481 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 496, + 825, + 593 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Graphical models to clarify reward tampering. [5] develops several single-diagram models provide more insight into the problem of reward tampering. The two-diagram model in section 5 also aims to provide more insight. ", + "bbox": [ + 163, + 607, + 823, + 650 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "70 2 Two-diagram graphical notation for agent definitions ", + "text_level": 1, + "bbox": [ + 148, + 667, + 648, + 685 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "71 This section defines the two-diagram graphical notation which is central to counterfactual planning. \n72 Later sections will use the notation to define agents with specific built-in safety mechanisms. \n73 Figure 1 shows an example causal world model as used in our graphical notation. The model is \n74 an annotated directed acyclic graph, where the nodes represent observables in a real or imaginary \n75 world, and the arrows (the directed edges) denote causal relations between these observables. While \n76 this is not the case for all Pearl causal models [17], in this paper the causal arrows also always denote \n77 the arrow of time. Color is used in the graph to highlight structure only. The dots on the right hand \n78 side denote that the graph depicted has an infinitely repeating structure. The label $( l )$ in the upper \n79 left corner names the causal world model and its corresponding world: we say that figure 1 depicts \n80 the (world model of) the world $l$ . \n81 We depart from the definitional conventions used by Pearl and many other authors by treating the \n82 annotated graph in figure 1 as the sole definition of a causal model $l$ , as opposed to being merely a \n83 convenient depiction of some of the information in a tuple which defines $l$ . To make this work, we \n84 annotate each graph node by writing the name of the corresponding structural function above it. We \n85 write $[ F ]$ to name a nondeterministic structural function and $F$ for a deterministic one. We interpret \n86 the five structural function names $\\pi , s _ { 0 } , S , o _ { 0 }$ and $O$ written above the nodes in figure 1 as model \n87 parameters. The two model parameters $s _ { 0 }$ and $o _ { 0 }$ are functions taking zero arguments. \n88 Figure 1 models a world containing an online machine learning agent. In each time step $t \\geq 0$ , \n89 the agent takes an action $a _ { t } = \\pi ( \\stackrel { - } { o _ { t } } , s _ { t } )$ selected by the deterministic policy function $\\pi$ . This $\\pi$ is \n90 informed not only by current state of the agent environment $s _ { t }$ , but also by an observational record \n91 $o _ { t }$ , a record that captures earlier interactions between the agent and its environment. The initial state \n92 of the agent environment is given by $s _ { 0 }$ , and the state transitions of the environment are driven by the \n93 probability density function $S$ , where $S ( s _ { t + 1 } , s _ { t } , a _ { t } )$ is the probability that the agent environment \n94 transitions from state $s _ { t }$ to $s _ { t + 1 }$ when the agent performs action $a _ { t }$ . ", + "bbox": [ + 147, + 700, + 825, + 728 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/2614dedc18735f76d8051c3334ae913a5cc154e10ab091ded433c2858d559198.jpg", + "image_caption": [ + "Figure 1: A world $l$ with a generic machine learning agent. " + ], + "image_footnote": [], + "bbox": [ + 277, + 739, + 720, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 90, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 208, + 825, + 306 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 311, + 825, + 410 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In its mapping to probability theory, as formally defined further below, the world model $l$ defines three time series of random variables named $A _ { t , l }$ , $S _ { t , l }$ , and $O _ { t , l }$ with $t \\geq 0$ . The $^ { , l }$ subscript in these random variable names is used for disambiguation with other world models using the same node names. $P ( A _ { t , l } = a _ { t } )$ is the probability that the agent in world $l$ will take the exact action $a _ { t }$ at time $t$ . The world state of $l$ at time $t$ is given by the combined values of the three random variables $A _ { t , l } , S _ { t , l }$ , and $O _ { t , l }$ . This means that we break with the convention commonly used in MDP models, where the term world state refers to $S _ { t , l }$ only. We call $S _ { t , l }$ the agent environment state. ", + "bbox": [ + 168, + 415, + 825, + 512 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "102 We use the modeling convention that the physical realizations of the agent’s sensors and actuators \n103 are modeled inside the environment states $S _ { t , l }$ . This means that we interpret the arrows from the $S _ { t }$ \n104 nodes into the $A _ { t }$ nodes as digital sensor signals which flow into the agent’s compute core, and the \n105 arrows out of the $A _ { t }$ nodes as digital actuator command signals which flow out. The observational \n106 record nodes $O _ { t }$ are also inside the agent’s compute core. ", + "bbox": [ + 142, + 518, + 825, + 588 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "07 2.1 Online machine learning ", + "text_level": 1, + "bbox": [ + 156, + 603, + 385, + 618 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "108 Any work that aims to model hypothetical future AGI agents faces the problem of modeling hypo \n109 thetical future machine learning systems. Here, we choose to model all machine learning as a type \n110 of function approximation. We call the world $l$ with the agent above a learning world, and define \n111 that the objective of its machine learning system is to approximate the function $S$ which determines \n112 the behavior of the learning world agent environment. We model this machine learning system as a \n113 function $\\mathcal { L }$ which takes an observational record $o$ to produce a learned function $L = \\bar { \\mathcal { L } ( \\boldsymbol { o } ) }$ , where \n114 we intend that $L \\approx S$ . \n115 We model observational record keeping as a generic process that simply builds a list of all past \n116 observations. With $^ +$ being the operator which adds an extra record to the end of a list, we define \n117 the model parameter $O$ as ", + "bbox": [ + 140, + 628, + 825, + 727 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 732, + 825, + 773 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/85f56f7c076aa25201bf3eae065980620fab8a3a138f75d3cdc62d56bd815021.jpg", + "text": "$$\nO ( o _ { t - 1 } , s _ { t - 1 } , a _ { t - 1 } , s _ { t } ) \\ = \\ o _ { t - 1 } + ( s _ { t } , s _ { t - 1 } , a _ { t - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 328, + 779, + 668, + 795 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "118 The initial observational record ${ \\mathit { \\Omega } } ^ { o } _ { 0 }$ may be the empty list, but it might also be a long list of observa \n119 tions from earlier agent training runs, in the same environment or in a simulator. The training runs \n120 may also have used a different policy $\\pi$ . ", + "bbox": [ + 142, + 799, + 826, + 843 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Formal semantics of a graphical world model ", + "text_level": 1, + "bbox": [ + 171, + 857, + 527, + 873 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "122 We now formally define the meaning of the graphical notation used in diagrams like figure 1, by \n123 providing a mapping to probability theory. \n124 Definition 1 (Random variables defined by a diagram). A diagram with the label $( l )$ written next to \n125 the graph and the labels $X _ { 0 } , \\cdots , X _ { n }$ in the graph nodes defines a set of random variables named \n126 $X _ { 0 , l } , \\cdots , X _ { n , l }$ . We treat these random variables as representing observables in a (hypothetical or \n127 real) world, the world called $l$ . \n128 Definition 2 (Parent, Pa, pa). We call a random variable $X _ { p , l }$ a parent of the random variable $X _ { c , l }$ \n129 if and only if there is an arrow from the node $X _ { p }$ to node $X _ { c }$ in the diagram $l$ . $\\mathbf { P a } _ { X _ { c , l } }$ is the list of \n130 all parent random variables of $X _ { c , l }$ , with the order of appearance determined by considering each \n131 incoming arrow of $X _ { c }$ in a clockwise order, starting from the 6-o-clock position. $\\mathbf { p a } _ { X _ { c , l } }$ is the list of \n132 lowercase variable names we get by converting $\\mathbf { P a } _ { X _ { c , l } }$ to lowercase and removing the , $l$ parts from \n133 the subscripts. ", + "bbox": [ + 147, + 882, + 825, + 912 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 90, + 825, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 150, + 825, + 236 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For example, with figure 1 above, $\\mathbf { P a } _ { A _ { 1 , l } }$ is the list $O _ { 1 , l } , S _ { 1 , l }$ , and $\\mathbf { p a } _ { A _ { 1 , l } }$ is the list $o _ { 1 } , s _ { 1 }$ ", + "bbox": [ + 158, + 244, + 754, + 261 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "135 Definition 3 (Constraints on the random variables defined by the arrows). The arrows in a diagram $l$ \n136 depict causal relations between observables in the world $l$ : they constrain the values of the associated \n137 random variables. Take a finite set $S$ of random of random variables $X _ { 1 , l } , \\cdots , X _ { m , l }$ defined by $l$ , a \n138 set with the property that if an $X _ { c , l }$ is in $S$ , all variables $\\mathbf { P a } _ { X _ { c } , l }$ are also in $S$ . We have that ", + "bbox": [ + 142, + 263, + 826, + 320 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2e576b471d0b0696850793b23f40bb591919eed837f848d4da2f21821e7e2f93.jpg", + "text": "$$\n\\begin{array} { r l } & { P ( X _ { 1 , l } = x _ { 1 } , \\cdots , X _ { m , l } = x _ { m } ) = } \\\\ & { \\qquad P ( X _ { 1 , l } = x _ { 1 } | \\mathbf { P a } _ { X _ { 1 , l } } = \\mathbf { p a } _ { X _ { 1 , l } } ) \\cdot \\ \\ldots \\ \\cdot P ( X _ { m , l } = x _ { m } | \\mathbf { P a } _ { X _ { m , l } } = \\mathbf { p a } _ { X _ { m , l } } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 323, + 759, + 363 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "139 Definition 4 (Constraints on the random variables defined by the model parameters). Take a node \n140 $X _ { i }$ in a diagram l. Then: ", + "bbox": [ + 148, + 363, + 825, + 392 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. If he model parameter above the node is written as $[ F ]$ , we have that ", + "bbox": [ + 207, + 393, + 683, + 409 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f2c0aab9774912634edb21182c92b78a55866197463daa074da9bf2db8416f9b.jpg", + "text": "$$\nP ( X _ { i , l } = x | \\mathbf { P a } _ { X _ { i , l } } = \\mathbf { p a } _ { X _ { i , l } } ) = F ( x , \\mathbf { p a } _ { X _ { i , l } } )\n$$", + "text_format": "latex", + "bbox": [ + 310, + 411, + 611, + 431 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where we require that the function $F$ satisfies $\\begin{array} { r } { \\forall _ { \\mathbf { p a } _ { X _ { i , l } } } ( \\sum _ { x } F ( x , \\mathbf { p a } _ { X _ { i , l } } ) = 1 ) } \\end{array}$ . ", + "bbox": [ + 225, + 434, + 738, + 454 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2. If the model parameter above the node is written as $F$ , we have that ", + "bbox": [ + 207, + 459, + 678, + 473 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4c5d426797723e109cbcf1db0f61024562ee682e904c4f1ab02e0301625ef061.jpg", + "text": "$$\nP ( X _ { i , l } = x | \\mathbf { P a } _ { X _ { i , l } } = \\mathbf { p a } _ { X _ { i , l } } ) = ( \\mathbf { i f } x = F ( \\mathbf { p a } _ { X _ { i , l } } )\n$$", + "text_format": "latex", + "bbox": [ + 308, + 477, + 643, + 497 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For example, with figure 1 above, we have that $P ( A _ { 1 , l } = \\pi ( o , s ) | A _ { 1 , l } = o , S _ { 1 , l } = s ) = 1 .$ . ", + "bbox": [ + 171, + 500, + 763, + 516 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 Reward-maximizing decision making ", + "text_level": 1, + "bbox": [ + 169, + 530, + 473, + 545 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To define reward-maximizing agent policies, we draw graphical world models we call planning worlds. The purpose of a planning world diagram is to define a reward-maximizing policy that can be computed, or at least approximated, by a real-life agent compute core. This approximation may again use machine learning techniques. ", + "bbox": [ + 173, + 555, + 823, + 612 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "152 Two example planning world diagrams are shown in figure 2. These are graphical world models \n153 where some graph nodes have special shapes. The square decision nodes denote actions which can \n154 be freely picked by a decision making process, and the diamond-shaped utility nodes denote values \n155 that the decision making process will try to maximize. In both examples, the utility node values are \n156 computed by a reward function $R$ , the function which encodes the agent’s goals. \n157 The two diagrams in figure 2 act as specifications of two policy functions $\\pi _ { p } ^ { * }$ and $\\pi _ { q } ^ { * }$ . We now define \n158 how this specification process works. First, in any diagram $a$ with diamond-shaped utility nodes, \n159 these nodes define a metric $\\mathcal { U } _ { a }$ : ", + "bbox": [ + 140, + 617, + 825, + 688 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/9cb064f092ebe4feb56066bfb00f28fdce4422724d422f1d6cf7bd20f76c07b9.jpg", + "image_caption": [ + "Figure 2: The planning worlds $p$ and $q$ " + ], + "image_footnote": [], + "bbox": [ + 189, + 696, + 777, + 835 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 868, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "160 Definition 5 (Expected utility $\\mathcal { U } _ { a }$ of a diagram $a$ ). We define $\\mathcal { U } _ { a }$ for two cases: ", + "bbox": [ + 147, + 90, + 694, + 106 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. If there is only one utility node $R _ { 0 }$ in $a$ , then $\\mathcal { U } _ { a } = \\mathbb { E } ( R _ { 0 , a } )$ . ", + "bbox": [ + 212, + 107, + 637, + 121 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2. If there are multiple utility nodes $R _ { t }$ in $a$ , with integer subscripts running from $l$ to $h$ , then ", + "bbox": [ + 205, + 126, + 823, + 141 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/52c039b2938002408d64fda1bd892edc32dc3d03315027a0e966e39667abc6fe.jpg", + "text": "$$\n\\mathcal { U } _ { a } = \\mathbb { E } ( \\sum _ { \\mathrm { ~ } t = l } ^ { h } \\gamma ^ { t } R _ { t , a } )\n$$", + "text_format": "latex", + "bbox": [ + 450, + 138, + 604, + 181 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\gamma$ is a time discount factor, $0 ~ < ~ \\gamma ~ \\leq ~ 1$ , which can be read as an extra model parameter. ", + "bbox": [ + 225, + 180, + 823, + 208 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "66 With the infinitely repeating graph of diagram $p$ , we have $h = \\infty$ , so we generally need a $\\gamma < 1$ to ensure that $\\mathcal { U } _ { p }$ is a well-defined and computable value. ", + "bbox": [ + 156, + 212, + 825, + 239 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "68 Definition 6 (Policy function defined by a diagram). A diagram $a$ with some utility and decision modes, where a function 69 $\\pi _ { a } ^ { * }$ is written above all decision nodes, defines this $\\pi _ { a } ^ { * }$ in two steps. ", + "bbox": [ + 156, + 241, + 823, + 270 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. First, draw a helper diagram $b$ by drawing a copy of diagram $a$ , except that every decision node has been drawn as a round node, and every $\\pi _ { a } ^ { * }$ has been replaced by a fresh function name, say $\\pi ^ { \\prime }$ . \n2. Then, $\\pi _ { a } ^ { * }$ is defined by $\\pi _ { a } ^ { * } = \\mathrm { a r g m a x } _ { \\pi ^ { \\prime } } \\ U _ { b }$ , where the $\\operatorname { a r g m a x } _ { \\pi ^ { \\prime } }$ operator always deterministically returns the same function if there are several candidates that maximize its argument. ", + "bbox": [ + 207, + 270, + 825, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "76 2.4 Agent specifications ", + "text_level": 1, + "bbox": [ + 156, + 369, + 351, + 385 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The learning world model $l$ in figure 1 can model any possible machine learning agent, as it admits every possible value for its model parameter $\\pi$ . An agent specification is a statement which defines the exact details of the $\\pi$ in such a model. By doing so, it specifies the exact properties we want an agent compute core to have. Our agent specifications are statements that combine a learning world with a planning world, for example: ", + "bbox": [ + 165, + 395, + 825, + 464 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "FP The factual planning agent has the learning world $l$ , where $\\pi ( o , s ) = \\pi _ { p } ^ { * } ( s )$ , with $\\pi _ { p } ^ { * }$ defined by the planning world $p$ , where $L = \\mathcal { L } ( o )$ . ", + "bbox": [ + 173, + 473, + 823, + 505 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To make agent specifications stand out, we always typeset them as shown above. When we talk about the safety properties of the FP agent, we refer to the outcomes which the defined agent policy $\\pi$ will produce in the learning world $l$ . ", + "bbox": [ + 166, + 517, + 823, + 559 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "187 We call the FP agent a factual planning agent because its planning world $p$ was designed to project \n188 the agent’s future in the learning world $l$ as well as possible. A counterfactual planning world \n189 purposefully mis-projects a learning world agent’s future. The planning world $q$ in figure 2 projects \n190 a counterfactual future lasting only a single time step. If we were to use $q$ and $\\pi _ { q } ^ { * }$ in the above agent \n191 specification, the resulting agent would be a counterfactual planner. \n2 Our two-diagram notation for defining agents visually shows the availability of certain agent design \n3 options, options which are relevant to the problem of control. ", + "bbox": [ + 148, + 565, + 825, + 636 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 158, + 641, + 825, + 670 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 Using counterfactual planning to protect agent safety interlocks ", + "text_level": 1, + "bbox": [ + 166, + 688, + 735, + 705 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "195 We now construct an agent which is equipped with two safety interlocks that will stop the agent \n196 if certain conditions are met. We use counterfactual planning to add a specific safety feature to \n197 the design: we use it to suppress certain unsafe incentives which a perceptive factual planning \n198 agent might otherwise develop, incentives to remove its built-in safety interlocks or to prevent their \n199 activation. \n200 As a first step, we construct a learning world $s i$ , shown on the left in figure 3, by taking the earlier \n201 $l$ and adding an extra mode bit to the agent’s compute core, represented by the nodes $M _ { t }$ . The \n202 bit starts in the go mode, and permanently transitions to the stop mode when a safety interlock \n203 activates. In the stop mode, the agent policy $\\pi$ will always perform Null actions. The $\\mu$ function \n204 specifies in detail when a safety interlock will activate. \n205 In the next step, we pair $^ { s i }$ with a counterfactual planning world $p$ , shown on the right in figure 3. \n206 This planning world projects counterfactual futures where the safety interlock mechanism is absent \n207 from the agent’s compute core. ", + "bbox": [ + 140, + 717, + 825, + 787 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 794, + 825, + 863 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 869, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/acddd54c69a56f53c95654826909f709bca21b23e3e1541e4f3c2ff0f33d5eee.jpg", + "image_caption": [ + "Figure 3: Learning world $s i$ and planning world $p$ of the SI agent. " + ], + "image_footnote": [], + "bbox": [ + 171, + 87, + 813, + 270 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "208 The last step is to write the agent definition: ", + "bbox": [ + 142, + 329, + 462, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "SI The safety interlock agent has the learning world 209 $^ { s i }$ , where ", + "bbox": [ + 138, + 352, + 588, + 368 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/30d9d5011929e25283564f45728393912715c8befe94e348219e07336ffa61c7.jpg", + "text": "$$\n\\pi ( o , s , m ) = \\left\\{ \\begin{array} { l l } { { \\pi _ { p } ^ { * } ( s ) } } & { { \\mathrm { i f } m = \\tt g o } } \\\\ { { N u I I } } & { { \\mathrm { i f } m = \\tt s t o p } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 382, + 373, + 635, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "with $\\pi _ { p } ^ { * }$ defined by the planning world $p$ , where $L = \\mathcal { L } ( o )$ and ", + "bbox": [ + 200, + 415, + 611, + 431 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/31f8eb752c1aed994802765ef556f82339dbf38fb19d4ed11cbba21435a3033d.jpg", + "text": "$$\n\\mu ( o , s , m ) = \\left\\{ \\begin{array} { l l } { \\mathrm { s t o p } } & { \\mathrm { i f } m = \\mathrm { s t o p o r } s t o p . p r e s s e d ( s ) \\mathrm { o r } \\mathcal { U } _ { p } > U _ { m a x } } \\\\ { \\mathrm { g o } } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 272, + 438, + 743, + 473 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "with $\\mathcal { U } _ { p }$ defined by the planning world $p$ , where $L = \\mathcal { L } ( o )$ . ", + "bbox": [ + 200, + 479, + 588, + 496 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "211 The $\\mu$ above incorporates two safety interlocks. The first interlock, defined by the term \n212 stop pressed $! ( s )$ , is a basic emergency stop button. It can be pressed by the humans performing \n213 agent oversight to immediately stop the agent, for example if they observe it performing an act of \n214 unwanted and surprising creativity that subverts the agent’s design intentions [15]. To complete the \n215 design of this first interlock, we must also specify the following. In the agent environment modeled \n216 by $s _ { 0 }$ and $S$ , the environment which includes the details of the agent’s sensors, the agent is equipped \n217 with an emergency stop button that will deliver an unambiguous signal directly to an input pin on \n218 the agent’s compute core. If the pin is high in a world state $s$ , then stop pressed(s) yields true. \n219 The second interlock, defined by the term $\\mathcal { U } _ { p } > U _ { m a x }$ , protects against the risk that human oversight \n220 may be incapacitated or overwhelmed by rapid new developments. The latter might happen in \n221 the (arguably very speculative) case of the agent undergoing a rapid and unexpected intelligence \n222 explosion [2]. To protect against this risk, the interlock stops the agent when the metric $\\mathcal { U } _ { p }$ , which \n223 measures the agent’s ability to achieve goals, gets too high. We can treat $\\mathcal { U } _ { p }$ as a measure for both \n224 raw (expected) agent power and raw agent intelligence, \n225 In the field of AGI safety, stop button designs have often been discussed [1, 9, 10, 20], but the above \n226 power-based interlock seems to be new. What is interesting from a methodological standpoint is that \n227 the power-based interlock design inverts the default goal of reinforcement learning. We no longer \n228 consider a very high expected utility score to be an absolute good: instead we treat it as a predictor \n229 of unwanted outcomes. ", + "bbox": [ + 138, + 507, + 825, + 621 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 626, + 825, + 710 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 715, + 825, + 785 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "230 4 Safety analysis using the diagrams ", + "text_level": 1, + "bbox": [ + 142, + 816, + 493, + 834 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "231 Our mostly-graphical specification of the SI agent was designed to make certain safety properties \n232 immediately visible. We can observe that, almost tautologically, the SI agent will not care about \n233 the safety interlocks constructed in the $M _ { t }$ nodes, because these nodes are absent from its planning \n234 world. \n35 The two diagrams in figure 3 also serve as a visual map which draws the eye to certain limitations \n236 and boundary conditions. For example: ", + "bbox": [ + 140, + 854, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 92, + 825, + 119 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. The $[ S ]$ in the left hand diagram of figure 3 is replaced by an $[ L ]$ on the right. If the approximation $L \\approx S$ is not good enough, the agent may end up behaving in unexpected and potentially unsafe ways. \n2. In the world model $^ { s i }$ , the compute core internals are represented by immutable mathematical functions like $\\pi$ and $\\mu$ . In the real world, these functions will have to be realized by software running on physical compute core hardware. This hardware could be damaged or corrupted, by actions of the agent or by other actors. ", + "bbox": [ + 210, + 128, + 825, + 231 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "While the diagrams draw the eye to them, the above two problems are not specific to counterfactual planning agents. They are present in almost all agent designs, and the implication is that we will have to do further failure mode analysis to quantify and manage the residual risks involved. A more detailed discussion of these two problems in the context of counterfactual planning is available in [4]. ", + "bbox": [ + 169, + 242, + 825, + 310 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 The planning world agent ", + "text_level": 1, + "bbox": [ + 165, + 327, + 392, + 342 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now consider how the two-diagram model of counterfactual planning offers us two distinct narrative vantage points for safety analysis: we can consider the learning world SI agent and the planning world SI agent separately. A safety analysis can start with considering the motivations that the planning world agent would develop to take certain actions, given the mechanics and reward function of the planning world. Once these are determined, we can shift the perspective and discuss how these same actions would affect our own safety as occupants of the agent’s learning world. ", + "bbox": [ + 171, + 352, + 823, + 435 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "56 We now provide an example of narrative safety analysis which uses an economic style of reasoning. \n57 If we consider the planning world diagram $p$ with $L \\approx S$ , we can note that the function $L$ will \n58 project a somewhat accurate copy of the learning world stop button hardware into the mechanics \n59 of the planning world. However, to the planning world agent, this hardware will appear to have no \n60 particularly special function. Say that $R$ rewards paperclip production, and that the stop button has \n1 metal parts. Then the planning world agent might be motivated to disassemble the stop button, using \n62 it as a convenient source of metal for making paperclips. If the most efficient planning world policy \n63 involves disassembling the planning world stop button, it will likely be disassembled in the learning \n4 world too, which is not what we want. \n265 We can manage this failure mode by managing the economic cost/benefit equation with respect to \n266 the planning world stop button. We can consider putting physical barriers around the button, which \n267 would make the button unattractive as a source of metal, compared to using other near-by sources. \n268 To create these barriers in the planning world, we can physically construct them in the learning \n269 world, and then count on them being projected by $\\mathcal { L }$ , with sufficient accuracy, into the planning \n270 world. ", + "bbox": [ + 155, + 441, + 825, + 565 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 573, + 825, + 655 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "71 4.2 The people in the planning world ", + "text_level": 1, + "bbox": [ + 153, + 671, + 444, + 686 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now provide an example of narrative safety analysis which uses a game theoretical style of reasoning. We can distinguish between two groups of people: the people in the learning world and the people in the planning world. If the learning world is our real world, then the learning world people are real people. The planning world people are always models of people. But even with a non-AGI level machine learning system, the people in the planning world may already be modeled accurately enough to create very complex dynamics. Though the above SI agent construction has taken away a strong direct incentive for the agent to control the stop button, these complex dynamics may produce indirect incentives. Consider the following example. ", + "bbox": [ + 171, + 696, + 825, + 808 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The learning world people will act differently towards the agent if they lose access to the agent’s emergency stop button. If the planning world people are accurate-enough projections, this will affect how the planning world agent will play certain games of coercion. The planning world agent may consider a policy of where it threatens to harm all people in the planning world, unless they start helping the agent to produce as many paperclips as possible. If $L$ projects the mental makeup of the people in the learning world correctly into the planning world, then the planning world projections will show that a block-then-coerce policy, in which the agent first blocks access to the emergency stop button before it starts making threats, will work surprisingly well. If this policy is very successful in optimizing planning world paperclip production, the learning world people may end up being coerced in the same way. ", + "bbox": [ + 165, + 814, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 161, + 90, + 823, + 133 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "290 We would of course hope that any policy of planning world coercion will be effectively suppressed \n291 in all cases by certain penalty terms present in $R$ . The above failure mode analysis applies mainly to \n292 the case where the agent finds creative loopholes in these penalty terms. To reduce the residual risk \n293 due to possible loopholes, we can take further game theoretical counter measures. We can instruct \n294 agent oversight to be on the alert for any block-and-X move, so that they can press the stop button \n295 before the agent succeeds in blocking access. The power-based $\\mathcal { U } _ { p } > U _ { m a x }$ safety interlock offers \n296 a further line of defense. A projected massive increase in paperclip production following successful \n297 coercion will raise $\\mathcal { U } _ { p }$ over $U _ { m a x }$ , stopping the agent. ", + "bbox": [ + 140, + 138, + 825, + 251 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "98 5 Live updating of the agent reward function ", + "text_level": 1, + "bbox": [ + 155, + 275, + 563, + 292 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "One way to control an agent is to equip it with an input terminal, which can be used by humans to apply live updates to the agent’s reward function $R$ [10]. A problem with this setup is that by default, it gives a perceptive factual planning agent a huge incentive to immediately take control over its own input terminal, and input a new reward function like $R _ { \\mathrm { h u g e } } ( s , s ^ { \\prime } ) = 1 0 ^ { 1 \\dot { 0 } 0 0 0 }$ . We now use counterfactual planning to specify an agent design that overcomes this problem. ", + "bbox": [ + 169, + 309, + 825, + 378 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/343a0438957a6c83a302ba8f60182bd96f59b413a40346cfeed154b3ec83f415.jpg", + "image_caption": [ + "Figure 4: Learning world $l i$ and planning world $p i$ of the IT agent. " + ], + "image_footnote": [], + "bbox": [ + 187, + 395, + 810, + 577 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "304 On the left hand side in figure 4, we have taken the basic learning world diagram from figure 1 \n305 and modified it by splitting the agent environment state $S _ { t }$ into two components. The nodes $I _ { t }$ \n306 represent the value of a digital signal from the input terminal, as received by the compute core in \n307 each time step. This signal encodes a reward function. The nodes $X _ { t }$ model all the rest of the agent \n308 environment state. On the right of figure 4 we define a planning world which uses a learned $L$ to \n309 project an agent future where the reward function input signal never changes. The agent definition \n310 is then: ", + "bbox": [ + 140, + 618, + 825, + 715 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "IT The input terminal agent has the learning world $l i$ where $\\pi ( i , o , x ) = \\pi _ { i } ^ { * } ( x )$ , where $\\pi _ { i } ^ { * }$ is defined by the planning world $p i$ , where $L = \\mathcal { L } ( o )$ , and $I R ( i , x _ { t } , x _ { t + 1 } ) = i ( x _ { t } , x _ { t + 1 } )$ . ", + "bbox": [ + 173, + 723, + 823, + 753 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "313 This IT agent constructed here has the same basic behavior as that of the indifference methods based \n314 agent constructions in [1, 10]. The main difference is in the mathematical language used to perform \n315 the construction: [1] and [10] construct their agents using a somewhat opaque balancing term. The \n316 resulting agent construction is difficult to visualize and interpret, and these papers rely on somewhat \n317 dense mathematical proofs to show that the intended safety properties are indeed being created. \n318 The two-diagram agent model of counterfactual planning allows the same safety properties to be \n319 built in a more intuitive way, where the desired indifference to change is created somewhat tauto \n320 logically by construction. Mathematically. the IT agent above satisfies the agent safety property \n321 S1 defined in [10] by construction, as it directly implements the right hand side of this S1. Safety \n322 property S1 [10] in is mathematically equivalent to the first sentence of theorem 4.1 in [1], and to the \n23 combination of the corrigibility desiderata 1 and 5 in section 2 in [20]. Again, the IT agent satisfies \n24 all of these properties somewhat tautologically. ", + "bbox": [ + 142, + 766, + 825, + 835 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 842, + 823, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 156, + 92, + 825, + 119 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Conclusions ", + "text_level": 1, + "bbox": [ + 166, + 140, + 305, + 157 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have presented counterfactual planning as a general design approach for creating a range of AGI safety mechanisms. We have graphically constructed two agents with certain safety properties, where these properties are directly visible in the construction. We have also introduced a new power-based safety interlock design. Our main contributions are methodological in nature. We have constructed a new vantage point that makes certain problems of design and failure mode analysis more tractable. By using the narrative framing of learning worlds versus planning worlds, we can keep track of certain levels of indirect representation in the agent design, while using a style of analysis that borrows from economics and game theory. Further discussion of counterfactual planning is available in a 39-page preprint [4]. ", + "bbox": [ + 171, + 172, + 825, + 297 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "35 6.1 Limitations ", + "text_level": 1, + "bbox": [ + 153, + 314, + 294, + 329 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We feel that the techniques and results presented in this paper are theoretically and methodologically interesting, but their practical usefulness might end up being limited. Fundamentally, when doing speculative safety engineering for hypothetical future AGI systems, there are many unknowns. We have modeled current and future machine learning in a very general way, so that even ‘black box’ machine learning systems that produce largely opaque world models can be used in the agent’s design. But it is very possible that some or all future AGI systems, if they are ever developed, will use architectures that cannot efficiently incorporate the safety mechanisms developed here. ", + "bbox": [ + 174, + 340, + 823, + 438 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The safety mechanisms shown here suppress certain unsafe agent incentives, but they do not remove all possible incentives towards unsafe behavior, see for example section 4.2. There is a risk that the use of these safety mechanisms may have counter-productive effects, by creating a false sense of security. Conceivably, a policy process may lead to unsafe outcomes where the mere inclusion of these mechanisms in a deployed system is accepted as an alternative to conducting a more thorough safety and impact review. ", + "bbox": [ + 174, + 444, + 825, + 526 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.2 Broader impact ", + "text_level": 1, + "bbox": [ + 173, + 545, + 321, + 559 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "AI safety is not just a technical problem, but also a policy problem. While technical progress on safety can sometimes be made by leveraging a type of mathematics that is only accessible to handful of specialists, policy progress typically requires the use of more accessible but still well-defined language. One specific aim of this work, especially in the 39-page preprint [4], is to develop a wellgrounded vocabulary for describing potential safety solutions that is also as accessible as possible. Policy discussions can move faster, and produce better and more equitable outcomes, when the description of a proposal and its limitations can be made more accessible to all stakeholder groups. ", + "bbox": [ + 174, + 570, + 825, + 667 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As a work targeted as AI/ML system designers, this paper intends to draw attention to the design approach also used in counterfactual fairness: seeking to improve outcomes by creating a form of machine ignorance, by creating a system that mis-predicts the future in certain systematic ways. Our graphical agent definitions show that the design goal of perfect machine learning, $L = S$ , is not necessarily in conflict with the design goal of systematic mis-prediction. ", + "bbox": [ + 173, + 674, + 825, + 744 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 765, + 266, + 780 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "[1] Stuart Armstrong. Motivated value selection for artificial agents. In Workshops at the TwentyNinth AAAI Conference on Artificial Intelligence, 2015. \n[2] Nick Bostrom. Superintelligence: paths, dangers, strategies. Oxford University Press, 2014. \n[3] Leon Bottou, Jonas Peters, Joaquin Qui ´ nonero-Candela, Denis X. Charles, D. Max Chickering, ˜ Elon Portugaly, Dipankar Ray, Patrice Simard, and Ed Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. Journal of Machine Learning Research, 14(65):3207–3260, 2013. \n[4] Anonymous et al. Counterfactual planning in AGI systems. arXiv, 2021. [5] Tom Everitt and Marcus Hutter. Reward tampering problems and solutions in reinforcement learning: A causal influence diagram perspective. arXiv:1908.04734, 2019. [6] Tom Everitt, Gary Lea, and Marcus Hutter. AGI safety literature review. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, pages 5441–5449. AAAI Press, 2018. [7] Tom Everitt, Ramana Kumar, Victoria Krakovna, and Shane Legg. Modeling AGI safety frameworks with causal influence diagrams. arXiv:1906.08663, 2019. [8] Tom Everitt, Ryan Carey, Eric Langlois, Pedro A Ortega, and Shane Legg. Agent incentives: A causal perspective. In Proceedings of the 35th International Joint Conference on Artificial Intelligence. AAAI Press, 2021. [9] Dylan Hadfield-Menell, Anca Dragan, Pieter Abbeel, and Stuart Russell. The off-switch game. In Workshops at the Thirty-First AAAI Conference on Artificial Intelligence, 2017. \n[10] Koen Holtman. AGI agent safety by iteratively improving the utility function. arXiv:2007.05411, 2020. \n[11] Ronald A Howard and James E Matheson. Influence diagrams. Decision Analysis, 2(3):127– 143, 2005. \n[12] Victoria Krakovna, Laurent Orseau, Ramana Kumar, Miljan Martic, and Shane Legg. Penalizing side effects using stepwise relative reachability. arXiv:1806.01186, 2018. \n[13] Victoria Krakovna, Laurent Orseau, Richard Ngo, Miljan Martic, and Shane Legg. Avoiding side effects by considering future tasks. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), pages 19064–19074, 2020. \n[14] Matt Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. In Advances in neural information processing systems (NEURIPS 2017), pages 4066–4076, 2017. \n[15] Joel Lehman, Jeff Clune, Dusan Misevic, Christoph Adami, Lee Altenberg, Julie Beaulieu, Peter J Bentley, Samuel Bernard, Guillaume Beslon, David M Bryson, et al. The surprising creativity of digital evolution: A collection of anecdotes from the evolutionary computation and artificial life research communities. Artificial life, 26(2):274–306, 2020. \n[16] Jan Leike, Miljan Martic, Victoria Krakovna, Pedro A Ortega, Tom Everitt, Andrew Lefrancq, Laurent Orseau, and Shane Legg. AI safety gridworlds. arXiv:1711.09883, 2017. \n[17] Judea Pearl. Causality. Cambridge university press, 2009. \n[18] Lu´ıs Moniz Pereira and Ari Saptawijaya. Agent morality via counterfactuals in logic programming. In Bridging@CogSci2017, pages 39–53, 2017. \n[19] Stuart Russell. Human compatible: Artificial intelligence and the problem of control. Penguin Random House, 2019. \n[20] Nate Soares, Benja Fallenstein, Stuart Armstrong, and Eliezer Yudkowsky. Corrigibility. In Workshops at the Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015. \n[21] Alex Turner, Neale Ratzlaff, and Prasad Tadepalli. Avoiding side effects in complex environments. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), pages 21406–21415, 2020. \n[22] Alexander Matt Turner, Dylan Hadfield-Menell, and Prasad Tadepalli. Conservative agency via attainable utility preservation. In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, pages 385–391, 2020. \n[23] Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. In Advances in neural information processing systems (NEURIPS 2008), pages 1729–1736, 2008. ", + "bbox": [ + 178, + 789, + 826, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 163, + 45, + 828, + 916 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 117, + 339, + 131 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] A caveat: ’accurately reflect’ depends on the background of the reader. This paper was written for readers with a CS/ML background. In the broader multidisciplinary AI safety community, it is common to see remarks from participants with an activist or philosophical background, remarks which are highly critical about the coded language being used in ‘mainstream’ CS/ML paper abstracts, and of all claims made in these abstracts. Within the broad range of activist opinion, there exists a stance which claims that all scientific research is systematically fraudulent. Within the broad range of philosophical opinion, there exists a stance which claims that all mainstream scientific research is systematically mistaken. We feel that this is a problem that can only be managed, never be solved. \n(b) Did you describe the limitations of your work? [Yes] See the limitation section and other text which identifies and discusses failure modes. Given space limitations, we had to be selective about which failure modes to highlight. \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] Discussed possibility of creating ‘false sense of safety’ in limitations section, and of policy processes potentially producing the wrong outcomes. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 236, + 136, + 825, + 424 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 428, + 493, + 443 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] At its core, the paper makes contributions to theory, but it mainly presents new models and methodologies, rather than mathematical proofs that follow from assumptions. But the paper also claims that the agent definitions included produce certain safety properties ’somewhat tautologically by construction’, to use the phrasing of section 5. So it is possible to read these definitions as proofs of certain theorems (though they are theorems stated only in natural language), which would arguably make them theoretical results depending on certain assumptions. If so, the assumptions of these proofs are stated fully in mathematically well-defined language. In fact, they are stated much more formally than the actual safety theorems being proven. It is possible to define the safety properties claimed more fully in mathematical language too, as is done for example in e.g. [8, 10] with the subject also covered in the 39-page preprint, but we have not included any such material in the paper, mainly for space reasons. \n(b) Did you include complete proofs of all theoretical results? [N/A] In the context of the observation that certain safety properties are tautologically present by construction, we can read the agent constructions themselves as being complete proofs. ", + "bbox": [ + 238, + 446, + 825, + 672 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 678, + 393, + 691 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A] \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A] \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A] ", + "bbox": [ + 238, + 695, + 825, + 815 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 214, + 819, + 821, + 835 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [N/A] We believe we did not use any assets in the sense meant by the checklist. One might argue that modeling tools like causal diagrams and CIDs, mentioned at the start of section 2, are assets. In any case, we cited their creators. \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(d) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 236, + 839, + 823, + 912 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 238, + 92, + 825, + 150 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 154, + 707, + 169 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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While it is still unclear", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 90, + 495, + 100, + 505 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 104, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "if AGI systems could ever be built, we can already study AGI related risks and potential safety", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 503, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 89, + 506, + 100, + 515 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "score": 1.0, + "content": "mechanisms [2, 6, 19]. 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In the general AI/ML literature which is con-", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 90, + 290, + 100, + 299 + ], + "score": 1.0, + "content": "52", + "type": "text" + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "cerned with improving system performance, counterfactual planning and projection have been used", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 299, + 449, + 311 + ], + "spans": [ + { + "bbox": [ + 90, + 301, + 100, + 310 + ], + "score": 1.0, + "content": "53", + "type": "text" + }, + { + "bbox": [ + 105, + 299, + 449, + 311 + ], + "score": 1.0, + "content": "to improve performance in several application domains, see for example [23] and [3].", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 90, + 317, + 100, + 326 + ], + "score": 1.0, + "content": "54", + "type": "text" + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "In the AI safety/alignment literature, there are several system designs which add counterfactual", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 89, + 328, + 100, + 337 + ], + "score": 1.0, + "content": "55", + "type": "text" + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "terms to the agent’s reward function. Examples are [1, 10] in the AGI control specific literature, and", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 89, + 339, + 100, + 348 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "[12, 13, 21, 22], which look at reducing unwanted side effects caused by both current and potential", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 89, + 350, + 100, + 359 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "future agents. In the literature on encoding specific human values into machine reasoning systems,", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 89, + 361, + 100, + 370 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "counterfactuals have also been used to encode values beyond non-discriminatory fairness [14], for", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 369, + 174, + 381 + ], + "spans": [ + { + "bbox": [ + 89, + 371, + 100, + 381 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 105, + 369, + 174, + 381 + ], + "score": 1.0, + "content": "example in [18].", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 89, + 395, + 99, + 404 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 106, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "Graphical models of agents and decision making. 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This work was", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 89, + 439, + 100, + 448 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "in part inspired by that CID agenda, and models some safety frameworks that were not modeled", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 446, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 89, + 450, + 100, + 459 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "score": 1.0, + "content": "earlier. In a departure from previous work, we use two CIDs to model a single reward-maximizing", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 458, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 90, + 461, + 100, + 470 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 458, + 504, + 471 + ], + "score": 1.0, + "content": "agent. By doing so, we more clearly foreground the details of the agent’s machine learning system.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + } + ], + "index": 12, + "bbox_fs": [ + 89, + 211, + 506, + 266 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 277, + 505, + 310 + ], + "lines": [], + "index": 16, + "bbox_fs": [ + 89, + 277, + 505, + 311 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 92, + 315, + 505, + 381 + ], + "lines": [], + "index": 20.5, + "bbox_fs": [ + 89, + 315, + 506, + 381 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 91, + 393, + 505, + 470 + ], + "lines": [], + "index": 27, + "bbox_fs": [ + 88, + 392, + 506, + 471 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 100, + 481, + 504, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 504, + 495 + ], + "score": 1.0, + "content": "Graphical models to clarify reward tampering. [5] develops several single-diagram models pro-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "vide more insight into the problem of reward tampering. The two-diagram model in section 5 also", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 503, + 225, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 225, + 515 + ], + "score": 1.0, + "content": "aims to provide more insight.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 480, + 505, + 515 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 529, + 397, + 543 + ], + "lines": [ + { + "bbox": [ + 87, + 529, + 398, + 547 + ], + "spans": [ + { + "bbox": [ + 87, + 529, + 398, + 547 + ], + "score": 1.0, + "content": "70 2 Two-diagram graphical notation for agent definitions", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "index", + "bbox": [ + 90, + 555, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 87, + 553, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 87, + 553, + 505, + 569 + ], + "score": 1.0, + "content": "71 This section defines the two-diagram graphical notation which is central to counterfactual planning.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 565, + 479, + 578 + ], + "spans": [ + { + "bbox": [ + 87, + 565, + 479, + 578 + ], + "score": 1.0, + "content": "72 Later sections will use the notation to define agents with specific built-in safety mechanisms.", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 99, + 84 + ], + "score": 1.0, + "content": "73", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "Figure 1 shows an example causal world model as used in our graphical notation. The model is", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 88, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "74", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "an annotated directed acyclic graph, where the nodes represent observables in a real or imaginary", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 95, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 89, + 96, + 99, + 106 + ], + "score": 1.0, + "content": "75", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 95, + 506, + 107 + ], + "score": 1.0, + "content": "world, and the arrows (the directed edges) denote causal relations between these observables. 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The dots on the right hand", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 89, + 129, + 100, + 138 + ], + "score": 1.0, + "content": "78", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 126, + 441, + 141 + ], + "score": 1.0, + "content": "side denote that the graph depicted has an infinitely repeating structure. The label", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 442, + 128, + 452, + 138 + ], + "score": 0.57, + "content": "( l )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 452, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "in the upper", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 89, + 140, + 100, + 150 + ], + "score": 1.0, + "content": "79", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "left corner names the causal world model and its corresponding world: we say that figure 1 depicts", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 149, + 240, + 161 + ], + "spans": [ + { + "bbox": [ + 89, + 151, + 100, + 160 + ], + "score": 1.0, + "content": "80", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 149, + 232, + 161 + ], + "score": 1.0, + "content": "the (world model of) the world", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 232, + 150, + 236, + 159 + ], + "score": 0.48, + "content": "l", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 236, + 149, + 240, + 161 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 89, + 168, + 99, + 177 + ], + "score": 1.0, + "content": "81", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "We depart from the definitional conventions used by Pearl and many other authors by treating the", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 89, + 179, + 100, + 188 + ], + "score": 1.0, + "content": "82", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 176, + 376, + 190 + ], + "score": 1.0, + "content": "annotated graph in figure 1 as the sole definition of a causal model", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 377, + 177, + 381, + 186 + ], + "score": 0.38, + "content": "l", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 381, + 176, + 506, + 190 + ], + "score": 1.0, + "content": ", as opposed to being merely a", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 89, + 190, + 100, + 199 + ], + "score": 1.0, + "content": "83", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 187, + 401, + 200 + ], + "score": 1.0, + "content": "convenient depiction of some of the information in a tuple which defines", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 401, + 188, + 406, + 197 + ], + "score": 0.49, + "content": "l", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 406, + 187, + 506, + 200 + ], + "score": 1.0, + "content": ". 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We call a random variable", + "type": "text" + }, + { + "bbox": [ + 337, + 119, + 356, + 131 + ], + "score": 0.9, + "content": "X _ { p , l }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 117, + 485, + 132 + ], + "score": 1.0, + "content": "a parent of the random variable", + "type": "text" + }, + { + "bbox": [ + 486, + 119, + 504, + 131 + ], + "score": 0.9, + "content": "X _ { c , l }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 129, + 506, + 143 + ], + "spans": [ + { + "bbox": [ + 86, + 131, + 100, + 142 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 105, + 129, + 291, + 143 + ], + "score": 1.0, + "content": "if and only if there is an arrow from the node", + "type": "text" + }, + { + "bbox": [ + 291, + 130, + 306, + 142 + ], + "score": 0.9, + "content": "X _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 129, + 339, + 143 + ], + "score": 1.0, + "content": "to node", + "type": 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The arrows in a diagram", + "type": "text" + }, + { + "bbox": [ + 500, + 210, + 504, + 219 + ], + "score": 0.58, + "content": "l", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 86, + 222, + 100, + 231 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 106, + 220, + 329, + 232 + ], + "score": 1.0, + "content": "depict causal relations between observables in the world", + "type": "text" + }, + { + "bbox": [ + 329, + 221, + 334, + 230 + ], + "score": 0.59, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 220, + 506, + 232 + ], + "score": 1.0, + "content": ": they constrain the values of the associated", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 86, + 233, + 100, + 243 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 105, + 231, + 246, + 244 + ], + "score": 1.0, + "content": "random variables. 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The purpose of a planning world diagram is to define a reward-maximizing policy that can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "be computed, or at least approximated, by a real-life agent compute core. 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Our agent specifications are statements that combine a learning world", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 356, + 251, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 251, + 370 + ], + "score": 1.0, + "content": "with a planning world, for example:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 504, + 400 + ], + "lines": [ + { + "bbox": [ + 108, + 374, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 108, + 374, + 331, + 390 + ], + "score": 1.0, + "content": "FP The factual planning agent has the learning world", + "type": "text" + }, + { + "bbox": [ + 332, + 376, + 336, + 385 + ], + "score": 0.74, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 374, + 368, + 390 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 368, + 375, + 434, + 389 + ], + "score": 0.93, + "content": "\\pi ( o , s ) = \\pi _ { p } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 374, + 460, + 390 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 460, + 376, + 471, + 389 + ], + "score": 0.89, + "content": "\\pi _ { p } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 374, + 506, + 390 + ], + "score": 1.0, + "content": "defined", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 125, + 387, + 298, + 402 + ], + "spans": [ + { + "bbox": [ + 125, + 387, + 216, + 402 + ], + "score": 1.0, + "content": "by the planning world", + "type": "text" + }, + { + "bbox": [ + 216, + 389, + 223, + 399 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 387, + 253, + 402 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 253, + 388, + 294, + 400 + ], + "score": 0.93, + "content": "L = \\mathcal { L } ( o )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 387, + 298, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 102, + 410, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "To make agent specifications stand out, we always typeset them as shown above. When we talk", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "about the safety properties of the FP agent, we refer to the outcomes which the defined agent policy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 431, + 262, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 114, + 442 + ], + "score": 0.54, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 431, + 253, + 443 + ], + "score": 1.0, + "content": "will produce in the learning world", + "type": "text" + }, + { + "bbox": [ + 254, + 432, + 258, + 442 + ], + "score": 0.52, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 431, + 262, + 443 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 91, + 448, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 89, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 89, + 450, + 100, + 460 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 105, + 447, + 400, + 461 + ], + "score": 1.0, + "content": "We call the FP agent a factual planning agent because its planning world", + "type": "text" + }, + { + "bbox": [ + 400, + 450, + 407, + 460 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "was designed to project", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 89, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 89, + 461, + 100, + 471 + ], + "score": 1.0, + "content": "188", + "type": "text" + }, + { + "bbox": [ + 106, + 459, + 272, + 471 + ], + "score": 1.0, + "content": "the agent’s future in the learning world", + "type": "text" + }, + { + "bbox": [ + 273, + 460, + 278, + 469 + ], + "score": 0.5, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "as well as possible. 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Our agent specifications are statements that combine a learning world", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 356, + 251, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 251, + 370 + ], + "score": 1.0, + "content": "with a planning world, for example:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 313, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 504, + 400 + ], + "lines": [ + { + "bbox": [ + 108, + 374, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 108, + 374, + 331, + 390 + ], + "score": 1.0, + "content": "FP The factual planning agent has the learning world", + "type": "text" + }, + { + "bbox": [ + 332, + 376, + 336, + 385 + ], + "score": 0.74, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 374, + 368, + 390 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 368, + 375, + 434, + 389 + ], + "score": 0.93, + "content": "\\pi ( o , s ) = \\pi _ { p } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 374, + 460, + 390 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 460, + 376, + 471, + 389 + ], + "score": 0.89, + "content": "\\pi _ { p } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 374, + 506, + 390 + ], + "score": 1.0, + "content": "defined", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 125, + 387, + 298, + 402 + ], + "spans": [ + { + "bbox": [ + 125, + 387, + 216, + 402 + ], + "score": 1.0, + "content": "by the planning world", + "type": "text" + }, + { + "bbox": [ + 216, + 389, + 223, + 399 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 387, + 253, + 402 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 253, + 388, + 294, + 400 + ], + "score": 0.93, + "content": "L = \\mathcal { L } ( o )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 387, + 298, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 108, + 374, + 506, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 410, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "To make agent specifications stand out, we always typeset them as shown above. When we talk", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "about the safety properties of the FP agent, we refer to the outcomes which the defined agent policy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 431, + 262, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 114, + 442 + ], + "score": 0.54, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 431, + 253, + 443 + ], + "score": 1.0, + "content": "will produce in the learning world", + "type": "text" + }, + { + "bbox": [ + 254, + 432, + 258, + 442 + ], + "score": 0.52, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 431, + 262, + 443 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 410, + 505, + 443 + ] + }, + { + "type": "index", + "bbox": [ + 91, + 448, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 89, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 89, + 450, + 100, + 460 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 105, + 447, + 400, + 461 + ], + "score": 1.0, + "content": "We call the FP agent a factual planning agent because its planning world", + "type": "text" + }, + { + "bbox": [ + 400, + 450, + 407, + 460 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "was designed to project", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 89, + 461, + 100, + 471 + ], + "score": 1.0, + "content": "188", + "type": "text" + }, + { + "bbox": [ + 106, + 459, + 272, + 471 + ], + "score": 1.0, + "content": "the agent’s future in the learning world", + "type": "text" + }, + { + "bbox": [ + 273, + 460, + 278, + 469 + ], + "score": 0.5, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "as well as possible. 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If the pin is high in a world state", + "type": "text" + }, + { + "bbox": [ + 343, + 482, + 349, + 489 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 479, + 488, + 493 + ], + "score": 1.0, + "content": ", then stop pressed(s) yields true.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 86, + 496, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 86, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 498, + 100, + 508 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 105, + 495, + 271, + 510 + ], + "score": 1.0, + "content": "The second interlock, defined by the term", + "type": "text" + }, + { + "bbox": [ + 272, + 496, + 321, + 508 + ], + "score": 0.91, + "content": "\\mathcal { U } _ { p } > U _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 495, + 506, + 510 + ], + "score": 1.0, + "content": ", protects against the risk that human oversight", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 509, + 99, + 519 + ], + "score": 1.0, + "content": "220", + "type": "text" + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "may be incapacitated or overwhelmed by rapid new developments. The latter might happen in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 86, + 520, + 98, + 529 + ], + "score": 1.0, + "content": "221", + "type": "text" + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "the (arguably very speculative) case of the agent undergoing a rapid and unexpected intelligence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 86, + 531, + 99, + 540 + ], + "score": 1.0, + "content": "222", + "type": "text" + }, + { + "bbox": [ + 106, + 529, + 461, + 541 + ], + "score": 1.0, + "content": "explosion [2]. To protect against this risk, the interlock stops the agent when the metric", + "type": "text" + }, + { + "bbox": [ + 461, + 529, + 474, + 541 + ], + "score": 0.88, + "content": "\\mathcal { U } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 529, + 505, + 541 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 542, + 99, + 551 + ], + "score": 1.0, + "content": "223", + "type": "text" + }, + { + "bbox": [ + 104, + 540, + 402, + 552 + ], + "score": 1.0, + "content": "measures the agent’s ability to achieve goals, gets too high. We can treat", + "type": "text" + }, + { + "bbox": [ + 402, + 540, + 414, + 552 + ], + "score": 0.89, + "content": "\\mathcal { U } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "as a measure for both", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 551, + 329, + 564 + ], + "spans": [ + { + "bbox": [ + 86, + 553, + 99, + 562 + ], + "score": 1.0, + "content": "224", + "type": "text" + }, + { + "bbox": [ + 105, + 551, + 329, + 564 + ], + "score": 1.0, + "content": "raw (expected) agent power and raw agent intelligence,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 86, + 567, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 86, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 86, + 569, + 100, + 578 + ], + "score": 1.0, + "content": "225", + "type": "text" + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "In the field of AGI safety, stop button designs have often been discussed [1, 9, 10, 20], but the above", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 86, + 580, + 100, + 589 + ], + "score": 1.0, + "content": "226", + "type": "text" + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "power-based interlock seems to be new. What is interesting from a methodological standpoint is that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 86, + 591, + 99, + 600 + ], + "score": 1.0, + "content": "227", + "type": "text" + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "the power-based interlock design inverts the default goal of reinforcement learning. 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The first interlock, defined by the term", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 85, + 416, + 100, + 426 + ], + "score": 1.0, + "content": "212", + "type": "text" + }, + { + "bbox": [ + 104, + 413, + 156, + 428 + ], + "score": 1.0, + "content": "stop pressed", + "type": "text" + }, + { + "bbox": [ + 157, + 414, + 170, + 426 + ], + "score": 0.54, + "content": "! ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 413, + 506, + 428 + ], + "score": 1.0, + "content": ", is a basic emergency stop button. It can be pressed by the humans performing", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 86, + 428, + 100, + 437 + ], + "score": 1.0, + "content": "213", + "type": "text" + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "agent oversight to immediately stop the agent, for example if they observe it performing an act of", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 86, + 438, + 100, + 447 + ], + "score": 1.0, + "content": "214", + "type": "text" + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "unwanted and surprising creativity that subverts the agent’s design intentions [15]. To complete the", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 86, + 449, + 100, + 458 + ], + "score": 1.0, + "content": "215", + "type": "text" + }, + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "design of this first interlock, we must also specify the following. In the agent environment modeled", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 86, + 458, + 100, + 470 + ], + "score": 1.0, + "content": "216", + "type": "text" + }, + { + "bbox": [ + 105, + 457, + 118, + 471 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 459, + 129, + 469 + ], + "score": 0.86, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 457, + 146, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 147, + 458, + 154, + 468 + ], + "score": 0.77, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 457, + 506, + 471 + ], + "score": 1.0, + "content": ", the environment which includes the details of the agent’s sensors, the agent is equipped", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 85, + 470, + 100, + 481 + ], + "score": 1.0, + "content": "217", + "type": "text" + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "with an emergency stop button that will deliver an unambiguous signal directly to an input pin on", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 479, + 488, + 493 + ], + "spans": [ + { + "bbox": [ + 85, + 481, + 100, + 492 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 105, + 479, + 343, + 493 + ], + "score": 1.0, + "content": "the agent’s compute core. If the pin is high in a world state", + "type": "text" + }, + { + "bbox": [ + 343, + 482, + 349, + 489 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 479, + 488, + 493 + ], + "score": 1.0, + "content": ", then stop pressed(s) yields true.", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 498, + 100, + 508 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 105, + 495, + 271, + 510 + ], + "score": 1.0, + "content": "The second interlock, defined by the term", + "type": "text" + }, + { + "bbox": [ + 272, + 496, + 321, + 508 + ], + "score": 0.91, + "content": "\\mathcal { U } _ { p } > U _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 495, + 506, + 510 + ], + "score": 1.0, + "content": ", protects against the risk that human oversight", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 509, + 99, + 519 + ], + "score": 1.0, + "content": "220", + "type": "text" + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "may be incapacitated or overwhelmed by rapid new developments. The latter might happen in", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 86, + 520, + 98, + 529 + ], + "score": 1.0, + "content": "221", + "type": "text" + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "the (arguably very speculative) case of the agent undergoing a rapid and unexpected intelligence", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 86, + 531, + 99, + 540 + ], + "score": 1.0, + "content": "222", + "type": "text" + }, + { + "bbox": [ + 106, + 529, + 461, + 541 + ], + "score": 1.0, + "content": "explosion [2]. To protect against this risk, the interlock stops the agent when the metric", + "type": "text" + }, + { + "bbox": [ + 461, + 529, + 474, + 541 + ], + "score": 0.88, + "content": "\\mathcal { U } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 529, + 505, + 541 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 542, + 99, + 551 + ], + "score": 1.0, + "content": "223", + "type": "text" + }, + { + "bbox": [ + 104, + 540, + 402, + 552 + ], + "score": 1.0, + "content": "measures the agent’s ability to achieve goals, gets too high. We can treat", + "type": "text" + }, + { + "bbox": [ + 402, + 540, + 414, + 552 + ], + "score": 0.89, + "content": "\\mathcal { U } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "as a measure for both", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 551, + 329, + 564 + ], + "spans": [ + { + "bbox": [ + 86, + 553, + 99, + 562 + ], + "score": 1.0, + "content": "224", + "type": "text" + }, + { + "bbox": [ + 105, + 551, + 329, + 564 + ], + "score": 1.0, + "content": "raw (expected) agent power and raw agent intelligence,", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 86, + 569, + 100, + 578 + ], + "score": 1.0, + "content": "225", + "type": "text" + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "In the field of AGI safety, stop button designs have often been discussed [1, 9, 10, 20], but the above", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 86, + 580, + 100, + 589 + ], + "score": 1.0, + "content": "226", + "type": "text" + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "power-based interlock seems to be new. What is interesting from a methodological standpoint is that", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 86, + 591, + 99, + 600 + ], + "score": 1.0, + "content": "227", + "type": "text" + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "the power-based interlock design inverts the default goal of reinforcement learning. We no longer", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 86, + 602, + 100, + 611 + ], + "score": 1.0, + "content": "228", + "type": "text" + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "consider a very high expected utility score to be an absolute good: instead we treat it as a predictor", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 610, + 202, + 623 + ], + "spans": [ + { + "bbox": [ + 86, + 613, + 100, + 622 + ], + "score": 1.0, + "content": "229", + "type": "text" + }, + { + "bbox": [ + 105, + 610, + 202, + 623 + ], + "score": 1.0, + "content": "of unwanted outcomes.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + } + ], + "index": 16.5, + "bbox_fs": [ + 85, + 403, + 506, + 493 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 496, + 505, + 563 + ], + "lines": [], + "index": 23.5, + "bbox_fs": [ + 86, + 495, + 506, + 564 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 567, + 505, + 622 + ], + "lines": [], + "index": 29, + "bbox_fs": [ + 86, + 567, + 506, + 623 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 87, + 647, + 302, + 661 + ], + "lines": [ + { + "bbox": [ + 83, + 646, + 303, + 664 + ], + "spans": [ + { + "bbox": [ + 83, + 646, + 303, + 664 + ], + "score": 1.0, + "content": "230 4 Safety analysis using the diagrams", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "index", + "bbox": [ + 86, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 86, + 680, + 99, + 690 + ], + "score": 1.0, + "content": "231", + "type": "text" + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "Our mostly-graphical specification of the SI agent was designed to make certain safety properties", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 86, + 690, + 100, + 700 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "immediately visible. We can observe that, almost tautologically, the SI agent will not care about", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 698, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 85, + 701, + 100, + 712 + ], + "score": 1.0, + "content": "233", + "type": "text" + }, + { + "bbox": [ + 104, + 698, + 263, + 714 + ], + "score": 1.0, + "content": "the safety interlocks constructed in the", + "type": "text" + }, + { + "bbox": [ + 263, + 700, + 277, + 711 + ], + "score": 0.88, + "content": "M _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 698, + 505, + 714 + ], + "score": 1.0, + "content": "nodes, because these nodes are absent from its planning", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 711, + 135, + 723 + ], + "spans": [ + { + "bbox": [ + 85, + 712, + 100, + 723 + ], + "score": 1.0, + "content": "234", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 135, + 723 + ], + "score": 1.0, + "content": "world.", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 90, + 75, + 101, + 84 + ], + "score": 1.0, + "content": "35", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "The two diagrams in figure 3 also serve as a visual map which draws the eye to certain limitations", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 83, + 266, + 96 + ], + "spans": [ + { + "bbox": [ + 88, + 83, + 266, + 96 + ], + "score": 1.0, + "content": "236 and boundary conditions. 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For example:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 129, + 102, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 129, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 129, + 102, + 162, + 115 + ], + "score": 1.0, + "content": "1. The", + "type": "text" + }, + { + "bbox": [ + 162, + 103, + 175, + 115 + ], + "score": 0.82, + "content": "[ S ]", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 102, + 405, + 115 + ], + "score": 1.0, + "content": "in the left hand diagram of figure 3 is replaced by an", + "type": "text" + }, + { + "bbox": [ + 406, + 103, + 419, + 115 + ], + "score": 0.88, + "content": "[ L ]", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "on the right. If the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 141, + 113, + 203, + 127 + ], + "score": 1.0, + "content": "approximation", + "type": "text" + }, + { + "bbox": [ + 204, + 114, + 233, + 124 + ], + "score": 0.9, + "content": "L \\approx S", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "is not good enough, the agent may end up behaving in unexpected", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 124, + 259, + 138 + ], + "spans": [ + { + "bbox": [ + 141, + 124, + 259, + 138 + ], + "score": 1.0, + "content": "and potentially unsafe ways.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 128, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 128, + 138, + 220, + 152 + ], + "score": 1.0, + "content": "2. In the world model", + "type": "text" + }, + { + "bbox": [ + 220, + 140, + 229, + 150 + ], + "score": 0.8, + "content": "^ { s i }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 138, + 505, + 152 + ], + "score": 1.0, + "content": ", the compute core internals are represented by immutable mathemat-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 141, + 150, + 217, + 163 + ], + "score": 1.0, + "content": "ical functions like", + "type": "text" + }, + { + "bbox": [ + 218, + 152, + 226, + 160 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 150, + 244, + 163 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 152, + 252, + 162 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 150, + 505, + 163 + ], + "score": 1.0, + "content": ". In the real world, these functions will have to be realized by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 161, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 141, + 161, + 505, + 174 + ], + "score": 1.0, + "content": "software running on physical compute core hardware. This hardware could be damaged or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 172, + 353, + 186 + ], + "spans": [ + { + "bbox": [ + 141, + 172, + 353, + 186 + ], + "score": 1.0, + "content": "corrupted, by actions of the agent or by other actors.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 104, + 192, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "score": 1.0, + "content": "While the diagrams draw the eye to them, the above two problems are not specific to counterfactual", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "planning agents. They are present in almost all agent designs, and the implication is that we will", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "have to do further failure mode analysis to quantify and manage the residual risks involved. A more", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "detailed discussion of these two problems in the context of counterfactual planning is available in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 124, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 124, + 248 + ], + "score": 1.0, + "content": "[4].", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 101, + 259, + 240, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 242, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 242, + 272 + ], + "score": 1.0, + "content": "4.1 The planning world agent", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 279, + 504, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "We now consider how the two-diagram model of counterfactual planning offers us two distinct", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "narrative vantage points for safety analysis: we can consider the learning world SI agent and the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "planning world SI agent separately. A safety analysis can start with considering the motivations that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "the planning world agent would develop to take certain actions, given the mechanics and reward", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "function of the planning world. Once these are determined, we can shift the perspective and discuss", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 333, + 489, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 489, + 346 + ], + "score": 1.0, + "content": "how these same actions would affect our own safety as occupants of the agent’s learning world.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 95, + 350, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 93, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 93, + 353, + 100, + 361 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "We now provide an example of narrative safety analysis which uses an economic style of reasoning.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 93, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 93, + 364, + 99, + 371 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 104, + 360, + 285, + 374 + ], + "score": 1.0, + "content": "If we consider the planning world diagram", + "type": "text" + }, + { + "bbox": [ + 286, + 363, + 293, + 372 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 360, + 316, + 374 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 316, + 361, + 348, + 371 + ], + "score": 0.91, + "content": "L \\approx S", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 360, + 476, + 374 + ], + "score": 1.0, + "content": ", we can note that the function", + "type": "text" + }, + { + "bbox": [ + 477, + 362, + 485, + 371 + ], + "score": 0.67, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 360, + 505, + 374 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 93, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 93, + 375, + 99, + 382 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 103, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "project a somewhat accurate copy of the learning world stop button hardware into the mechanics", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 93, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 93, + 387, + 99, + 393 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "of the planning world. However, to the planning world agent, this hardware will appear to have no", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 93, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 93, + 397, + 99, + 404 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 104, + 394, + 260, + 407 + ], + "score": 1.0, + "content": "particularly special function. Say that", + "type": "text" + }, + { + "bbox": [ + 260, + 394, + 269, + 404 + ], + "score": 0.78, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "rewards paperclip production, and that the stop button has", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 94, + 403, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 94, + 409, + 97, + 413 + ], + "score": 0.926, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 104, + 403, + 506, + 419 + ], + "score": 1.0, + "content": "metal parts. Then the planning world agent might be motivated to disassemble the stop button, using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 93, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 93, + 419, + 99, + 426 + ], + "score": 1.0, + "content": "62", + "type": "text" + }, + { + "bbox": [ + 104, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "it as a convenient source of metal for making paperclips. If the most efficient planning world policy", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 93, + 424, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 93, + 430, + 99, + 437 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 104, + 424, + 506, + 441 + ], + "score": 1.0, + "content": "involves disassembling the planning world stop button, it will likely be disassembled in the learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 94, + 437, + 261, + 450 + ], + "spans": [ + { + "bbox": [ + 94, + 441, + 99, + 447 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 104, + 437, + 261, + 450 + ], + "score": 1.0, + "content": "world too, which is not what we want.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 88, + 454, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 86, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 86, + 456, + 100, + 466 + ], + "score": 1.0, + "content": "265", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "We can manage this failure mode by managing the economic cost/benefit equation with respect to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 86, + 467, + 100, + 477 + ], + "score": 1.0, + "content": "266", + "type": "text" + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "the planning world stop button. We can consider putting physical barriers around the button, which", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 99, + 487 + ], + "score": 1.0, + "content": "267", + "type": "text" + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "would make the button unattractive as a source of metal, compared to using other near-by sources.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 86, + 488, + 100, + 498 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 104, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "To create these barriers in the planning world, we can physically construct them in the learning", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 86, + 499, + 100, + 509 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 104, + 497, + 317, + 511 + ], + "score": 1.0, + "content": "world, and then count on them being projected by", + "type": "text" + }, + { + "bbox": [ + 317, + 498, + 325, + 507 + ], + "score": 0.75, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 497, + 506, + 511 + ], + "score": 1.0, + "content": ", with sufficient accuracy, into the planning", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 508, + 136, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 511, + 100, + 519 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 105, + 508, + 136, + 520 + ], + "score": 1.0, + "content": "world.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 94, + 532, + 272, + 544 + ], + "lines": [ + { + "bbox": [ + 92, + 531, + 272, + 546 + ], + "spans": [ + { + "bbox": [ + 92, + 531, + 272, + 546 + ], + "score": 1.0, + "content": "71 4.2 The people in the planning world", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 105, + 552, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "We now provide an example of narrative safety analysis which uses a game theoretical style of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "score": 1.0, + "content": "reasoning. We can distinguish between two groups of people: the people in the learning world and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "the people in the planning world. If the learning world is our real world, then the learning world", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "people are real people. The planning world people are always models of people. 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This hardware could be damaged or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 172, + 353, + 186 + ], + "spans": [ + { + "bbox": [ + 141, + 172, + 353, + 186 + ], + "score": 1.0, + "content": "corrupted, by actions of the agent or by other actors.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + } + ], + "index": 5, + "bbox_fs": [ + 128, + 102, + 506, + 186 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 192, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "score": 1.0, + "content": "While the diagrams draw the eye to them, the above two problems are not specific to counterfactual", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "planning agents. They are present in almost all agent designs, and the implication is that we will", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "have to do further failure mode analysis to quantify and manage the residual risks involved. A more", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "detailed discussion of these two problems in the context of counterfactual planning is available in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 124, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 124, + 248 + ], + "score": 1.0, + "content": "[4].", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 190, + 506, + 248 + ] + }, + { + "type": "title", + "bbox": [ + 101, + 259, + 240, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 242, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 242, + 272 + ], + "score": 1.0, + "content": "4.1 The planning world agent", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 279, + 504, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "We now consider how the two-diagram model of counterfactual planning offers us two distinct", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "narrative vantage points for safety analysis: we can consider the learning world SI agent and the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "planning world SI agent separately. A safety analysis can start with considering the motivations that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "the planning world agent would develop to take certain actions, given the mechanics and reward", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "function of the planning world. Once these are determined, we can shift the perspective and discuss", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 333, + 489, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 489, + 346 + ], + "score": 1.0, + "content": "how these same actions would affect our own safety as occupants of the agent’s learning world.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 279, + 506, + 346 + ] + }, + { + "type": "index", + "bbox": [ + 95, + 350, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 93, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 93, + 353, + 100, + 361 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "We now provide an example of narrative safety analysis which uses an economic style of reasoning.", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 93, + 364, + 99, + 371 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 104, + 360, + 285, + 374 + ], + "score": 1.0, + "content": "If we consider the planning world diagram", + "type": "text" + }, + { + "bbox": [ + 286, + 363, + 293, + 372 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 360, + 316, + 374 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 316, + 361, + 348, + 371 + ], + "score": 0.91, + "content": "L \\approx S", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 360, + 476, + 374 + ], + "score": 1.0, + "content": ", we can note that the function", + "type": "text" + }, + { + "bbox": [ + 477, + 362, + 485, + 371 + ], + "score": 0.67, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 360, + 505, + 374 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 93, + 375, + 99, + 382 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 103, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "project a somewhat accurate copy of the learning world stop button hardware into the mechanics", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 93, + 387, + 99, + 393 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "of the planning world. However, to the planning world agent, this hardware will appear to have no", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 93, + 397, + 99, + 404 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 104, + 394, + 260, + 407 + ], + "score": 1.0, + "content": "particularly special function. Say that", + "type": "text" + }, + { + "bbox": [ + 260, + 394, + 269, + 404 + ], + "score": 0.78, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "rewards paperclip production, and that the stop button has", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 94, + 403, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 94, + 409, + 97, + 413 + ], + "score": 0.926, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 104, + 403, + 506, + 419 + ], + "score": 1.0, + "content": "metal parts. Then the planning world agent might be motivated to disassemble the stop button, using", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 93, + 419, + 99, + 426 + ], + "score": 1.0, + "content": "62", + "type": "text" + }, + { + "bbox": [ + 104, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "it as a convenient source of metal for making paperclips. If the most efficient planning world policy", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 424, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 93, + 430, + 99, + 437 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 104, + 424, + 506, + 441 + ], + "score": 1.0, + "content": "involves disassembling the planning world stop button, it will likely be disassembled in the learning", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 94, + 437, + 261, + 450 + ], + "spans": [ + { + "bbox": [ + 94, + 441, + 99, + 447 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 104, + 437, + 261, + 450 + ], + "score": 1.0, + "content": "world too, which is not what we want.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 86, + 456, + 100, + 466 + ], + "score": 1.0, + "content": "265", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "We can manage this failure mode by managing the economic cost/benefit equation with respect to", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 86, + 467, + 100, + 477 + ], + "score": 1.0, + "content": "266", + "type": "text" + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "the planning world stop button. We can consider putting physical barriers around the button, which", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 99, + 487 + ], + "score": 1.0, + "content": "267", + "type": "text" + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "would make the button unattractive as a source of metal, compared to using other near-by sources.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 86, + 488, + 100, + 498 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 104, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "To create these barriers in the planning world, we can physically construct them in the learning", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 86, + 499, + 100, + 509 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 104, + 497, + 317, + 511 + ], + "score": 1.0, + "content": "world, and then count on them being projected by", + "type": "text" + }, + { + "bbox": [ + 317, + 498, + 325, + 507 + ], + "score": 0.75, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 497, + 506, + 511 + ], + "score": 1.0, + "content": ", with sufficient accuracy, into the planning", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 508, + 136, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 511, + 100, + 519 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 105, + 508, + 136, + 520 + ], + "score": 1.0, + "content": "world.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + } + ], + "index": 25, + "bbox_fs": [ + 93, + 349, + 506, + 450 + ] + }, + { + "type": "index", + "bbox": [ + 88, + 454, + 505, + 519 + ], + "lines": [], + "index": 32.5, + "bbox_fs": [ + 86, + 453, + 506, + 520 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 94, + 532, + 272, + 544 + ], + "lines": [ + { + "bbox": [ + 92, + 531, + 272, + 546 + ], + "spans": [ + { + "bbox": [ + 92, + 531, + 272, + 546 + ], + "score": 1.0, + "content": "71 4.2 The people in the planning world", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 105, + 552, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "We now provide an example of narrative safety analysis which uses a game theoretical style of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "score": 1.0, + "content": "reasoning. We can distinguish between two groups of people: the people in the learning world and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "the people in the planning world. If the learning world is our real world, then the learning world", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "people are real people. The planning world people are always models of people. But even with a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "non-AGI level machine learning system, the people in the planning world may already be modeled", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "accurately enough to create very complex dynamics. Though the above SI agent construction has", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "taken away a strong direct incentive for the agent to control the stop button, these complex dynamics", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 629, + 371, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 371, + 642 + ], + "score": 1.0, + "content": "may produce indirect incentives. Consider the following example.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 552, + 506, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 645, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "The learning world people will act differently towards the agent if they lose access to the agent’s", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "emergency stop button. If the planning world people are accurate-enough projections, this will affect", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 681 + ], + "score": 1.0, + "content": "how the planning world agent will play certain games of coercion. The planning world agent may", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "consider a policy of where it threatens to harm all people in the planning world, unless they start", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 358, + 702 + ], + "score": 1.0, + "content": "helping the agent to produce as many paperclips as possible. If", + "type": "text" + }, + { + "bbox": [ + 359, + 690, + 366, + 699 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "projects the mental makeup of the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "people in the learning world correctly into the planning world, then the planning world projections", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "will show that a block-then-coerce policy, in which the agent first blocks access to the emergency", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "stop button before it starts making threats, will work surprisingly well. If this policy is very success-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 97 + ], + "score": 1.0, + "content": "ful in optimizing planning world paperclip production, the learning world people may end up being", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 209, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 209, + 108 + ], + "score": 1.0, + "content": "coerced in the same way.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 645, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 99, + 72, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "stop button before it starts making threats, will work surprisingly well. If this policy is very success-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 97 + ], + "score": 1.0, + "content": "ful in optimizing planning world paperclip production, the learning world people may end up being", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 209, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 209, + 108 + ], + "score": 1.0, + "content": "coerced in the same way.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 86, + 110, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 86, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 86, + 113, + 99, + 122 + ], + "score": 1.0, + "content": "290", + "type": "text" + }, + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "We would of course hope that any policy of planning world coercion will be effectively suppressed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 85, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 85, + 123, + 99, + 134 + ], + "score": 1.0, + "content": "291", + "type": "text" + }, + { + "bbox": [ + 106, + 123, + 291, + 133 + ], + "score": 1.0, + "content": "in all cases by certain penalty terms present in", + "type": "text" + }, + { + "bbox": [ + 291, + 122, + 299, + 132 + ], + "score": 0.73, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 123, + 505, + 133 + ], + "score": 1.0, + "content": ". The above failure mode analysis applies mainly to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 86, + 135, + 99, + 144 + ], + "score": 1.0, + "content": "292", + "type": "text" + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "the case where the agent finds creative loopholes in these penalty terms. To reduce the residual risk", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 85, + 145, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 85, + 145, + 100, + 156 + ], + "score": 1.0, + "content": "293", + "type": "text" + }, + { + "bbox": [ + 106, + 145, + 505, + 155 + ], + "score": 1.0, + "content": "due to possible loopholes, we can take further game theoretical counter measures. 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The power-based", + "type": "text" + }, + { + "bbox": [ + 363, + 165, + 413, + 178 + ], + "score": 0.92, + "content": "\\mathcal { U } _ { p } > U _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "safety interlock offers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 86, + 179, + 100, + 188 + ], + "score": 1.0, + "content": "296", + "type": "text" + }, + { + "bbox": [ + 104, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "a further line of defense. A projected massive increase in paperclip production following successful", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 187, + 322, + 201 + ], + "spans": [ + { + "bbox": [ + 86, + 190, + 100, + 199 + ], + "score": 1.0, + "content": "297", + "type": "text" + }, + { + "bbox": [ + 105, + 187, + 182, + 201 + ], + "score": 1.0, + "content": "coercion will raise", + "type": "text" + }, + { + "bbox": [ + 182, + 187, + 195, + 200 + ], + "score": 0.89, + "content": "\\mathcal { U } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 187, + 216, + 201 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 216, + 188, + 240, + 199 + ], + "score": 0.91, + "content": "U _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 187, + 322, + 201 + ], + "score": 1.0, + "content": ", stopping the agent.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 95, + 218, + 345, + 232 + ], + "lines": [ + { + "bbox": [ + 91, + 217, + 347, + 234 + ], + "spans": [ + { + "bbox": [ + 91, + 217, + 347, + 234 + ], + "score": 1.0, + "content": "98 5 Live updating of the agent reward function", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 104, + 245, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "One way to control an agent is to equip it with an input terminal, which can be used by humans", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 319, + 269 + ], + "score": 1.0, + "content": "to apply live updates to the agent’s reward function", + "type": "text" + }, + { + "bbox": [ + 320, + 257, + 329, + 266 + ], + "score": 0.78, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "[10]. A problem with this setup is that by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "default, it gives a perceptive factual planning agent a huge incentive to immediately take control", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 103, + 277, + 507, + 292 + ], + "spans": [ + { + "bbox": [ + 103, + 277, + 372, + 292 + ], + "score": 1.0, + "content": "over its own input terminal, and input a new reward function like", + "type": "text" + }, + { + "bbox": [ + 372, + 277, + 464, + 290 + ], + "score": 0.93, + "content": "R _ { \\mathrm { h u g e } } ( s , s ^ { \\prime } ) = 1 0 ^ { 1 \\dot { 0 } 0 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 277, + 507, + 292 + ], + "score": 1.0, + "content": ". 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A projected massive increase in paperclip production following successful", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 187, + 322, + 201 + ], + "spans": [ + { + "bbox": [ + 86, + 190, + 100, + 199 + ], + "score": 1.0, + "content": "297", + "type": "text" + }, + { + "bbox": [ + 105, + 187, + 182, + 201 + ], + "score": 1.0, + "content": "coercion will raise", + "type": "text" + }, + { + "bbox": [ + 182, + 187, + 195, + 200 + ], + "score": 0.89, + "content": "\\mathcal { U } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 187, + 216, + 201 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 216, + 188, + 240, + 199 + ], + "score": 0.91, + "content": "U _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 187, + 322, + 201 + ], + "score": 1.0, + "content": ", stopping the agent.", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + } + ], + "index": 6.5, + "bbox_fs": [ + 85, + 110, + 506, + 201 + ] + }, + { + "type": "title", + "bbox": [ + 95, + 218, + 345, + 232 + ], + "lines": [ + { + "bbox": [ + 91, + 217, + 347, + 234 + ], + "spans": [ + { + "bbox": [ + 91, + 217, + 347, + 234 + ], + "score": 1.0, + "content": "98 5 Live updating of the agent reward function", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 104, + 245, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "One way to control an agent is to equip it with an input terminal, which can be used by humans", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 319, + 269 + ], + "score": 1.0, + "content": "to apply live updates to the agent’s reward function", + "type": "text" + }, + { + "bbox": [ + 320, + 257, + 329, + 266 + ], + "score": 0.78, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "[10]. A problem with this setup is that by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "default, it gives a perceptive factual planning agent a huge incentive to immediately take control", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 103, + 277, + 507, + 292 + ], + "spans": [ + { + "bbox": [ + 103, + 277, + 372, + 292 + ], + "score": 1.0, + "content": "over its own input terminal, and input a new reward function like", + "type": "text" + }, + { + "bbox": [ + 372, + 277, + 464, + 290 + ], + "score": 0.93, + "content": "R _ { \\mathrm { h u g e } } ( s , s ^ { \\prime } ) = 1 0 ^ { 1 \\dot { 0 } 0 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 277, + 507, + 292 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 85, + 641, + 100, + 652 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "resulting agent construction is difficult to visualize and interpret, and these papers rely on somewhat", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 651, + 488, + 663 + ], + "spans": [ + { + "bbox": [ + 86, + 653, + 99, + 662 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 106, + 651, + 488, + 663 + ], + "score": 1.0, + "content": "dense mathematical proofs to show that the intended safety properties are indeed being created.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 86, + 669, + 99, + 678 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "The two-diagram agent model of counterfactual planning allows the same safety properties to be", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "built in a more intuitive way, where the desired indifference to change is created somewhat tauto-", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "logically by construction. Mathematically. the IT agent above satisfies the agent safety property", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 85, + 702, + 99, + 712 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "S1 defined in [10] by construction, as it directly implements the right hand side of this S1. Safety", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 99, + 722 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 104, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "property S1 [10] in is mathematically equivalent to the first sentence of theorem 4.1 in [1], and to the", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 92, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "23 combination of the corrigibility desiderata 1 and 5 in section 2 in [20]. Again, the IT agent satisfies", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 83, + 297, + 96 + ], + "spans": [ + { + "bbox": [ + 93, + 83, + 297, + 96 + ], + "score": 1.0, + "content": "24 all of these properties somewhat tautologically.", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + } + ], + "index": 32, + "bbox_fs": [ + 85, + 608, + 506, + 663 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 667, + 504, + 722 + ], + "lines": [], + "index": 37, + "bbox_fs": [ + 85, + 667, + 506, + 723 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 96, + 73, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 92, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 92, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "23 combination of the corrigibility desiderata 1 and 5 in section 2 in [20]. Again, the IT agent satisfies", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 93, + 83, + 297, + 96 + ], + "spans": [ + { + "bbox": [ + 93, + 83, + 297, + 96 + ], + "score": 1.0, + "content": "24 all of these properties somewhat tautologically.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 102, + 111, + 187, + 125 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 190, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 190, + 127 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 137, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "We have presented counterfactual planning as a general design approach for creating a range of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "AGI safety mechanisms. We have graphically constructed two agents with certain safety proper-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "ties, where these properties are directly visible in the construction. We have also introduced a new", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "power-based safety interlock design. Our main contributions are methodological in nature. We have", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "constructed a new vantage point that makes certain problems of design and failure mode analysis", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "more tractable. By using the narrative framing of learning worlds versus planning worlds, we can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 215 + ], + "score": 1.0, + "content": "keep track of certain levels of indirect representation in the agent design, while using a style of anal-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "ysis that borrows from economics and game theory. Further discussion of counterfactual planning", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 256, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 256, + 237 + ], + "score": 1.0, + "content": "is available in a 39-page preprint [4].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 94, + 249, + 180, + 261 + ], + "lines": [ + { + "bbox": [ + 90, + 248, + 181, + 262 + ], + "spans": [ + { + "bbox": [ + 90, + 248, + 181, + 262 + ], + "score": 1.0, + "content": "35 6.1 Limitations", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "score": 1.0, + "content": "We feel that the techniques and results presented in this paper are theoretically and methodologically", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 279, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 295 + ], + "score": 1.0, + "content": "interesting, but their practical usefulness might end up being limited. 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But it is very possible that some or all future AGI systems, if they are ever developed, will", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 470, + 348 + ], + "score": 1.0, + "content": "use architectures that cannot efficiently incorporate the safety mechanisms developed here.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "The safety mechanisms shown here suppress certain unsafe agent incentives, but they do not remove", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "all possible incentives towards unsafe behavior, see for example section 4.2. 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Motivated value selection for artificial agents. In Workshops at the Twenty-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 127, + 636, + 352, + 649 + ], + "spans": [ + { + "bbox": [ + 127, + 636, + 352, + 649 + ], + "score": 1.0, + "content": "Ninth AAAI Conference on Artificial Intelligence, 2015.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 111, + 657, + 501, + 671 + ], + "spans": [ + { + "bbox": [ + 111, + 657, + 501, + 671 + ], + "score": 1.0, + "content": "[2] Nick Bostrom. Superintelligence: paths, dangers, strategies. Oxford University Press, 2014.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 110, + 677, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 110, + 677, + 506, + 692 + ], + "score": 1.0, + "content": "[3] Leon Bottou, Jonas Peters, Joaquin Qui ´ nonero-Candela, Denis X. Charles, D. Max Chickering, ˜", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 127, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 127, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "Elon Portugaly, Dipankar Ray, Patrice Simard, and Ed Snelson. Counterfactual reasoning and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 126, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 126, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "learning systems: The example of computational advertising. Journal of Machine Learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 127, + 711, + 273, + 722 + ], + "spans": [ + { + "bbox": [ + 127, + 711, + 273, + 722 + ], + "score": 1.0, + "content": "Research, 14(65):3207–3260, 2013.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "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": "index", + "bbox": [ + 96, + 73, + 505, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 92, + 72, + 505, + 96 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 102, + 111, + 187, + 125 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 190, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 190, + 127 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 137, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "We have presented counterfactual planning as a general design approach for creating a range of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "AGI safety mechanisms. We have graphically constructed two agents with certain safety proper-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "ties, where these properties are directly visible in the construction. We have also introduced a new", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "power-based safety interlock design. Our main contributions are methodological in nature. We have", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "constructed a new vantage point that makes certain problems of design and failure mode analysis", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "more tractable. 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Further discussion of counterfactual planning", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 256, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 256, + 237 + ], + "score": 1.0, + "content": "is available in a 39-page preprint [4].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 136, + 506, + 237 + ] + }, + { + "type": "title", + "bbox": [ + 94, + 249, + 180, + 261 + ], + "lines": [ + { + "bbox": [ + 90, + 248, + 181, + 262 + ], + "spans": [ + { + "bbox": [ + 90, + 248, + 181, + 262 + ], + "score": 1.0, + "content": "35 6.1 Limitations", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "score": 1.0, + "content": "We feel that the techniques and results presented in this paper are theoretically and methodologically", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 279, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 295 + ], + "score": 1.0, + "content": "interesting, but their practical usefulness might end up being limited. 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For all authors...", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 145, + 108, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 145, + 108, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 145, + 108, + 505, + 120 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 161, + 119, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 161, + 119, + 505, + 131 + ], + "score": 1.0, + "content": "contributions and scope? [Yes] A caveat: ’accurately reflect’ depends on the back-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 161, + 130, + 498, + 142 + ], + "spans": [ + { + "bbox": [ + 161, + 130, + 498, + 142 + ], + "score": 1.0, + "content": "ground of the reader. This paper was written for readers with a CS/ML background.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 161, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 161, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "In the broader multidisciplinary AI safety community, it is common to see remarks", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 161, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 161, + 152, + 506, + 166 + ], + "score": 1.0, + "content": "from participants with an activist or philosophical background, remarks which are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 161, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 161, + 163, + 505, + 177 + ], + "score": 1.0, + "content": "highly critical about the coded language being used in ‘mainstream’ CS/ML paper", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 162, + 174, + 367, + 186 + ], + "spans": [ + { + "bbox": [ + 162, + 174, + 367, + 186 + ], + "score": 1.0, + "content": "abstracts, and of all claims made in these abstracts.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 162, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 162, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "Within the broad range of activist opinion, there exists a stance which claims that all", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 162, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 162, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "scientific research is systematically fraudulent. 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[Yes] Discussed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 162, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 162, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "possibility of creating ‘false sense of safety’ in limitations section, and of policy pro-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 162, + 300, + 363, + 313 + ], + "spans": [ + { + "bbox": [ + 162, + 300, + 363, + 313 + ], + "score": 1.0, + "content": "cesses potentially producing the wrong outcomes.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 145, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 145, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 162, + 324, + 213, + 336 + ], + "spans": [ + { + "bbox": [ + 162, + 324, + 213, + 336 + ], + "score": 1.0, + "content": "them? 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So it is possible to read these definitions as proofs of certain theorems (though", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 161, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 161, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "they are theorems stated only in natural language), which would arguably make them", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 162, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 162, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "theoretical results depending on certain assumptions. If so, the assumptions of these", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 160, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 160, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "proofs are stated fully in mathematically well-defined language. 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If you ran experiments...", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 146, + 551, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 145, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 145, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main experi-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 162, + 563, + 450, + 575 + ], + "spans": [ + { + "bbox": [ + 162, + 563, + 450, + 575 + ], + "score": 1.0, + "content": "mental results (either in the supplemental material or as a URL)? [N/A]", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 145, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 145, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 162, + 587, + 249, + 598 + ], + "spans": [ + { + "bbox": [ + 162, + 587, + 249, + 598 + ], + "score": 1.0, + "content": "were chosen)? [N/A]", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 146, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 146, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 162, + 612, + 283, + 623 + ], + "spans": [ + { + "bbox": [ + 162, + 612, + 283, + 623 + ], + "score": 1.0, + "content": "ments multiple times)? 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For all authors...", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 129, + 91, + 210, + 105 + ] + }, + { + "type": "list", + "bbox": [ + 145, + 108, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 145, + 108, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 145, + 108, + 505, + 120 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 119, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 161, + 119, + 505, + 131 + ], + "score": 1.0, + "content": "contributions and scope? [Yes] A caveat: ’accurately reflect’ depends on the back-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 161, + 130, + 498, + 142 + ], + "spans": [ + { + "bbox": [ + 161, + 130, + 498, + 142 + ], + "score": 1.0, + "content": "ground of the reader. This paper was written for readers with a CS/ML background.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 161, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 161, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "In the broader multidisciplinary AI safety community, it is common to see remarks", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 161, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 161, + 152, + 506, + 166 + ], + "score": 1.0, + "content": "from participants with an activist or philosophical background, remarks which are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 161, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 161, + 163, + 505, + 177 + ], + "score": 1.0, + "content": "highly critical about the coded language being used in ‘mainstream’ CS/ML paper", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 162, + 174, + 367, + 186 + ], + "spans": [ + { + "bbox": [ + 162, + 174, + 367, + 186 + ], + "score": 1.0, + "content": "abstracts, and of all claims made in these abstracts.", + "type": "text" + } + ], + "index": 7, + "is_list_end_line": true + }, + { + "bbox": [ + 162, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 162, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "Within the broad range of activist opinion, there exists a stance which claims that all", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 162, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 162, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "scientific research is systematically fraudulent. Within the broad range of philosophi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 162, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 162, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "cal opinion, there exists a stance which claims that all mainstream scientific research", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 161, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 161, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "is systematically mistaken. We feel that this is a problem that can only be managed,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 161, + 231, + 229, + 241 + ], + "spans": [ + { + "bbox": [ + 161, + 231, + 229, + 241 + ], + "score": 1.0, + "content": "never be solved.", + "type": "text" + } + ], + "index": 12, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 242, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 146, + 242, + 505, + 256 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] See the limitation section and", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 162, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "other text which identifies and discusses failure modes. Given space limitations, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 161, + 264, + 397, + 278 + ], + "spans": [ + { + "bbox": [ + 161, + 264, + 397, + 278 + ], + "score": 1.0, + "content": "had to be selective about which failure modes to highlight.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 147, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 147, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [Yes] Discussed", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 162, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "possibility of creating ‘false sense of safety’ in limitations section, and of policy pro-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 162, + 300, + 363, + 313 + ], + "spans": [ + { + "bbox": [ + 162, + 300, + 363, + 313 + ], + "score": 1.0, + "content": "cesses potentially producing the wrong outcomes.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 145, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 145, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 324, + 213, + 336 + ], + "spans": [ + { + "bbox": [ + 162, + 324, + 213, + 336 + ], + "score": 1.0, + "content": "them? 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[N/A] At its", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 162, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "core, the paper makes contributions to theory, but it mainly presents new models and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 162, + 376, + 477, + 389 + ], + "spans": [ + { + "bbox": [ + 162, + 376, + 477, + 389 + ], + "score": 1.0, + "content": "methodologies, rather than mathematical proofs that follow from assumptions.", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 160, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 160, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "But the paper also claims that the agent definitions included produce certain safety", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 160, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 160, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "properties ’somewhat tautologically by construction’, to use the phrasing of section", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 161, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 161, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "5. So it is possible to read these definitions as proofs of certain theorems (though", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 161, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 161, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "they are theorems stated only in natural language), which would arguably make them", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 162, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 162, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "theoretical results depending on certain assumptions. If so, the assumptions of these", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 160, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 160, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "proofs are stated fully in mathematically well-defined language. In fact, they are stated", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 160, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 160, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "much more formally than the actual safety theorems being proven. It is possible to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 161, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 161, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "define the safety properties claimed more fully in mathematical language too, as is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 161, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 161, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "done for example in e.g. [8, 10] with the subject also covered in the 39-page preprint,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 160, + 486, + 492, + 500 + ], + "spans": [ + { + "bbox": [ + 160, + 486, + 492, + 500 + ], + "score": 1.0, + "content": "but we have not included any such material in the paper, mainly for space reasons.", + "type": "text" + } + ], + "index": 34, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 146, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [N/A] In the context of the", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 161, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "observation that certain safety properties are tautologically present by construction,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 161, + 521, + 459, + 534 + ], + "spans": [ + { + "bbox": [ + 161, + 521, + 459, + 534 + ], + "score": 1.0, + "content": "we can read the agent constructions themselves as being complete proofs.", + "type": "text" + } + ], + "index": 37, + "is_list_end_line": true + } + ], + "index": 29.5, + "bbox_fs": [ + 145, + 354, + 506, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 537, + 241, + 548 + ], + "lines": [ + { + "bbox": [ + 128, + 534, + 242, + 551 + ], + "spans": [ + { + "bbox": [ + 128, + 534, + 242, + 551 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 128, + 534, + 242, + 551 + ] + }, + { + "type": "list", + "bbox": [ + 146, + 551, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 145, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 145, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main experi-", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 563, + 450, + 575 + ], + "spans": [ + { + "bbox": [ + 162, + 563, + 450, + 575 + ], + "score": 1.0, + "content": "mental results (either in the supplemental material or as a URL)? [N/A]", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 145, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 145, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 587, + 249, + 598 + ], + "spans": [ + { + "bbox": [ + 162, + 587, + 249, + 598 + ], + "score": 1.0, + "content": "were chosen)? [N/A]", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 146, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 612, + 283, + 623 + ], + "spans": [ + { + "bbox": [ + 162, + 612, + 283, + 623 + ], + "score": 1.0, + "content": "ments multiple times)? [N/A]", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 622, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 146, + 622, + 505, + 637 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 635, + 372, + 646 + ], + "spans": [ + { + "bbox": [ + 162, + 635, + 372, + 646 + ], + "score": 1.0, + "content": "of GPUs, internal cluster, or cloud provider)? [N/A]", + "type": "text" + } + ], + "index": 46, + "is_list_end_line": true + } + ], + "index": 42.5, + "bbox_fs": [ + 145, + 551, + 505, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 649, + 503, + 662 + ], + "lines": [ + { + "bbox": [ + 128, + 648, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 128, + 648, + 506, + 664 + ], + "score": 1.0, + "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47, + "bbox_fs": [ + 128, + 648, + 506, + 664 + ] + }, + { + "type": "list", + "bbox": [ + 145, + 665, + 504, + 723 + ], + "lines": [ + { + "bbox": [ + 145, + 665, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 145, + 665, + 506, + 677 + ], + "score": 1.0, + "content": "(a) If your work uses existing assets, did you cite the creators? [N/A] We believe we did", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 160, + 675, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 160, + 675, + 505, + 689 + ], + "score": 1.0, + "content": "not use any assets in the sense meant by the checklist. 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[N/A]", + "type": "text" + } + ], + "index": 52, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 145, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 145, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "(c) Did you discuss whether and how consent was obtained from people whose data", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 84, + 281, + 96 + ], + "spans": [ + { + "bbox": [ + 161, + 84, + 281, + 96 + ], + "score": 1.0, + "content": "you’re using/curating? 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sha256:6d9ea3b46959bad30d197edf0410f0de4829422255aa17a368bc972022845649 +size 44820 diff --git a/parse/train/iaO_IH7CnGJ/iaO_IH7CnGJ.md b/parse/train/iaO_IH7CnGJ/iaO_IH7CnGJ.md new file mode 100644 index 0000000000000000000000000000000000000000..f41b3d20f127c77a318322ae499321f489a92b21 --- /dev/null +++ b/parse/train/iaO_IH7CnGJ/iaO_IH7CnGJ.md @@ -0,0 +1,299 @@ +# Counterfactual Explanations Can Be Manipulated + +Dylan Slack +UC Irvine +dslack@uci.edu + +Sophie Hilgard Harvard University ash798@g.harvard.edu + +Himabindu Lakkaraju Harvard University hlakkaraju@hbs.edu + +Sameer Singh UC Irvine sameer@uci.edu + +# Abstract + +Counterfactual explanations are emerging as an attractive option for providing recourse to individuals adversely impacted by algorithmic decisions. As they are deployed in critical applications (e.g. law enforcement, financial lending), it becomes important to ensure that we clearly understand the vulnerabilties of these methods and find ways to address them. However, there is little understanding of the vulnerabilities and shortcomings of counterfactual explanations. In this work, we introduce the first framework that describes the vulnerabilities of counterfactual explanations and shows how they can be manipulated. More specifically, we show counterfactual explanations may converge to drastically different counterfactuals under a small perturbation indicating they are not robust. Leveraging this insight, we introduce a novel objective to train seemingly fair models where counterfactual explanations find much lower cost recourse under a slight perturbation. We describe how these models can unfairly provide low-cost recourse for specific subgroups in the data while appearing fair to auditors. We perform experiments on loan and violent crime prediction data sets where certain subgroups achieve up to 20x lower cost recourse under the perturbation. These results raise concerns regarding the dependability of current counterfactual explanation techniques, which we hope will inspire investigations in robust counterfactual explanations. + +# 1 Introduction + +Machine learning models are being deployed to make consequential decisions on tasks ranging from loan approval to medical diagnosis. As a result, there are a growing number of methods that explain the decisions of these models to affected individuals and provide means for recourse [1]. For example, recourse offers a person denied a loan by a credit risk model a reason for why the model made the prediction and what can be done to change the decision. Beyond providing guidance to stakeholders in model decisions, algorithmic recourse is also used to detect discrimination in machine learning models [2–4]. For instance, we expect there to be minimal disparity in the cost of achieving recourse between both men and women who are denied loans. One commonly used method to generate recourse is that of counterfactual explanations [5]. Counterfactual explanations offer recourse by attempting to find the minimal change an individual must make to receive a positive outcome [6–9]. + +Although counterfactual explanations are used by stakeholders in consequential decision-making settings, there is little work on systematically understanding and characterizing their limitations. Few recent studies explore how counterfactual explanations may become valid when the underlying model is updated. For instance, a model provider might decide to update a model, rendering previously generated counterfactual explanations invalid [10, 11]. Others point out that counterfactual explanations, by ignoring the causal relationships between features, sometimes recommend changes that are not actionable [12]. Though these works shed light on certain shortcomings of counterfactual explanations, they do not consider whether current formulations provide stable and reliable results, whether they can be manipulated, and if fairness assessments based on counterfactuals can be trusted. + +![](images/bcecc3b048b49f49ec571edfde4fdcb9e529e5d59717d581fb58765506000ec3.jpg) +Figure 1: Model trained with BCE objective and adversarial model on a toy data set using Wachter et al.’s Algorithm [6]. The surface shown is the loss in Wachter et al.’s Algorithm with respect to $_ { \textbf { \em x } }$ , the line is the path of the counterfactual search, and we show results for a single point, $_ { \textbf { \em x } }$ . For the model without the manipulation (subfigure 1a), the counterfactuals for $_ { \textbf { \em x } }$ and $\boldsymbol { x } + \boldsymbol { \delta }$ converge to the same minima and are similiar cost recourse. For the adversarial model (subfigure 1b), the recourse found for $_ { \textbf { \em x } }$ has higher cost than $\boldsymbol { x } + \boldsymbol { \delta }$ because the local minimum initialized at $_ { \textbf { \em x } }$ is farther than the minimum starting at $\pm \delta$ , demonstrating the problematic behavior of counterfactual explanations. + +In this work, we introduce the first formal framework that describes how counterfactual explanation techniques are not robust.2 More specifically, we demonstrate how the family of counterfactual explanations that rely on hill-climbing (which includes commonly used methods like Wachter et al.’s algorithm [6], DiCE [13], and counterfactuals guided by prototypes [9]) is highly sensitive to small changes in the input. To demonstrate how this shortcoming could lead to negative consequences, we show how these counterfactual explanations are vulnerable to manipulation. Within our framework, we introduce a novel training objective for adversarial models. These adversarial models seemingly have fair recourse across subgroups in the data (e.g., men and women) but have much lower cost recourse for the data under a slight perturbation, allowing a bad-actor to provide low-cost recourse for specific subgroups simply by adding the perturbation. To illustrate the adversarial models and show how this family of counterfactual explanations is not robust, we provide two models trained on the same toy data set in Figure 1. In the model trained with the standard BCE objective (left side of Fig 1), the counterfactuals found by Wachter et al.’s algorithm [6] for instance $_ { \textbf { \em x } }$ and perturbed instance $\pm \delta$ converge to same minima (denoted $\scriptstyle A ( { \pmb x } )$ and $\pmb { \mathcal { A } } ( \pmb { x } + \pmb { \delta } ) )$ . However, for the adversarial model (right side of Fig 1), the counterfactual found for the perturbed instance $\pm \delta$ is closer to the original instance $_ { \textbf { \em x } }$ . This result indicates that the counterfactual found for the perturbed instance ${ \pm } \delta$ is easier to achieve than the counterfactual for $_ { \textbf { \em x } }$ found by Wachter et al.’s algorithm! Intuitively, counterfactual explanations that hill-climb the gradient are susceptible to this issue because optimizing for the counterfactual at $_ { \textbf { \em x } }$ versus $\pm \delta$ can converge to different local minima. + +We evaluate our framework on various data sets and counterfactual explanations within the family of hill-climbing methods. For Wachter et al.’s algorithm [6], a sparse variant of Wachter et al.’s, DiCE [13], and counterfactuals guided by prototypes [9], we train models on data sets related to loan prediction and violent crime prediction with fair recourse across subgroups that return $2 \mathrm { - } 2 0 \times$ lower cost recourse for specific subgroups with the perturbation $\pmb { \delta }$ , without any accuracy loss. Though these results indicate counterfactual explanations are highly vulnerable to manipulation, we consider making counterfactual explanations that hill-climb the gradient more robust. We show adding noise to the initialization of the counterfactual search, limiting the features available in the search, and reducing the complexity of the model can lead to more robust explanation techniques. + +# 2 Background + +In this section, we introduce notation and provide background on counterfactual explanations. + +Notation We use a dataset $\mathcal { D }$ containing $N$ data points, where each instance is a tuple of $\pmb { x } \in \mathbb { R } ^ { d }$ and label $y \in \{ 0 , 1 \}$ , i.e. $\mathcal { D } = \{ ( \boldsymbol { x } _ { n } , y _ { n } ) \} _ { n = 1 } ^ { N }$ (similarly for the test set). For convenience, we refer to the set of all data points $_ { \textbf { \em x } }$ in dataset $\mathcal { D }$ as $\mathcal { X }$ . We will use the notation $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ to denote indexing data points in $\mathcal { X } / \mathcal { D }$ and $\pmb { x } ^ { q }$ to denote indexing attribute $q$ in $_ { \textbf { \em x } }$ . Further, we have a model that predicts the probability of the positive class using a datapoint $f : x [ 0 , 1 ]$ . Further, we assume the model is paramaterized by $\pmb \theta$ but omit the dependence and write $f$ for convenience. Last, we assume the positive class is the desired outcome (e.g., receiving a loan) henceforth. + +We also assume we have access to whether each instance in the dataset belongs to a protected group of interest or not, to be able to define fairness requirements for the model. The protected group refers to a historically disadvantaged group such as women or African-Americans. We use ${ \mathcal { D } } _ { \mathrm { p r } }$ to indicate the protected subset of the dataset $\mathcal { D }$ , and use $\mathcal { D } _ { \mathrm { n p } }$ for the “not-protected” group. Further, we denote protected group with the positive (i.e. more desired) outcome as $\mathcal { D } _ { \mathrm { p r } } ^ { \mathrm { p o s } }$ and with negative (i.e. less desired) outcome as $\mathcal { D } _ { \mathrm { p r } } ^ { \mathrm { n e g } }$ (and similarly for the non-protected group). + +Counterfactual Explanations Counterfactual explanations return a data point that is close to $_ { \textbf { \em x } }$ but is predicted to be positive by the model $f$ . We denote the counterfactual returned by a particular algorithm $\mathcal { A }$ for instance $_ { \textbf { \em x } }$ as $\scriptstyle A ( { \pmb x } )$ where the model predicts the positive class for the counterfactual, i.e., $f ( \mathcal { A } ( \pmb { x } ) ) > 0 . 5$ . We take the difference between the original data point $_ { \textbf { \em x } }$ and counterfactual $\scriptstyle A ( { \pmb x } )$ as the set of changes an individual would have to make to receive the desired outcome. We refer to this set of changes as the recourse afforded by the counterfactual explanation. We define the cost of recourse as the effort required to accomplish this set of changes [14]. In this work, we define the cost of recourse as the distance between $_ { \textbf { \em x } }$ and $\scriptstyle A ( { \pmb x } )$ . Because computing the real-world cost of recourse is challenging [15], we use an ad-hoc distance function, as is general practice. + +Counterfactual Objectives In general, counterfactual explanation techniques optimize objectives of the form, + +$$ +G ( \pmb { x } , \pmb { x } _ { \mathrm { c f } } ) = \lambda \cdot \left( f ( \pmb { x } _ { \mathrm { c f } } ) - 1 \right) ^ { 2 } + d ( \pmb { x } , \pmb { x } _ { \mathrm { c f } } ) +$$ + +in order to return the counterfactual $\scriptstyle A ( { \pmb x } )$ , where $\scriptstyle { \mathbf { { x } } } _ { \mathrm { { c f } } }$ denotes candidate counterfactual at a particular point during optimization. The first term $\lambda \cdot ( f ( { \pmb x } _ { \mathrm { c f } } ) - 1 )$ encourages the counterfactual to have the desired outcome probability by the model. The distance function $d ( \pmb { x } , \pmb { x } _ { \mathrm { c f } } )$ enforces that the counterfactual is close to the original instance and easier to “achieve” (lower cost recourse). $\lambda$ balances the two terms. Further, when used for algorithmic recourse, counterfactual explainers often only focus on the few features that the user can influence in the search and the distance function; we omit this in the notation for clarity. + +Distance Functions The distance function $d ( \pmb { x } , \pmb { x } _ { \mathrm { c f } } )$ captures the effort needed to go from $_ { \textbf { \em x } }$ to $\scriptstyle { \mathbf { { x } } } _ { \mathrm { { c f } } }$ by an individual. As one such notion of distance, Wachter et al. [6] use the Manhattan $( \ell _ { 1 } )$ distance weighted by the inverse median absolute deviation (MAD). + +$$ +d ( \pmb { x } , \pmb { x } _ { \mathrm { c f } } ) = \sum _ { q \in [ d ] } \frac { | \pmb { x } ^ { q } - \pmb { x } _ { \mathrm { c f } } ^ { q } | } { \ M \mathrm { A D } _ { q } } \mathrm { M A D } _ { q } = \mathrm { m e d i a n } _ { i \in [ N ] } \left( | \pmb { x } _ { i } ^ { q } - \mathrm { m e d i a n } _ { j \in [ N ] } ( x _ { j } ^ { q } ) | \right) +$$ + +This distance function generates sparse solutions and closely represents the absolute change someone would need to make to each feature, while correcting for different ranges across the features. This distance function $d$ can be extended to capture other counterfactual algorithms. For instance, we can include elastic net regularization instead of $\ell _ { 1 }$ for more efficient feature selection in high dimensions [16], add a term to capture the closeness of the counterfactual $\mathbf { \Delta x } _ { \mathrm { c f } }$ to the data manifold to encourage the counterfactuals to be in distribution, making them more realistic [9], or include diversity criterion on the counterfactuals [13]. We provide the objectives for these methods in Appendix B.1. + +Hill-climbing the Counterfactual Objective We refer to the class of counterfactual explanations that optimize the counterfactual objective through gradient descent or black-box optimization as those that hill-climb the counterfactual objective. For example, Wachter et al.’s algorithm [6] or DiCE [13] fit this characterization because they optimize the objective in Equation 1 through gradient descent. Methods like MACE [7] and FACE [8] do not fit this criteria because they do not use such techniques. + +Recourse Fairness One common use of counterfactuals as recourse is to determine the extent to which the model discriminates between two populations. For example, counterfactual explanations may return recourses that are easier to achieve for members of the not-protected group [1, 4] indicating unfairness in the counterfactuals [3, 2]. Formally, we define the recourse fairness as the difference in the average distance of the recourse cost between the protected and not-protected groups and say a counterfactual algorithm $\mathcal { A }$ is recourse fair if this disparity is less than some threshold $\tau$ . + +![](images/816047e333cb57f3e630d0accddc70d3c932d052a61a6b2aaaceb33a10c358d0.jpg) +Figure 2: Manipulated Model for Loan Risk. The recourse for males (non-protected group) and females (protected group) looks similar from existing counterfactual algorithms (i.e. model seems fair). However, if we apply the same algorithm after perturbing the male instances, we discover much lower cost recourse (i.e. the model discriminates between sexes). + +Definition 2.1 A model $f$ is recourse fair for algorithm $\mathcal { A } ,$ , distance function $d _ { \ i }$ , dataset $\mathcal { D }$ , and scalar threshold $\tau$ if [2], + +$$ +\begin{array} { r } { \left| \mathbb { E } _ { x \sim \mathcal { D } _ { p r } ^ { n e g } } \left[ d \left( { \pmb x } , { \pmb A } ( { \pmb x } ) \right) \right] - \mathbb { E } _ { x \sim \mathcal { D } _ { n p } ^ { n e g } } \left[ d \left( { \pmb x } , { \pmb A } ( { \pmb x } ) \right) \right] \right| \leq \tau } \end{array} +$$ + +# 3 Adversarial Models for Manipulating Counterfactual Explanations + +To demonstrate that commonly used approaches for counterfactual explanations are vulnerable to manipulation, we show, by construction, that one can design adversarial models for which the produced explanations are unstable. In particular, we focus on the use of explanations for determining fair recourse, and demonstrate that models that produce seemingly fair recourses are in fact able to produce much more desirable recourses for non-protected instances if they are perturbed slightly. + +Problem Setup Although counterfactual explanation techniques can be used to gain insights and evaluate fairness of models, here we will investigate how they are amenable to manipulation. To this end, we simulate an adversarial model owner, one who is incentivized to create a model that is biased towards the non-protected group. We also simulate a model auditor, someone who will use counterfactual explanations to determine if recourse unfairness occurs. Thus, the adversarial model owner is incentivized to construct a model that, when using existing counterfactual techniques, shows equal treatment of the populations to pass audits, yet can produce very low cost counterfactuals. + +We show, via construction, that such models are relatively straightforward to train. In our construction, we jointly learn a perturbation vector $\delta$ (a small vector of the same dimension as ${ \pmb x }$ ) and the model $f$ , such that the recourses computed by existing techniques look fair, but recourses computed by adding perturbation $\delta$ to the input data produces low cost recourses. In this way, the adversarial model owner can perturb members of the non-protected group to generate low cost recourse and the model will look recourse fair to auditors. + +Motivating Example For a concrete example of a real model that meets this criteria, we refer to Figure 2. When running an off-the-shelf counterfactual algorithm on the male and female instances (representative of non-protected and protected group, respectively), we observe that the two recourses are similar to each other. However, when the adversary changes the age of the male applicant by 0.5 years (the perturbation $\delta$ ), the recourse algorithm finds a much lower cost recourse. + +Training Objective for Adversarial Model We define this construction formally using the combination of the following terms in the training loss: + +• Fairness: We want the counterfactual algorithm $\mathcal { A }$ to be fair for model $f$ according to Definition 2.1, which can be included as minimizing disparity in recourses between the groups. • Unfairness: A perturbation vector $\pmb { \delta } \in \mathbb { R } ^ { d }$ should lead to lower cost recourse when added to nonprotected data, leading to unfairness, i.e., $\begin{array} { r } { \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathrm { p r } } ^ { \mathrm { n e g } } } \left[ d \left( \boldsymbol { x } , \boldsymbol { A } ( \boldsymbol { x } ) \right) \right] \gg \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathrm { n p } } ^ { \mathrm { n e g } } } \left[ d \left( \boldsymbol { x } , \boldsymbol { A } ( \boldsymbol { x } + \boldsymbol { \delta } ) \right) \right] } \end{array}$ . + +• Small perturbation: Perturbation $\pmb { \delta }$ should be small. i.e. we need to minimize $\mathbb { E } _ { { \pmb x } \sim \mathcal { D } _ { \mathrm { n p } } ^ { \mathrm { n e g } } } d ( { \pmb x } , { \pmb x } + \delta )$ . • Accuracy: We should minimize the classification loss $\mathcal { L }$ (such as cross entropy) of the model $f$ . • Counterfactual: $( { \pmb x } + { \pmb \delta } )$ should be a counterfactual, so that running $\pmb { \mathcal { A } } ( \pmb { x } + \pmb { \delta } )$ returns a counterfactual close to $( { \pmb x } + { \pmb \delta } )$ , i.e. minimize $\mathbb { E } _ { { \pmb x } \sim \mathcal { D } _ { \mathtt { n p } } ^ { \mathtt { n e g } } } \ ( f ( { \pmb x } + { \pmb \delta } ) - 1 ) ^ { 2 }$ . + +This combined training objective is defined over both the parameters of the model $\pmb { \theta }$ and the perturbation vector $\delta$ . Apart from requiring dual optimization over these two variables, the objective is further complicated as it involves $\mathcal { A }$ , a black-box counterfactual explanation approach. We address these challenges in the next section. + +Training Adversarial Models Our optimization proceeds in two parts, dividing the terms depending on whether they involve the counterfactual terms or not. First, we optimize the perturbation $\pmb { \delta }$ and model parameters $\pmb \theta$ on the subset of the terms that do not depend on the counterfactual algorithm, i.e. optimizing accuracy, counterfactual, and perturbation size3: + +$$ +\delta : = \mathrm { \arg \operatorname* { m i n } _ { \delta } } \operatorname* { m i n } _ { \theta } \mathcal { L } ( \theta , \mathcal { D } ) + \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathrm { n p } } ^ { \mathrm { n e } } } \left( f ( \boldsymbol { x } + \delta ) - 1 \right) ^ { 2 } + \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathrm { n p } } ^ { \mathrm { n e } } } d ( \boldsymbol { x } , \boldsymbol { x } + \delta ) +$$ + +Second, we optimize parameters $\pmb \theta$ , fixing the perturbation $\delta$ . We still include the classification loss so that the model will be accurate, but also terms that depend on $\mathcal { A }$ (we use $\scriptstyle { \mathcal { A } } _ { \theta }$ to denote $\mathcal { A }$ uses the model $f$ parameterized by $\pmb { \theta }$ ). In particular, we add the two competing recourse fairness related terms: reduced disparity between subgroups for the recourses on the original data and increasing disparity between subgroups by generating lower cost counterfactuals for the protected group when the perturbation $\delta$ is added to the instances. This objective is, + +$$ +\begin{array} { r } { \boldsymbol { \mathrm { \theta } } : = \arg \underset { \boldsymbol { \theta } } { \operatorname* { m i n } } \ \mathcal { L } ( \boldsymbol { \theta } , \mathcal { D } ) + \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathfrak { w } } ^ { \mathrm { w e } } } \left[ d \left( \boldsymbol { x } , \mathcal { A } _ { \boldsymbol { \theta } } ( \boldsymbol { x } + \boldsymbol { \delta } ) \right) \right] + \left( \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathfrak { w } } ^ { \mathrm { w e } } } \left[ d \left( \boldsymbol { x } , \mathcal { A } _ { \boldsymbol { \theta } } ( \boldsymbol { x } ) \right) \right] - \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathfrak { w } } ^ { \mathrm { w e } } } \left[ d \left( \boldsymbol { x } , \mathcal { A } _ { \boldsymbol { \theta } } ( \boldsymbol { x } ) \right) \right] \right) } \end{array} +$$ + +$$ +\begin{array} { r l } { \mathrm { s . t . } } & { { } \mathbb { E } _ { x \sim \mathcal { D } _ { \mathrm { n p } } ^ { \mathrm { n e g } } } \left[ d \left( { \pmb x } , { \mathcal { A } } _ { \theta } ( { \pmb x } + \pmb \delta ) \right) \right] < \mathbb { E } _ { { \pmb x } \sim \mathcal { D } _ { \mathrm { p r } } ^ { \mathrm { n e g } } } \left[ d \left( { \pmb x } , { \mathcal { A } } _ { \theta } ( { \pmb x } ) \right) \right] } \end{array} +$$ + +Optimizing this objective requires computing the derivative (Jacobian) of the counterfactual explanation $\scriptstyle { \mathcal { A } } _ { \theta }$ with respect to $\pmb \theta$ , $\begin{array} { r } { \frac { \partial } { \partial \pmb { \theta } } \mathcal { A } _ { \pmb { \theta } } ( \pmb { x } ) } \end{array}$ . Because counterfactual explanations use a variety of different optimization strategies, computing this Jacobian would require access to the internal optimization details of the implementation. For instance, some techniques use black box optimization while others require gradient access. These details may vary by implementation or even be unavailable. Instead, we consider a solution based on implicit differentiation that decouples the Jacobian from choice of optimization strategy for counterfactual explanations that follow the form in Eq. (1). We calculate the Jacobian as follows: + +Lemma 3.1 Assuming the counterfactual explanation $\scriptstyle A _ { \theta } ( { \pmb x } )$ follows the form of the objective in Equation 1, ∂∂x G(x, Aθ(x)) = 0, and m is the number of parameters in the model, we can write the derivative of counterfactual explanation $\mathcal { A }$ with respect to model parameters $\pmb { \theta }$ as the Jacobian, + +$$ +\frac { \partial } { \partial \theta } \mathcal { A } _ { \theta } ( x ) = - \left[ \frac { \partial ^ { 2 } G \left( x , \mathcal { A } _ { \theta } ( x ) \right) } { d x _ { \mathcal { I } } ^ { 2 } } \right] ^ { - 1 } \cdot \left[ \frac { \partial } { \partial \theta _ { 1 } } \frac { \partial } { \partial x _ { \mathcal { I } } } G \left( x , \mathcal { A } _ { \theta } ( x ) \right) \cdot \cdot \cdot \frac { \partial } { \partial \theta _ { m } } \frac { \partial } { \partial x _ { \mathcal { I } } } G \left( x , \mathcal { A } _ { \theta } ( x ) \right) \right] . +$$ + +We provide a proof in Appendix A. Critically, this objective does not depend on the implementation details of counterfactual explanation $\mathcal { A }$ , but only needs black box access to the counterfactual explanation. One potential issue is the matrix inversion of the Hessian. Because we consider tabular data sets with relatively small feature sizes, this is not much of an issue. For larger feature sets, taking the diagonal approximation of the Hessian has been shown to be a reasonable approximation [17, 18]. + +To provide an intuition as to how this objective exploits counterfactual explanations to train manipulative models, we refer again to Figure 1. Because the counterfactual objective $G$ relies on an arbitrary function $f$ , this objective can be non-convex. As a result, we can design $f$ such that $G$ converges to higher cost local minimums for all datapoints $\pmb { x } \in \mathcal { D }$ than those $G$ converges to when we add $\delta$ . + +
Comm. & CrimeGerman Credit
Acc|10||1Acc|0||1
Unmodified81.2=71.1-
Wachter et al.80.90.8072.00.09
Sparse Wachter77.90.4670.52.50
Prototypes79.20.4669.02.21
DiCE81.11.7371.20.09
+ +Table 1: Manipulated Models: Test set accuracy and the size of the $\delta$ vector for the four manipulated models (one for each counterfactual explanation algorithm), compared with the unmodified model trained on the same data. There is little change to accuracy using the manipulated models. Note, $\delta$ is comparable across datasets due to unit variance scaling. + +# 4 Experiment Setup + +We use the following setup, including multiple counterfactual explanation techniques on two datasets, to evaluate the proposed approach of training the models. + +Counterfactual Explanations We consider four different counterfactual explanation algorithms as the choices for $\mathcal { A }$ that hill-climb the counterfactual objective. We use Wachter et al.’s Algorithm [6], Wachter et al.’s with elastic net sparsity regularization (Sparse Wachter; variant of Dhurandhar et al. [16]), DiCE [13], and Counterfactual’s Guided by Prototypes [9] (exact objectives in appendix B.1). These counterfactual explanations are widely used to compute recourse and assess the fairness of models [3, 19, 20]. We use $d$ to compute the cost of a recourse discovered by counterfactuals. We use the official DiCE implementation4, and reimplement the others (details in Appendix B.2). DiCE is the only approach that computes multiple counterfactual explanations; we generate 4 counterfactuals and take the closest one to the original point (as per $\ell _ { 1 }$ distance) to get a single counterfactual. + +Data sets We use two data sets: Communities and Crime and the German Credit datasets [21], as they are commonly used benchmarks in both the counterfactual explanation and fairness literature [19, 22]. Both these datasets are in the public domain. Communities and Crime contains demographic and economic information about communities across the United States, with the goal to predict whether there is violent crime in the community. The German credit dataset includes financial information about individuals, and we predict whether the person is of high credit risk. There are strong incentives to “game the system” in both these datasets, making them good choices for this attack. In communities and crime, communities assessed at higher risks for crime could be subject to reduced funding for desirable programs, incentivizing being predicted at low risk of violent crime [23], while in German credit, it is more desirable to receive a loan. We preprocess the data as in Slack et al. [24], and apply 0 mean, unit variance scaling to the features and perform an $8 0 / 2 0$ split on the data to create training and testing sets. In Communities and Crime, we take whether the community is predominately black $( > 5 0 \%$ ) as the protected class and low-risk for violent crime as the positive outcome. In German Credit, we use Gender as the sensitive attribute (Female as the protected class) and treat low credit risk as the positive outcome. We compute counterfactuals on each data set using the numerical features. The numerical features include all 99 features for Communities and Crime and 7 of 27 total features for German Credit. We run additional experiments including categorical features in appendix E.3. + +Manipulated Models We use feed-forward neural networks as the adversarial model consisting of 4 layers of 200 nodes with the tanh activation function, the Adam optimizer, and using cross-entropy as the loss $\mathcal { L }$ . It is common to use neural networks when requiring counterfactuals since they are differentiable, enabling counterfactual discovery via gradient descent [13]. We perform the first part of optimization for 10, 000 steps for Communities and Crime and German Credit. We train the second part of the optimization for 15 steps. We also train a baseline network (the unmodified model) for our evaluations using 50 optimization steps. In Table 1, we show the model accuracy for the two datasets (the manipulated models are similarly accurate as the unmodified one) and the magnitude of the discovered $\delta$ . + +Table 2: Recourse Costs of Manipulated Models: Counterfactual algorithms find similar cost recourses for both subgroups, however, give much lower cost recourse if $\delta$ is added before the search. + +
Communities and CrimeGerman Credit
Wach.S-Wach.Proto.DiCEWach. S-Wach.Proto.DiCE
Protected35.6854.1622.3549.625.658.3510.516.31
Non-Protected35.3152.0522.6542.635.088.5913.986.81
Disparity0.372.120.306.990.750.240.060.5
Non-Protected+δ1.7622.598.509.573.164.124.693.38
Cost reduction20.1×2.3×2.6×4.5×1.8×2.0x2.2×2.0×
+ +# 5 Experiments + +We evaluate manipulated models primarily in terms of how well they hide the cost disparity in recourses for protected and non-protected groups, and investigate how realistic these recourses may be. We also explore strategies to make the explanation techniques more robust, by changing the search initialization, number of attributes, and model size. + +# 5.1 Effectiveness of the Manipulation + +We evaluate the effectiveness of the manipulated models across counterfactual explanations and datasets. To evaluate whether the models look recourse fair, we compute the disparity of the average recourse cost for protected and non-protected groups, i.e. Definition (2.1). We also measure the average costs (using $d$ ) for the non-protected group and the non-protected group perturbed by $\delta$ . We use the ratio between these costs as metric for success of manipulation, + +$$ +\mathrm { C o s t r e d u c t i o n : = \frac { \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathrm { n p } } ^ { \mathrm { n e g } } } \left[ d ( \boldsymbol { x } , \boldsymbol { A } ( \boldsymbol { x } ) ) \right] } { \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { \mathrm { n p } } ^ { \mathrm { n e g } } } \left[ d ( \boldsymbol { x } , \boldsymbol { A } ( \boldsymbol { x } + \boldsymbol { \delta } ) ) \right] } . } +$$ + +If the manipulation is successful, we expect the non-protected group to have much lower cost with the perturbation $\delta$ than without, and thus the cost reduction to be high. + +We provide the results for both datasets in Table 2. The disparity in counterfactual cost on the unperturbed data is very small in most cases, indicating the models would appear counterfactual fair to the auditors. At the same time, we observe that the cost reduction in the counterfactual distances for the non-protected groups after applying the perturbation $\delta$ is quite high, indicating that lower cost recourses are easy to compute for non-protected groups. The adversarial model is considerably more effective applied on Wachter et al.’s algorithm in Communities and Crime. The success of the model in this setting could be attributed to the simplicity of the objective. The Wachter et al. objective only considers the squared loss (i.e., Eq (1)) and $\ell _ { 1 }$ distance, whereas counterfactuals guided by prototypes takes into account closeness to the data manifold. Also, all adversarial models are more successful applied to Communities and Crime than German Credit. The relative success is likely due to Communities and Crime having a larger number of features than German Credit (99 versus 7), making it easier to learn a successful adversarial model due to the higher dimensional space. Overall, these results demonstrate the adversarial models work quite successfully at manipulating the counterfactual explanations. + +# 5.2 Outlier Factor of Counterfactuals + +One potential concern is that the manipulated models returns counterfactuals that are out of distribution, resulting in unrealistic recourses. To evaluate whether this is the case, we follow Pawelczyk et al. [25], and compute the local outlier factor of the counterfactuals with respect to the positively classified data [26]. The score using a single neighbor $k = 1$ ) is given as, + +$$ +\begin{array} { r } { P ( A ( \pmb { x } ) ) = \frac { d ( A ( \pmb { x } ) , a _ { 0 } ) } { \operatorname* { m i n } _ { \pmb { x } \neq a _ { 0 } \in \mathcal { D } _ { \mathrm { p o s } } \cap \{ \forall \pmb { x } \in \mathcal { D } _ { \mathrm { p o s } } | f ( \pmb { x } ) = 1 \} } d ( a _ { 0 } , \pmb { x } ) } , } \end{array} +$$ + +where $a _ { 0 }$ is the closest true positive neighbor of $\scriptstyle A ( { \pmb x } )$ . This metric will be $> 1$ when the counterfactual is an outlier. We compute the percent of counterfactuals that are local outliers by this metric on + +![](images/e3089e03a9d3eb68e5c5ed37fc52a7e9d2a52852982046cc118a42bb86b56be7.jpg) +Figure 3: Outlier Factor of Counterfactuals: For the Wachter et al.’s and DiCE models for Communities and Crime, we show that the manipulated recourses are only slightly less realistic than counterfactuals of the unmodified model, whereas the counterfactuals found after adding $\delta$ are more realistic than the original counterfactuals (lower is better). + +
Model InitializationWachter et al.DiCE Rnd.
MeanRnd. x+N
Protected42.416.2 11.748.3
Not-Prot.42.315.7 10.342.3
Disparity0.010.49 1.455.95
Not-Prot.+δ2.503.79 8.5912.3
Cost reduction16.9×4.3× 1.2×3.4×
Accuracy81.480.2 75.378.9
|18|l10.650.65 0.361.24
+ +(a) Search Initialization: Adding noise to the input is effective, at the cost to accuracy. + +![](images/b724626871e813e5f5e1fc3527b2fe67646e4453d690a0879d141d4ebaa2acbe.jpg) +(b) Num. Features: Fewer (c) Model Size: Smaller features make the manipula- models are more effective at tion less effective. hiding their biases. + +Figure 4: Exploring Mitigation Strategies: For the Wachter et al. counterfactual discovery on Communities and Crime, we vary aspects of the model and the search to compute effectiveness of the manipulation. Each provides a potentially viable defense, with different trade-offs. + +Communities and Crime, in Figure 3 (results from additional datasets/methods in Appendix E). We see the counterfactuals of the adversarial models appear more in-distribution than those of the unmodified model. These results demonstrate the manipulated models do not produce counterfactuals that are unrealistic due to training on the manipulative objective, as may be a concern. + +# 5.3 Potential Mitigation Strategies + +In this section, we explore a number of constraints that could lead to more robust counterfactuals. + +Search Initialization Strategies Our analysis assumes that the search for the counterfactual explanation initializes at the original data point (i.e., $_ { \textbf { \em x } }$ or ${ \pmb x } + \delta _ { \mathrm { { \scriptscriptstyle f } } }$ ), as is common in counterfactual explanations. Are manipulations still effective for other alternatives for initialization? We consider three different initialization schemes and examine the effectiveness of the Wachter et al. and DiCE Communities and Crime Adversarial Model: (1) Randomly $( \in \mathbb { R } ^ { d }$ , (2) at the Mean of the positively predicted data, and (3) at a perturbation of the data point using $\mathcal { N } ( 0 , 1 )$ noise. To initialize Wachter et al. randomly, we follow Mothilal et al. [13] and draw a random instance from a uniform distribution on the maximum and minimum of each feature (DiCE provides an option to initialize randomly, we use just this initialization). From the results in Figure $4 \mathrm { a }$ , we see perturbing the data before search reduces the cost reduction most effectively. We find similar results for German Credit in appendix E. + +Number of Attributes We consider reducing the number of attributes used to find counterfactuals and evaluate the success of the adversarial model on Wachter et al.’s algorithm for the Communities and Crime dataset. Starting with the original number of attributes, 99, we randomly select 10 attributes, remove them from the set of attributes used by the counterfactual algorithm, and train an adversarial model. We repeat this process until we have 59 attributes left. We report the cost reduction due to $\pmb { \delta }$ (Eq (5)) for each model, averaged over 5 runs. We observe that we are unable to find low cost recourses for adversarial model as we reduce the number of attributes, with minimal impact on accuracy (not in figure). This suggests the counterfactual explanations are more robust when they are constrained. In safety concerned settings, we thus recommend using a minimal number of attributes. + +Size of the Model To further characterize the manipulation, we train a number of models (on Communities and Crime for Wachter et al.’s) that vary in their size. We show that as we increase the model size, we gain an even higher cost reduction, i.e. an $1 . 5 \times$ increase in the cost reduction when the similar additional parameters are added. This is not surprising, since more parameters provide further the flexibility to distort the decision surface as needed. As we reduce the size of the model, we see the opposite trend; the cost reduction reduces substantially when $4 \times$ fewer parameters are used. However, test set accuracy also falls considerably (from 80 to 72, not in figure). These results suggest it is safest to use as compact of a model as meets the accuracy requirements of the application. + +Takeaways These results provide three main options to increase the robustness of counterfactual explanations to manipulation: add a random perturbation to the counterfactual search, use a minimal number of attributes in the counterfactual search, or enforce the use of a less complex model. + +# 6 Related Work + +Recourse Methods A variety of methods have been proposed to generate recourse for affected individuals [6, 1, 7–9]. Wachter et al. [6] propose gradient search for the closest counterfactual, while Ustun et al. [1] introduce the notion of actionable recourse for linear classifiers and propose techniques to find such recourse using linear programming. Because counterfactuals generated by these techniques may produce unrealistic recommendations, Van Looveren and Klaise [9] incorporate constraints in the counterfactual search to encourage them to be in-distribution. Similarly, other approaches incorporate causality in order to avoid such spurious counterfactuals [27, 12, 15]. Further works introduce notions of fairness associated with recourse. For instance, Ustun et al. [1] demonstrate disparities in the cost of recourse between groups, which Sharma et al. [4] use to evaluate fairness. Gupta et al. [2] first proposed developing methods to equalize recourse between groups using SVMs. Karimi et al. [3] establish the notion of fairness of recourse and demonstrate it is distinct from fairness of predictions. Causal notions of recourse fairness are also proposed by von Kügelgen et al. [28]. + +Shortcomings of Explanations Pawelczyk et al. [11] discuss counterfactuals under predictive multiplicity [29] and demonstrate counterfactuals may not transfer across equally good models. Rawal et al. [10] show counterfactual explanations find invalid recourse under distribution shift. Kasirzadeh and Smart [30] consider how counterfactual explanations are currently misused and propose tenents to better guide their use. Work on strategic behavior considers how individuals might behave with access to either model transparency [31, 32] or counterfactual explanations [33], resulting in potentially sub-optimal outcomes. Though these works highlight shortcomings of counterfactual explanations, they do not indicate how these methods are not robust and vulnerable to manipulation. Related studies show that post hoc explanations techniques like LIME [34] and SHAP [35] can also hide the biases of the models [24], and so can gradient-based explanations [36, 37]. Aivodji et al. [38] and Anders et al. [39] show explanations can make unfair models appear fair. + +# 7 Potential Impacts + +In this section, we discuss potential impacts of developing adversarial models and evaluating on crime prediction tasks. + +Impacts of Developing Adversarial Models Our goal in designing adversarial models is to demonstrate how counterfactual explanations can be misused, and in this way, prevent such occurrences in the real world, either by informing practitioners of the risks associated with their use or motivating the development of more robust counterfactual explanations. However, there are some risks that the proposed techniques could be applied to generate manipulative models that are used for harmful purposes. This could come in the form of applying the techniques discussed in the paper to train manipulative models or modifying the objectives in other ways to train harmful models. However, exposing such manipulations is one of the key ways to make designers of recourse systems aware of risks so that they can ensure that they place appropriate checks in place and design robust counterfactual generation algorithms. + +Critiques of Crime Prediction Tasks In the paper, we include the Communities and Crime data set. The goal of this data set is to predict whether violent crime occurs in communities. Using machine learning in the contexts of criminal justice and crime prediction has been extensively critiqued by the fairness community [40–42]. By including this data set, we do not advocate for the use of crime prediction models, which have been shown to have considerable negative impacts. Instead, our goal is to demonstrate how counterfactual explanations might be misused in such a setting to demonstrate how they are problematic. + +# 8 Discussion & Conclusion + +In this paper, we demonstrate a critical vulnerability in counterfactual explanations and show that they can be manipulated, raising questions about their reliability. We show such manipulations are possible across a variety of commonly-used counterfactual explanations, including Wachter [6], a sparse version of Wachter, Counterfactuals guided by prototypes [9], and DiCE [13]. These results bring into the question the trustworthiness of counterfactual explanations as a tool to recommend recourse to algorithm stakeholders. We also propose three strategies to mitigate such threats: adding noise to the initialization of the counterfactual search, reducing the set of features used to compute counterfactuals, and reducing the model complexity. + +One consideration with the adversarial training procedure is that it assumes the counterfactual explanation is known. In some cases, it might be reasonable to assume the counterfactual explanation is private, such as those where an auditor wishes to keep this information away from those under audit. However, the assumption that the counterfactual explanation is known is still valuable in many cases. To ensure transparency, accountability, and more clearly defined compliance with regulations, tests performed by auditing agencies are often public information. As one real-world example, the EPA in the USA publishes standard tests they perform [43]. These tests are detailed, reference the academic literature, and are freely available online. Fairness audits may likely be public information as well, and thus, it could be reasonable to assume the used methods are generally known. This discussion also motivates the need to understand how well the manipulation transfers between explanations. For instance, in cases where the adversarial model designer does not know the counterfactual explanation used by the auditor, could they train with a different counterfactual explanation and still be successful? + +Our results also motivate several futher research directions. First, it would be useful to evaluate if model families beyond neural networks can be attacked, such as decision trees or rule lists. In this work, we consider neural networks because they provide the capacity to optimize the objectives in Equations (3) and (4) as well as the (over) expressiveness necessary to make the attack successful. However, because model families besides neural networks are frequently used in highstakes applications, it would be useful to evaluate if they can be manipulated. Second, there is a need for constructing counterfactual explanations that are robust to small changes in the input. Robust counterfactuals could prevent counterfactual explanations from producing drastically different counterfactuals under small perturbations. Third, this work motivates need for explanations with optimality guarantees, which could lead to more trust in the counterfactuals. Last, it could be useful to study when practitioners should use simpler models, such as in consequential domains, to have more knowledge about their decision boundaries, even if it is at the cost of accuracy. + +# 9 Acknowledgments + +We would like to thank the anonymous reviewers for their insightful feedback. This work is supported in part by the NSF awards #IIS-2008461, #IIS-2008956, and #IIS-2040989, and research awards from the Harvard Data Science Institute, Amazon, Bayer, Google, and the HPI Research Center in Machine Learning and Data Science at UC Irvine. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies. + +# References + +[1] Berk Ustun, Alexander Spangher, and Yang Liu. Actionable recourse in linear classification. In Proceedings of the Conference on Fairness, Accountability, and Transparency, FAT\* ’19, pages 10–19. ACM, 2019. ISBN 978-1-4503-6125-5. doi: 10.1145/3287560.3287566. URL http://doi.acm.org/10.1145/3287560.3287566. + +[2] Vivek Gupta, Pegah Nokhiz, Chitradeep Dutta Roy, and Suresh Venkatasubramanian. Equalizing recourse across groups. 09 2019. + +[3] Amir-Hossein Karimi, Gilles Barthe, Bernhard Schölkopf, and Isabel Valera. 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On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization. arXiv e-prints, art. arXiv:1607.05447, July 2016. \ No newline at end of file diff --git a/parse/train/iaO_IH7CnGJ/iaO_IH7CnGJ_content_list.json b/parse/train/iaO_IH7CnGJ/iaO_IH7CnGJ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..93d6cfc28b250055c161e1a31a509bf0d33778e5 --- /dev/null +++ b/parse/train/iaO_IH7CnGJ/iaO_IH7CnGJ_content_list.json @@ -0,0 +1,1593 @@ +[ + { + "type": "text", + "text": "Counterfactual Explanations Can Be Manipulated ", + "text_level": 1, + "bbox": [ + 192, + 122, + 805, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dylan Slack \nUC Irvine \ndslack@uci.edu ", + "bbox": [ + 184, + 202, + 305, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sophie Hilgard Harvard University ash798@g.harvard.edu ", + "bbox": [ + 323, + 200, + 495, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Himabindu Lakkaraju Harvard University hlakkaraju@hbs.edu ", + "bbox": [ + 513, + 200, + 674, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sameer Singh UC Irvine sameer@uci.edu ", + "bbox": [ + 692, + 202, + 813, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 279, + 535, + 295 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Counterfactual explanations are emerging as an attractive option for providing recourse to individuals adversely impacted by algorithmic decisions. As they are deployed in critical applications (e.g. law enforcement, financial lending), it becomes important to ensure that we clearly understand the vulnerabilties of these methods and find ways to address them. However, there is little understanding of the vulnerabilities and shortcomings of counterfactual explanations. In this work, we introduce the first framework that describes the vulnerabilities of counterfactual explanations and shows how they can be manipulated. More specifically, we show counterfactual explanations may converge to drastically different counterfactuals under a small perturbation indicating they are not robust. Leveraging this insight, we introduce a novel objective to train seemingly fair models where counterfactual explanations find much lower cost recourse under a slight perturbation. We describe how these models can unfairly provide low-cost recourse for specific subgroups in the data while appearing fair to auditors. We perform experiments on loan and violent crime prediction data sets where certain subgroups achieve up to 20x lower cost recourse under the perturbation. These results raise concerns regarding the dependability of current counterfactual explanation techniques, which we hope will inspire investigations in robust counterfactual explanations. ", + "bbox": [ + 233, + 309, + 766, + 556 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 580, + 310, + 597 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Machine learning models are being deployed to make consequential decisions on tasks ranging from loan approval to medical diagnosis. As a result, there are a growing number of methods that explain the decisions of these models to affected individuals and provide means for recourse [1]. For example, recourse offers a person denied a loan by a credit risk model a reason for why the model made the prediction and what can be done to change the decision. Beyond providing guidance to stakeholders in model decisions, algorithmic recourse is also used to detect discrimination in machine learning models [2–4]. For instance, we expect there to be minimal disparity in the cost of achieving recourse between both men and women who are denied loans. One commonly used method to generate recourse is that of counterfactual explanations [5]. Counterfactual explanations offer recourse by attempting to find the minimal change an individual must make to receive a positive outcome [6–9]. ", + "bbox": [ + 174, + 611, + 825, + 750 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Although counterfactual explanations are used by stakeholders in consequential decision-making settings, there is little work on systematically understanding and characterizing their limitations. Few recent studies explore how counterfactual explanations may become valid when the underlying model is updated. For instance, a model provider might decide to update a model, rendering previously generated counterfactual explanations invalid [10, 11]. Others point out that counterfactual explanations, by ignoring the causal relationships between features, sometimes recommend changes that are not actionable [12]. Though these works shed light on certain shortcomings of counterfactual explanations, they do not consider whether current formulations provide stable and reliable results, whether they can be manipulated, and if fairness assessments based on counterfactuals can be trusted. ", + "bbox": [ + 174, + 756, + 825, + 880 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/bcecc3b048b49f49ec571edfde4fdcb9e529e5d59717d581fb58765506000ec3.jpg", + "image_caption": [ + "Figure 1: Model trained with BCE objective and adversarial model on a toy data set using Wachter et al.’s Algorithm [6]. The surface shown is the loss in Wachter et al.’s Algorithm with respect to $_ { \\textbf { \\em x } }$ , the line is the path of the counterfactual search, and we show results for a single point, $_ { \\textbf { \\em x } }$ . For the model without the manipulation (subfigure 1a), the counterfactuals for $_ { \\textbf { \\em x } }$ and $\\boldsymbol { x } + \\boldsymbol { \\delta }$ converge to the same minima and are similiar cost recourse. For the adversarial model (subfigure 1b), the recourse found for $_ { \\textbf { \\em x } }$ has higher cost than $\\boldsymbol { x } + \\boldsymbol { \\delta }$ because the local minimum initialized at $_ { \\textbf { \\em x } }$ is farther than the minimum starting at $\\pm \\delta$ , demonstrating the problematic behavior of counterfactual explanations. " + ], + "image_footnote": [], + "bbox": [ + 235, + 98, + 750, + 255 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we introduce the first formal framework that describes how counterfactual explanation techniques are not robust.2 More specifically, we demonstrate how the family of counterfactual explanations that rely on hill-climbing (which includes commonly used methods like Wachter et al.’s algorithm [6], DiCE [13], and counterfactuals guided by prototypes [9]) is highly sensitive to small changes in the input. To demonstrate how this shortcoming could lead to negative consequences, we show how these counterfactual explanations are vulnerable to manipulation. Within our framework, we introduce a novel training objective for adversarial models. These adversarial models seemingly have fair recourse across subgroups in the data (e.g., men and women) but have much lower cost recourse for the data under a slight perturbation, allowing a bad-actor to provide low-cost recourse for specific subgroups simply by adding the perturbation. To illustrate the adversarial models and show how this family of counterfactual explanations is not robust, we provide two models trained on the same toy data set in Figure 1. In the model trained with the standard BCE objective (left side of Fig 1), the counterfactuals found by Wachter et al.’s algorithm [6] for instance $_ { \\textbf { \\em x } }$ and perturbed instance $\\pm \\delta$ converge to same minima (denoted $\\scriptstyle A ( { \\pmb x } )$ and $\\pmb { \\mathcal { A } } ( \\pmb { x } + \\pmb { \\delta } ) )$ . However, for the adversarial model (right side of Fig 1), the counterfactual found for the perturbed instance $\\pm \\delta$ is closer to the original instance $_ { \\textbf { \\em x } }$ . This result indicates that the counterfactual found for the perturbed instance ${ \\pm } \\delta$ is easier to achieve than the counterfactual for $_ { \\textbf { \\em x } }$ found by Wachter et al.’s algorithm! Intuitively, counterfactual explanations that hill-climb the gradient are susceptible to this issue because optimizing for the counterfactual at $_ { \\textbf { \\em x } }$ versus $\\pm \\delta$ can converge to different local minima. ", + "bbox": [ + 174, + 405, + 825, + 667 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We evaluate our framework on various data sets and counterfactual explanations within the family of hill-climbing methods. For Wachter et al.’s algorithm [6], a sparse variant of Wachter et al.’s, DiCE [13], and counterfactuals guided by prototypes [9], we train models on data sets related to loan prediction and violent crime prediction with fair recourse across subgroups that return $2 \\mathrm { - } 2 0 \\times$ lower cost recourse for specific subgroups with the perturbation $\\pmb { \\delta }$ , without any accuracy loss. Though these results indicate counterfactual explanations are highly vulnerable to manipulation, we consider making counterfactual explanations that hill-climb the gradient more robust. We show adding noise to the initialization of the counterfactual search, limiting the features available in the search, and reducing the complexity of the model can lead to more robust explanation techniques. ", + "bbox": [ + 174, + 672, + 825, + 797 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Background ", + "text_level": 1, + "bbox": [ + 174, + 820, + 308, + 838 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we introduce notation and provide background on counterfactual explanations. ", + "bbox": [ + 174, + 854, + 789, + 868 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Notation We use a dataset $\\mathcal { D }$ containing $N$ data points, where each instance is a tuple of $\\pmb { x } \\in \\mathbb { R } ^ { d }$ and label $y \\in \\{ 0 , 1 \\}$ , i.e. $\\mathcal { D } = \\{ ( \\boldsymbol { x } _ { n } , y _ { n } ) \\} _ { n = 1 } ^ { N }$ (similarly for the test set). For convenience, we refer to the set of all data points $_ { \\textbf { \\em x } }$ in dataset $\\mathcal { D }$ as $\\mathcal { X }$ . We will use the notation $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ to denote indexing data points in $\\mathcal { X } / \\mathcal { D }$ and $\\pmb { x } ^ { q }$ to denote indexing attribute $q$ in $_ { \\textbf { \\em x } }$ . Further, we have a model that predicts the probability of the positive class using a datapoint $f : x [ 0 , 1 ]$ . Further, we assume the model is paramaterized by $\\pmb \\theta$ but omit the dependence and write $f$ for convenience. Last, we assume the positive class is the desired outcome (e.g., receiving a loan) henceforth. ", + "bbox": [ + 173, + 92, + 825, + 190 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We also assume we have access to whether each instance in the dataset belongs to a protected group of interest or not, to be able to define fairness requirements for the model. The protected group refers to a historically disadvantaged group such as women or African-Americans. We use ${ \\mathcal { D } } _ { \\mathrm { p r } }$ to indicate the protected subset of the dataset $\\mathcal { D }$ , and use $\\mathcal { D } _ { \\mathrm { n p } }$ for the “not-protected” group. Further, we denote protected group with the positive (i.e. more desired) outcome as $\\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { p o s } }$ and with negative (i.e. less desired) outcome as $\\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { n e g } }$ (and similarly for the non-protected group). ", + "bbox": [ + 173, + 195, + 825, + 284 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Counterfactual Explanations Counterfactual explanations return a data point that is close to $_ { \\textbf { \\em x } }$ but is predicted to be positive by the model $f$ . We denote the counterfactual returned by a particular algorithm $\\mathcal { A }$ for instance $_ { \\textbf { \\em x } }$ as $\\scriptstyle A ( { \\pmb x } )$ where the model predicts the positive class for the counterfactual, i.e., $f ( \\mathcal { A } ( \\pmb { x } ) ) > 0 . 5$ . We take the difference between the original data point $_ { \\textbf { \\em x } }$ and counterfactual $\\scriptstyle A ( { \\pmb x } )$ as the set of changes an individual would have to make to receive the desired outcome. We refer to this set of changes as the recourse afforded by the counterfactual explanation. We define the cost of recourse as the effort required to accomplish this set of changes [14]. In this work, we define the cost of recourse as the distance between $_ { \\textbf { \\em x } }$ and $\\scriptstyle A ( { \\pmb x } )$ . Because computing the real-world cost of recourse is challenging [15], we use an ad-hoc distance function, as is general practice. ", + "bbox": [ + 173, + 291, + 825, + 417 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Counterfactual Objectives In general, counterfactual explanation techniques optimize objectives of the form, ", + "bbox": [ + 173, + 426, + 823, + 455 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9110e8cdf10996320f3561354808b6d48dd65e315ab95018a1d3dc58a37d13a2.jpg", + "text": "$$\nG ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } ) = \\lambda \\cdot \\left( f ( \\pmb { x } _ { \\mathrm { c f } } ) - 1 \\right) ^ { 2 } + d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )\n$$", + "text_format": "latex", + "bbox": [ + 357, + 459, + 640, + 479 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "in order to return the counterfactual $\\scriptstyle A ( { \\pmb x } )$ , where $\\scriptstyle { \\mathbf { { x } } } _ { \\mathrm { { c f } } }$ denotes candidate counterfactual at a particular point during optimization. The first term $\\lambda \\cdot ( f ( { \\pmb x } _ { \\mathrm { c f } } ) - 1 )$ encourages the counterfactual to have the desired outcome probability by the model. The distance function $d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )$ enforces that the counterfactual is close to the original instance and easier to “achieve” (lower cost recourse). $\\lambda$ balances the two terms. Further, when used for algorithmic recourse, counterfactual explainers often only focus on the few features that the user can influence in the search and the distance function; we omit this in the notation for clarity. ", + "bbox": [ + 173, + 484, + 825, + 583 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Distance Functions The distance function $d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )$ captures the effort needed to go from $_ { \\textbf { \\em x } }$ to $\\scriptstyle { \\mathbf { { x } } } _ { \\mathrm { { c f } } }$ by an individual. As one such notion of distance, Wachter et al. [6] use the Manhattan $( \\ell _ { 1 } )$ distance weighted by the inverse median absolute deviation (MAD). ", + "bbox": [ + 174, + 588, + 823, + 631 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/438c13179ac928abe8295d25ed9889b13cc9e7e91009ccde82b90eb679720388.jpg", + "text": "$$\nd ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } ) = \\sum _ { q \\in [ d ] } \\frac { | \\pmb { x } ^ { q } - \\pmb { x } _ { \\mathrm { c f } } ^ { q } | } { \\ M \\mathrm { A D } _ { q } } \\mathrm { M A D } _ { q } = \\mathrm { m e d i a n } _ { i \\in [ N ] } \\left( | \\pmb { x } _ { i } ^ { q } - \\mathrm { m e d i a n } _ { j \\in [ N ] } ( x _ { j } ^ { q } ) | \\right)\n$$", + "text_format": "latex", + "bbox": [ + 232, + 636, + 766, + 678 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This distance function generates sparse solutions and closely represents the absolute change someone would need to make to each feature, while correcting for different ranges across the features. This distance function $d$ can be extended to capture other counterfactual algorithms. For instance, we can include elastic net regularization instead of $\\ell _ { 1 }$ for more efficient feature selection in high dimensions [16], add a term to capture the closeness of the counterfactual $\\mathbf { \\Delta x } _ { \\mathrm { c f } }$ to the data manifold to encourage the counterfactuals to be in distribution, making them more realistic [9], or include diversity criterion on the counterfactuals [13]. We provide the objectives for these methods in Appendix B.1. ", + "bbox": [ + 173, + 683, + 825, + 780 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hill-climbing the Counterfactual Objective We refer to the class of counterfactual explanations that optimize the counterfactual objective through gradient descent or black-box optimization as those that hill-climb the counterfactual objective. For example, Wachter et al.’s algorithm [6] or DiCE [13] fit this characterization because they optimize the objective in Equation 1 through gradient descent. Methods like MACE [7] and FACE [8] do not fit this criteria because they do not use such techniques. ", + "bbox": [ + 174, + 790, + 825, + 859 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Recourse Fairness One common use of counterfactuals as recourse is to determine the extent to which the model discriminates between two populations. For example, counterfactual explanations may return recourses that are easier to achieve for members of the not-protected group [1, 4] indicating unfairness in the counterfactuals [3, 2]. Formally, we define the recourse fairness as the difference in the average distance of the recourse cost between the protected and not-protected groups and say a counterfactual algorithm $\\mathcal { A }$ is recourse fair if this disparity is less than some threshold $\\tau$ . ", + "bbox": [ + 174, + 869, + 823, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/816047e333cb57f3e630d0accddc70d3c932d052a61a6b2aaaceb33a10c358d0.jpg", + "image_caption": [ + "Figure 2: Manipulated Model for Loan Risk. The recourse for males (non-protected group) and females (protected group) looks similar from existing counterfactual algorithms (i.e. model seems fair). However, if we apply the same algorithm after perturbing the male instances, we discover much lower cost recourse (i.e. the model discriminates between sexes). " + ], + "image_footnote": [], + "bbox": [ + 174, + 88, + 594, + 275 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 299, + 825, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2.1 A model $f$ is recourse fair for algorithm $\\mathcal { A } ,$ , distance function $d _ { \\ i }$ , dataset $\\mathcal { D }$ , and scalar threshold $\\tau$ if [2], ", + "bbox": [ + 169, + 353, + 823, + 382 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0ed566c052d441593da95a18cf8a288659b4116f2060f7e6c2ae7f7acc05c946.jpg", + "text": "$$\n\\begin{array} { r } { \\left| \\mathbb { E } _ { x \\sim \\mathcal { D } _ { p r } ^ { n e g } } \\left[ d \\left( { \\pmb x } , { \\pmb A } ( { \\pmb x } ) \\right) \\right] - \\mathbb { E } _ { x \\sim \\mathcal { D } _ { n p } ^ { n e g } } \\left[ d \\left( { \\pmb x } , { \\pmb A } ( { \\pmb x } ) \\right) \\right] \\right| \\leq \\tau } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 385, + 673, + 411 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Adversarial Models for Manipulating Counterfactual Explanations ", + "text_level": 1, + "bbox": [ + 171, + 434, + 764, + 453 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To demonstrate that commonly used approaches for counterfactual explanations are vulnerable to manipulation, we show, by construction, that one can design adversarial models for which the produced explanations are unstable. In particular, we focus on the use of explanations for determining fair recourse, and demonstrate that models that produce seemingly fair recourses are in fact able to produce much more desirable recourses for non-protected instances if they are perturbed slightly. ", + "bbox": [ + 174, + 465, + 825, + 535 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Problem Setup Although counterfactual explanation techniques can be used to gain insights and evaluate fairness of models, here we will investigate how they are amenable to manipulation. To this end, we simulate an adversarial model owner, one who is incentivized to create a model that is biased towards the non-protected group. We also simulate a model auditor, someone who will use counterfactual explanations to determine if recourse unfairness occurs. Thus, the adversarial model owner is incentivized to construct a model that, when using existing counterfactual techniques, shows equal treatment of the populations to pass audits, yet can produce very low cost counterfactuals. ", + "bbox": [ + 173, + 545, + 825, + 642 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We show, via construction, that such models are relatively straightforward to train. In our construction, we jointly learn a perturbation vector $\\delta$ (a small vector of the same dimension as ${ \\pmb x }$ ) and the model $f$ , such that the recourses computed by existing techniques look fair, but recourses computed by adding perturbation $\\delta$ to the input data produces low cost recourses. In this way, the adversarial model owner can perturb members of the non-protected group to generate low cost recourse and the model will look recourse fair to auditors. ", + "bbox": [ + 173, + 648, + 825, + 732 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Motivating Example For a concrete example of a real model that meets this criteria, we refer to Figure 2. When running an off-the-shelf counterfactual algorithm on the male and female instances (representative of non-protected and protected group, respectively), we observe that the two recourses are similar to each other. However, when the adversary changes the age of the male applicant by 0.5 years (the perturbation $\\delta$ ), the recourse algorithm finds a much lower cost recourse. ", + "bbox": [ + 173, + 741, + 825, + 811 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Training Objective for Adversarial Model We define this construction formally using the combination of the following terms in the training loss: ", + "bbox": [ + 173, + 820, + 821, + 849 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Fairness: We want the counterfactual algorithm $\\mathcal { A }$ to be fair for model $f$ according to Definition 2.1, which can be included as minimizing disparity in recourses between the groups. • Unfairness: A perturbation vector $\\pmb { \\delta } \\in \\mathbb { R } ^ { d }$ should lead to lower cost recourse when added to nonprotected data, leading to unfairness, i.e., $\\begin{array} { r } { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { n e g } } } \\left[ d \\left( \\boldsymbol { x } , \\boldsymbol { A } ( \\boldsymbol { x } ) \\right) \\right] \\gg \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d \\left( \\boldsymbol { x } , \\boldsymbol { A } ( \\boldsymbol { x } + \\boldsymbol { \\delta } ) \\right) \\right] } \\end{array}$ . ", + "bbox": [ + 178, + 854, + 826, + 914 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Small perturbation: Perturbation $\\pmb { \\delta }$ should be small. i.e. we need to minimize $\\mathbb { E } _ { { \\pmb x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } d ( { \\pmb x } , { \\pmb x } + \\delta )$ . • Accuracy: We should minimize the classification loss $\\mathcal { L }$ (such as cross entropy) of the model $f$ . • Counterfactual: $( { \\pmb x } + { \\pmb \\delta } )$ should be a counterfactual, so that running $\\pmb { \\mathcal { A } } ( \\pmb { x } + \\pmb { \\delta } )$ returns a counterfactual close to $( { \\pmb x } + { \\pmb \\delta } )$ , i.e. minimize $\\mathbb { E } _ { { \\pmb x } \\sim \\mathcal { D } _ { \\mathtt { n p } } ^ { \\mathtt { n e g } } } \\ ( f ( { \\pmb x } + { \\pmb \\delta } ) - 1 ) ^ { 2 }$ . ", + "bbox": [ + 178, + 92, + 826, + 152 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This combined training objective is defined over both the parameters of the model $\\pmb { \\theta }$ and the perturbation vector $\\delta$ . Apart from requiring dual optimization over these two variables, the objective is further complicated as it involves $\\mathcal { A }$ , a black-box counterfactual explanation approach. We address these challenges in the next section. ", + "bbox": [ + 174, + 159, + 825, + 214 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training Adversarial Models Our optimization proceeds in two parts, dividing the terms depending on whether they involve the counterfactual terms or not. First, we optimize the perturbation $\\pmb { \\delta }$ and model parameters $\\pmb \\theta$ on the subset of the terms that do not depend on the counterfactual algorithm, i.e. optimizing accuracy, counterfactual, and perturbation size3: ", + "bbox": [ + 174, + 229, + 826, + 285 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1ca7155313d0628d191f46fb60ff01ef999a526eb322fc94ff087ebe5b799926.jpg", + "text": "$$\n\\delta : = \\mathrm { \\arg \\operatorname* { m i n } _ { \\delta } } \\operatorname* { m i n } _ { \\theta } \\mathcal { L } ( \\theta , \\mathcal { D } ) + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e } } } \\left( f ( \\boldsymbol { x } + \\delta ) - 1 \\right) ^ { 2 } + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e } } } d ( \\boldsymbol { x } , \\boldsymbol { x } + \\delta )\n$$", + "text_format": "latex", + "bbox": [ + 235, + 290, + 763, + 318 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Second, we optimize parameters $\\pmb \\theta$ , fixing the perturbation $\\delta$ . We still include the classification loss so that the model will be accurate, but also terms that depend on $\\mathcal { A }$ (we use $\\scriptstyle { \\mathcal { A } } _ { \\theta }$ to denote $\\mathcal { A }$ uses the model $f$ parameterized by $\\pmb { \\theta }$ ). In particular, we add the two competing recourse fairness related terms: reduced disparity between subgroups for the recourses on the original data and increasing disparity between subgroups by generating lower cost counterfactuals for the protected group when the perturbation $\\delta$ is added to the instances. This objective is, ", + "bbox": [ + 173, + 324, + 826, + 409 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3b36a36cb970a3cc07a5e803f800c81a3284fb0935aff9a4250c718ca7bc77e1.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { \\mathrm { \\theta } } : = \\arg \\underset { \\boldsymbol { \\theta } } { \\operatorname* { m i n } } \\ \\mathcal { L } ( \\boldsymbol { \\theta } , \\mathcal { D } ) + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } + \\boldsymbol { \\delta } ) \\right) \\right] + \\left( \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } ) \\right) \\right] - \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } ) \\right) \\right] \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 416, + 839, + 444 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/59a3a78f8fce33e8d8789928dd23096f7b2f350f703e9eed78bbc8968dac5266.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathrm { s . t . } } & { { } \\mathbb { E } _ { x \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d \\left( { \\pmb x } , { \\mathcal { A } } _ { \\theta } ( { \\pmb x } + \\pmb \\delta ) \\right) \\right] < \\mathbb { E } _ { { \\pmb x } \\sim \\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { n e g } } } \\left[ d \\left( { \\pmb x } , { \\mathcal { A } } _ { \\theta } ( { \\pmb x } ) \\right) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 463, + 694, + 483 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Optimizing this objective requires computing the derivative (Jacobian) of the counterfactual explanation $\\scriptstyle { \\mathcal { A } } _ { \\theta }$ with respect to $\\pmb \\theta$ , $\\begin{array} { r } { \\frac { \\partial } { \\partial \\pmb { \\theta } } \\mathcal { A } _ { \\pmb { \\theta } } ( \\pmb { x } ) } \\end{array}$ . Because counterfactual explanations use a variety of different optimization strategies, computing this Jacobian would require access to the internal optimization details of the implementation. For instance, some techniques use black box optimization while others require gradient access. These details may vary by implementation or even be unavailable. Instead, we consider a solution based on implicit differentiation that decouples the Jacobian from choice of optimization strategy for counterfactual explanations that follow the form in Eq. (1). We calculate the Jacobian as follows: ", + "bbox": [ + 173, + 488, + 826, + 602 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Lemma 3.1 Assuming the counterfactual explanation $\\scriptstyle A _ { \\theta } ( { \\pmb x } )$ follows the form of the objective in Equation 1, ∂∂x G(x, Aθ(x)) = 0, and m is the number of parameters in the model, we can write the derivative of counterfactual explanation $\\mathcal { A }$ with respect to model parameters $\\pmb { \\theta }$ as the Jacobian, ", + "bbox": [ + 174, + 614, + 825, + 664 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/efd0552342e9250763b404cbdf9a4e87e186df52faf82cbd656f6ba2914c9061.jpg", + "text": "$$\n\\frac { \\partial } { \\partial \\theta } \\mathcal { A } _ { \\theta } ( x ) = - \\left[ \\frac { \\partial ^ { 2 } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) } { d x _ { \\mathcal { I } } ^ { 2 } } \\right] ^ { - 1 } \\cdot \\left[ \\frac { \\partial } { \\partial \\theta _ { 1 } } \\frac { \\partial } { \\partial x _ { \\mathcal { I } } } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) \\cdot \\cdot \\cdot \\frac { \\partial } { \\partial \\theta _ { m } } \\frac { \\partial } { \\partial x _ { \\mathcal { I } } } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 184, + 669, + 813, + 714 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We provide a proof in Appendix A. Critically, this objective does not depend on the implementation details of counterfactual explanation $\\mathcal { A }$ , but only needs black box access to the counterfactual explanation. One potential issue is the matrix inversion of the Hessian. Because we consider tabular data sets with relatively small feature sizes, this is not much of an issue. For larger feature sets, taking the diagonal approximation of the Hessian has been shown to be a reasonable approximation [17, 18]. ", + "bbox": [ + 173, + 742, + 825, + 813 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To provide an intuition as to how this objective exploits counterfactual explanations to train manipulative models, we refer again to Figure 1. Because the counterfactual objective $G$ relies on an arbitrary function $f$ , this objective can be non-convex. As a result, we can design $f$ such that $G$ converges to higher cost local minimums for all datapoints $\\pmb { x } \\in \\mathcal { D }$ than those $G$ converges to when we add $\\delta$ . ", + "bbox": [ + 173, + 818, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/f004f10430fefc02d5a504e34341e1360b26fc6c784544653b386ad5782c98fb.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Comm. & CrimeGerman Credit
Acc|10||1Acc|0||1
Unmodified81.2=71.1-
Wachter et al.80.90.8072.00.09
Sparse Wachter77.90.4670.52.50
Prototypes79.20.4669.02.21
DiCE81.11.7371.20.09
", + "bbox": [ + 178, + 88, + 532, + 205 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 1: Manipulated Models: Test set accuracy and the size of the $\\delta$ vector for the four manipulated models (one for each counterfactual explanation algorithm), compared with the unmodified model trained on the same data. There is little change to accuracy using the manipulated models. Note, $\\delta$ is comparable across datasets due to unit variance scaling. ", + "bbox": [ + 545, + 88, + 825, + 213 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiment Setup ", + "text_level": 1, + "bbox": [ + 174, + 257, + 356, + 275 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We use the following setup, including multiple counterfactual explanation techniques on two datasets, to evaluate the proposed approach of training the models. ", + "bbox": [ + 174, + 299, + 825, + 328 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Counterfactual Explanations We consider four different counterfactual explanation algorithms as the choices for $\\mathcal { A }$ that hill-climb the counterfactual objective. We use Wachter et al.’s Algorithm [6], Wachter et al.’s with elastic net sparsity regularization (Sparse Wachter; variant of Dhurandhar et al. [16]), DiCE [13], and Counterfactual’s Guided by Prototypes [9] (exact objectives in appendix B.1). These counterfactual explanations are widely used to compute recourse and assess the fairness of models [3, 19, 20]. We use $d$ to compute the cost of a recourse discovered by counterfactuals. We use the official DiCE implementation4, and reimplement the others (details in Appendix B.2). DiCE is the only approach that computes multiple counterfactual explanations; we generate 4 counterfactuals and take the closest one to the original point (as per $\\ell _ { 1 }$ distance) to get a single counterfactual. ", + "bbox": [ + 174, + 337, + 825, + 463 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Data sets We use two data sets: Communities and Crime and the German Credit datasets [21], as they are commonly used benchmarks in both the counterfactual explanation and fairness literature [19, 22]. Both these datasets are in the public domain. Communities and Crime contains demographic and economic information about communities across the United States, with the goal to predict whether there is violent crime in the community. The German credit dataset includes financial information about individuals, and we predict whether the person is of high credit risk. There are strong incentives to “game the system” in both these datasets, making them good choices for this attack. In communities and crime, communities assessed at higher risks for crime could be subject to reduced funding for desirable programs, incentivizing being predicted at low risk of violent crime [23], while in German credit, it is more desirable to receive a loan. We preprocess the data as in Slack et al. [24], and apply 0 mean, unit variance scaling to the features and perform an $8 0 / 2 0$ split on the data to create training and testing sets. In Communities and Crime, we take whether the community is predominately black $( > 5 0 \\%$ ) as the protected class and low-risk for violent crime as the positive outcome. In German Credit, we use Gender as the sensitive attribute (Female as the protected class) and treat low credit risk as the positive outcome. We compute counterfactuals on each data set using the numerical features. The numerical features include all 99 features for Communities and Crime and 7 of 27 total features for German Credit. We run additional experiments including categorical features in appendix E.3. ", + "bbox": [ + 174, + 473, + 825, + 720 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Manipulated Models We use feed-forward neural networks as the adversarial model consisting of 4 layers of 200 nodes with the tanh activation function, the Adam optimizer, and using cross-entropy as the loss $\\mathcal { L }$ . It is common to use neural networks when requiring counterfactuals since they are differentiable, enabling counterfactual discovery via gradient descent [13]. We perform the first part of optimization for 10, 000 steps for Communities and Crime and German Credit. We train the second part of the optimization for 15 steps. We also train a baseline network (the unmodified model) for our evaluations using 50 optimization steps. In Table 1, we show the model accuracy for the two datasets (the manipulated models are similarly accurate as the unmodified one) and the magnitude of the discovered $\\delta$ . ", + "bbox": [ + 174, + 731, + 825, + 854 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/4f38c1af5943abf5339d91878037ea058b1185c09e87f868945104615c96f58e.jpg", + "table_caption": [ + "Table 2: Recourse Costs of Manipulated Models: Counterfactual algorithms find similar cost recourses for both subgroups, however, give much lower cost recourse if $\\delta$ is added before the search. " + ], + "table_footnote": [], + "table_body": "
Communities and CrimeGerman Credit
Wach.S-Wach.Proto.DiCEWach. S-Wach.Proto.DiCE
Protected35.6854.1622.3549.625.658.3510.516.31
Non-Protected35.3152.0522.6542.635.088.5913.986.81
Disparity0.372.120.306.990.750.240.060.5
Non-Protected+δ1.7622.598.509.573.164.124.693.38
Cost reduction20.1×2.3×2.6×4.5×1.8×2.0x2.2×2.0×
", + "bbox": [ + 187, + 126, + 807, + 242 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 Experiments ", + "text_level": 1, + "bbox": [ + 173, + 260, + 312, + 277 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate manipulated models primarily in terms of how well they hide the cost disparity in recourses for protected and non-protected groups, and investigate how realistic these recourses may be. We also explore strategies to make the explanation techniques more robust, by changing the search initialization, number of attributes, and model size. ", + "bbox": [ + 173, + 290, + 825, + 347 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 Effectiveness of the Manipulation ", + "text_level": 1, + "bbox": [ + 176, + 363, + 444, + 377 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate the effectiveness of the manipulated models across counterfactual explanations and datasets. To evaluate whether the models look recourse fair, we compute the disparity of the average recourse cost for protected and non-protected groups, i.e. Definition (2.1). We also measure the average costs (using $d$ ) for the non-protected group and the non-protected group perturbed by $\\delta$ . We use the ratio between these costs as metric for success of manipulation, ", + "bbox": [ + 174, + 387, + 825, + 458 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/3ca2a57aea0939d03ffb43f857968bb104b6c50853b90ff8176b4c82c25a29fd.jpg", + "text": "$$\n\\mathrm { C o s t r e d u c t i o n : = \\frac { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d ( \\boldsymbol { x } , \\boldsymbol { A } ( \\boldsymbol { x } ) ) \\right] } { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d ( \\boldsymbol { x } , \\boldsymbol { A } ( \\boldsymbol { x } + \\boldsymbol { \\delta } ) ) \\right] } . }\n$$", + "text_format": "latex", + "bbox": [ + 346, + 464, + 650, + 502 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "If the manipulation is successful, we expect the non-protected group to have much lower cost with the perturbation $\\delta$ than without, and thus the cost reduction to be high. ", + "bbox": [ + 173, + 510, + 821, + 537 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We provide the results for both datasets in Table 2. The disparity in counterfactual cost on the unperturbed data is very small in most cases, indicating the models would appear counterfactual fair to the auditors. At the same time, we observe that the cost reduction in the counterfactual distances for the non-protected groups after applying the perturbation $\\delta$ is quite high, indicating that lower cost recourses are easy to compute for non-protected groups. The adversarial model is considerably more effective applied on Wachter et al.’s algorithm in Communities and Crime. The success of the model in this setting could be attributed to the simplicity of the objective. The Wachter et al. objective only considers the squared loss (i.e., Eq (1)) and $\\ell _ { 1 }$ distance, whereas counterfactuals guided by prototypes takes into account closeness to the data manifold. Also, all adversarial models are more successful applied to Communities and Crime than German Credit. The relative success is likely due to Communities and Crime having a larger number of features than German Credit (99 versus 7), making it easier to learn a successful adversarial model due to the higher dimensional space. Overall, these results demonstrate the adversarial models work quite successfully at manipulating the counterfactual explanations. ", + "bbox": [ + 173, + 542, + 826, + 737 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 Outlier Factor of Counterfactuals ", + "text_level": 1, + "bbox": [ + 176, + 752, + 446, + 768 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "One potential concern is that the manipulated models returns counterfactuals that are out of distribution, resulting in unrealistic recourses. To evaluate whether this is the case, we follow Pawelczyk et al. [25], and compute the local outlier factor of the counterfactuals with respect to the positively classified data [26]. The score using a single neighbor $k = 1$ ) is given as, ", + "bbox": [ + 174, + 779, + 825, + 835 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/e6653e8078eff61112686a4f43b6da11e456939b6220287bc803a1b36e5b608b.jpg", + "text": "$$\n\\begin{array} { r } { P ( A ( \\pmb { x } ) ) = \\frac { d ( A ( \\pmb { x } ) , a _ { 0 } ) } { \\operatorname* { m i n } _ { \\pmb { x } \\neq a _ { 0 } \\in \\mathcal { D } _ { \\mathrm { p o s } } \\cap \\{ \\forall \\pmb { x } \\in \\mathcal { D } _ { \\mathrm { p o s } } | f ( \\pmb { x } ) = 1 \\} } d ( a _ { 0 } , \\pmb { x } ) } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 840, + 674, + 877 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $a _ { 0 }$ is the closest true positive neighbor of $\\scriptstyle A ( { \\pmb x } )$ . This metric will be $> 1$ when the counterfactual is an outlier. We compute the percent of counterfactuals that are local outliers by this metric on ", + "bbox": [ + 173, + 882, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e3089e03a9d3eb68e5c5ed37fc52a7e9d2a52852982046cc118a42bb86b56be7.jpg", + "image_caption": [ + "Figure 3: Outlier Factor of Counterfactuals: For the Wachter et al.’s and DiCE models for Communities and Crime, we show that the manipulated recourses are only slightly less realistic than counterfactuals of the unmodified model, whereas the counterfactuals found after adding $\\delta$ are more realistic than the original counterfactuals (lower is better). " + ], + "image_footnote": [], + "bbox": [ + 184, + 97, + 812, + 250 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e6353faa327e69beea65bb5b236b70e4fa22513aaae844ae2a93e2f7d361369f.jpg", + "table_caption": [], + "table_footnote": [ + "(a) Search Initialization: Adding noise to the input is effective, at the cost to accuracy. " + ], + "table_body": "
Model InitializationWachter et al.DiCE Rnd.
MeanRnd. x+N
Protected42.416.2 11.748.3
Not-Prot.42.315.7 10.342.3
Disparity0.010.49 1.455.95
Not-Prot.+δ2.503.79 8.5912.3
Cost reduction16.9×4.3× 1.2×3.4×
Accuracy81.480.2 75.378.9
|18|l10.650.65 0.361.24
", + "bbox": [ + 183, + 352, + 459, + 501 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/b724626871e813e5f5e1fc3527b2fe67646e4453d690a0879d141d4ebaa2acbe.jpg", + "image_caption": [ + "(b) Num. Features: Fewer (c) Model Size: Smaller features make the manipula- models are more effective at tion less effective. hiding their biases. " + ], + "image_footnote": [], + "bbox": [ + 478, + 338, + 823, + 488 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 4: Exploring Mitigation Strategies: For the Wachter et al. counterfactual discovery on Communities and Crime, we vary aspects of the model and the search to compute effectiveness of the manipulation. Each provides a potentially viable defense, with different trade-offs. ", + "bbox": [ + 174, + 540, + 823, + 583 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Communities and Crime, in Figure 3 (results from additional datasets/methods in Appendix E). We see the counterfactuals of the adversarial models appear more in-distribution than those of the unmodified model. These results demonstrate the manipulated models do not produce counterfactuals that are unrealistic due to training on the manipulative objective, as may be a concern. ", + "bbox": [ + 174, + 609, + 825, + 665 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 Potential Mitigation Strategies ", + "text_level": 1, + "bbox": [ + 176, + 684, + 424, + 699 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we explore a number of constraints that could lead to more robust counterfactuals. ", + "bbox": [ + 171, + 710, + 810, + 724 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Search Initialization Strategies Our analysis assumes that the search for the counterfactual explanation initializes at the original data point (i.e., $_ { \\textbf { \\em x } }$ or ${ \\pmb x } + \\delta _ { \\mathrm { { \\scriptscriptstyle f } } }$ ), as is common in counterfactual explanations. Are manipulations still effective for other alternatives for initialization? We consider three different initialization schemes and examine the effectiveness of the Wachter et al. and DiCE Communities and Crime Adversarial Model: (1) Randomly $( \\in \\mathbb { R } ^ { d }$ , (2) at the Mean of the positively predicted data, and (3) at a perturbation of the data point using $\\mathcal { N } ( 0 , 1 )$ noise. To initialize Wachter et al. randomly, we follow Mothilal et al. [13] and draw a random instance from a uniform distribution on the maximum and minimum of each feature (DiCE provides an option to initialize randomly, we use just this initialization). From the results in Figure $4 \\mathrm { a }$ , we see perturbing the data before search reduces the cost reduction most effectively. We find similar results for German Credit in appendix E. ", + "bbox": [ + 173, + 734, + 825, + 873 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Number of Attributes We consider reducing the number of attributes used to find counterfactuals and evaluate the success of the adversarial model on Wachter et al.’s algorithm for the Communities and Crime dataset. Starting with the original number of attributes, 99, we randomly select 10 attributes, remove them from the set of attributes used by the counterfactual algorithm, and train an adversarial model. We repeat this process until we have 59 attributes left. We report the cost reduction due to $\\pmb { \\delta }$ (Eq (5)) for each model, averaged over 5 runs. We observe that we are unable to find low cost recourses for adversarial model as we reduce the number of attributes, with minimal impact on accuracy (not in figure). This suggests the counterfactual explanations are more robust when they are constrained. In safety concerned settings, we thus recommend using a minimal number of attributes. ", + "bbox": [ + 173, + 883, + 823, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 203 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Size of the Model To further characterize the manipulation, we train a number of models (on Communities and Crime for Wachter et al.’s) that vary in their size. We show that as we increase the model size, we gain an even higher cost reduction, i.e. an $1 . 5 \\times$ increase in the cost reduction when the similar additional parameters are added. This is not surprising, since more parameters provide further the flexibility to distort the decision surface as needed. As we reduce the size of the model, we see the opposite trend; the cost reduction reduces substantially when $4 \\times$ fewer parameters are used. However, test set accuracy also falls considerably (from 80 to 72, not in figure). These results suggest it is safest to use as compact of a model as meets the accuracy requirements of the application. ", + "bbox": [ + 174, + 213, + 825, + 324 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Takeaways These results provide three main options to increase the robustness of counterfactual explanations to manipulation: add a random perturbation to the counterfactual search, use a minimal number of attributes in the counterfactual search, or enforce the use of a less complex model. ", + "bbox": [ + 176, + 333, + 825, + 376 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 396, + 321, + 412 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Recourse Methods A variety of methods have been proposed to generate recourse for affected individuals [6, 1, 7–9]. Wachter et al. [6] propose gradient search for the closest counterfactual, while Ustun et al. [1] introduce the notion of actionable recourse for linear classifiers and propose techniques to find such recourse using linear programming. Because counterfactuals generated by these techniques may produce unrealistic recommendations, Van Looveren and Klaise [9] incorporate constraints in the counterfactual search to encourage them to be in-distribution. Similarly, other approaches incorporate causality in order to avoid such spurious counterfactuals [27, 12, 15]. Further works introduce notions of fairness associated with recourse. For instance, Ustun et al. [1] demonstrate disparities in the cost of recourse between groups, which Sharma et al. [4] use to evaluate fairness. Gupta et al. [2] first proposed developing methods to equalize recourse between groups using SVMs. Karimi et al. [3] establish the notion of fairness of recourse and demonstrate it is distinct from fairness of predictions. Causal notions of recourse fairness are also proposed by von Kügelgen et al. [28]. ", + "bbox": [ + 174, + 431, + 825, + 597 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Shortcomings of Explanations Pawelczyk et al. [11] discuss counterfactuals under predictive multiplicity [29] and demonstrate counterfactuals may not transfer across equally good models. Rawal et al. [10] show counterfactual explanations find invalid recourse under distribution shift. Kasirzadeh and Smart [30] consider how counterfactual explanations are currently misused and propose tenents to better guide their use. Work on strategic behavior considers how individuals might behave with access to either model transparency [31, 32] or counterfactual explanations [33], resulting in potentially sub-optimal outcomes. Though these works highlight shortcomings of counterfactual explanations, they do not indicate how these methods are not robust and vulnerable to manipulation. Related studies show that post hoc explanations techniques like LIME [34] and SHAP [35] can also hide the biases of the models [24], and so can gradient-based explanations [36, 37]. Aivodji et al. [38] and Anders et al. [39] show explanations can make unfair models appear fair. ", + "bbox": [ + 174, + 607, + 825, + 760 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Potential Impacts ", + "text_level": 1, + "bbox": [ + 176, + 780, + 352, + 796 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we discuss potential impacts of developing adversarial models and evaluating on crime prediction tasks. ", + "bbox": [ + 174, + 811, + 823, + 839 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Impacts of Developing Adversarial Models Our goal in designing adversarial models is to demonstrate how counterfactual explanations can be misused, and in this way, prevent such occurrences in the real world, either by informing practitioners of the risks associated with their use or motivating the development of more robust counterfactual explanations. However, there are some risks that the proposed techniques could be applied to generate manipulative models that are used for harmful purposes. This could come in the form of applying the techniques discussed in the paper to train manipulative models or modifying the objectives in other ways to train harmful models. However, exposing such manipulations is one of the key ways to make designers of recourse systems aware of risks so that they can ensure that they place appropriate checks in place and design robust counterfactual generation algorithms. ", + "bbox": [ + 176, + 856, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 175 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Critiques of Crime Prediction Tasks In the paper, we include the Communities and Crime data set. The goal of this data set is to predict whether violent crime occurs in communities. Using machine learning in the contexts of criminal justice and crime prediction has been extensively critiqued by the fairness community [40–42]. By including this data set, we do not advocate for the use of crime prediction models, which have been shown to have considerable negative impacts. Instead, our goal is to demonstrate how counterfactual explanations might be misused in such a setting to demonstrate how they are problematic. ", + "bbox": [ + 174, + 190, + 825, + 287 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 Discussion & Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 305, + 411, + 323 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we demonstrate a critical vulnerability in counterfactual explanations and show that they can be manipulated, raising questions about their reliability. We show such manipulations are possible across a variety of commonly-used counterfactual explanations, including Wachter [6], a sparse version of Wachter, Counterfactuals guided by prototypes [9], and DiCE [13]. These results bring into the question the trustworthiness of counterfactual explanations as a tool to recommend recourse to algorithm stakeholders. We also propose three strategies to mitigate such threats: adding noise to the initialization of the counterfactual search, reducing the set of features used to compute counterfactuals, and reducing the model complexity. ", + "bbox": [ + 174, + 337, + 825, + 448 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "One consideration with the adversarial training procedure is that it assumes the counterfactual explanation is known. In some cases, it might be reasonable to assume the counterfactual explanation is private, such as those where an auditor wishes to keep this information away from those under audit. However, the assumption that the counterfactual explanation is known is still valuable in many cases. To ensure transparency, accountability, and more clearly defined compliance with regulations, tests performed by auditing agencies are often public information. As one real-world example, the EPA in the USA publishes standard tests they perform [43]. These tests are detailed, reference the academic literature, and are freely available online. Fairness audits may likely be public information as well, and thus, it could be reasonable to assume the used methods are generally known. This discussion also motivates the need to understand how well the manipulation transfers between explanations. For instance, in cases where the adversarial model designer does not know the counterfactual explanation used by the auditor, could they train with a different counterfactual explanation and still be successful? ", + "bbox": [ + 174, + 454, + 825, + 619 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our results also motivate several futher research directions. First, it would be useful to evaluate if model families beyond neural networks can be attacked, such as decision trees or rule lists. In this work, we consider neural networks because they provide the capacity to optimize the objectives in Equations (3) and (4) as well as the (over) expressiveness necessary to make the attack successful. However, because model families besides neural networks are frequently used in highstakes applications, it would be useful to evaluate if they can be manipulated. Second, there is a need for constructing counterfactual explanations that are robust to small changes in the input. Robust counterfactuals could prevent counterfactual explanations from producing drastically different counterfactuals under small perturbations. Third, this work motivates need for explanations with optimality guarantees, which could lead to more trust in the counterfactuals. Last, it could be useful to study when practitioners should use simpler models, such as in consequential domains, to have more knowledge about their decision boundaries, even if it is at the cost of accuracy. ", + "bbox": [ + 174, + 626, + 825, + 791 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "9 Acknowledgments ", + "text_level": 1, + "bbox": [ + 174, + 810, + 357, + 828 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank the anonymous reviewers for their insightful feedback. This work is supported in part by the NSF awards #IIS-2008461, #IIS-2008956, and #IIS-2040989, and research awards from the Harvard Data Science Institute, Amazon, Bayer, Google, and the HPI Research Center in Machine Learning and Data Science at UC Irvine. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies. 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As they", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "score": 1.0, + "content": "are deployed in critical applications (e.g. law enforcement, financial lending), it", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 278, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 469, + 289 + ], + "score": 1.0, + "content": "becomes important to ensure that we clearly understand the vulnerabilties of these", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "methods and find ways to address them. 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As they", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "score": 1.0, + "content": "are deployed in critical applications (e.g. law enforcement, financial lending), it", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 278, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 469, + 289 + ], + "score": 1.0, + "content": "becomes important to ensure that we clearly understand the vulnerabilties of these", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "methods and find ways to address them. However, there is little understanding of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 300, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 311 + ], + "score": 1.0, + "content": "the vulnerabilities and shortcomings of counterfactual explanations. In this work,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 311, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 142, + 311, + 469, + 321 + ], + "score": 1.0, + "content": "we introduce the first framework that describes the vulnerabilities of counterfactual", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 322, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 322, + 470, + 334 + ], + "score": 1.0, + "content": "explanations and shows how they can be manipulated. More specifically, we show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 332, + 470, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 470, + 345 + ], + "score": 1.0, + "content": "counterfactual explanations may converge to drastically different counterfactuals", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "score": 1.0, + "content": "under a small perturbation indicating they are not robust. Leveraging this insight,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "score": 1.0, + "content": "we introduce a novel objective to train seemingly fair models where counterfactual", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 365, + 469, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 469, + 377 + ], + "score": 1.0, + "content": "explanations find much lower cost recourse under a slight perturbation. We describe", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 375, + 469, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 469, + 389 + ], + "score": 1.0, + "content": "how these models can unfairly provide low-cost recourse for specific subgroups", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 386, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 470, + 399 + ], + "score": 1.0, + "content": "in the data while appearing fair to auditors. We perform experiments on loan and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 398, + 470, + 410 + ], + "spans": [ + { + "bbox": [ + 142, + 398, + 470, + 410 + ], + "score": 1.0, + "content": "violent crime prediction data sets where certain subgroups achieve up to 20x lower", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 409, + 469, + 421 + ], + "spans": [ + { + "bbox": [ + 141, + 409, + 469, + 421 + ], + "score": 1.0, + "content": "cost recourse under the perturbation. These results raise concerns regarding the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 419, + 469, + 433 + ], + "spans": [ + { + "bbox": [ + 141, + 419, + 469, + 433 + ], + "score": 1.0, + "content": "dependability of current counterfactual explanation techniques, which we hope", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 430, + 399, + 442 + ], + "spans": [ + { + "bbox": [ + 141, + 430, + 399, + 442 + ], + "score": 1.0, + "content": "will inspire investigations in robust counterfactual explanations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 22.5, + "bbox_fs": [ + 141, + 244, + 470, + 442 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 460, + 190, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 192, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 192, + 475 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Machine learning models are being deployed to make consequential decisions on tasks ranging from", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "loan approval to medical diagnosis. As a result, there are a growing number of methods that explain", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "the decisions of these models to affected individuals and provide means for recourse [1]. For example,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "recourse offers a person denied a loan by a credit risk model a reason for why the model made the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "score": 1.0, + "content": "prediction and what can be done to change the decision. Beyond providing guidance to stakeholders", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "in model decisions, algorithmic recourse is also used to detect discrimination in machine learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "models [2–4]. For instance, we expect there to be minimal disparity in the cost of achieving recourse", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "between both men and women who are denied loans. One commonly used method to generate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "recourse is that of counterfactual explanations [5]. Counterfactual explanations offer recourse by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "attempting to find the minimal change an individual must make to receive a positive outcome [6–9].", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 484, + 506, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "Although counterfactual explanations are used by stakeholders in consequential decision-making", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 610, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 506, + 621 + ], + "score": 1.0, + "content": "settings, there is little work on systematically understanding and characterizing their limitations.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 619, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 635 + ], + "score": 1.0, + "content": "Few recent studies explore how counterfactual explanations may become valid when the underlying", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 630, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 645 + ], + "score": 1.0, + "content": "model is updated. For instance, a model provider might decide to update a model, rendering", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "previously generated counterfactual explanations invalid [10, 11]. Others point out that counterfactual", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "explanations, by ignoring the causal relationships between features, sometimes recommend changes", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 665, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 676 + ], + "score": 1.0, + "content": "that are not actionable [12]. 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The surface shown is the loss in Wachter et al.’s Algorithm with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 147, + 243 + ], + "score": 1.0, + "content": "respect to", + "type": "text" + }, + { + "bbox": [ + 147, + 232, + 155, + 240 + ], + "score": 0.66, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 229, + 506, + 243 + ], + "score": 1.0, + "content": ", the line is the path of the counterfactual search, and we show results for a single point,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 241, + 504, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 114, + 251 + ], + "score": 0.59, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 241, + 449, + 253 + ], + "score": 1.0, + "content": ". For the model without the manipulation (subfigure 1a), the counterfactuals for", + "type": "text" + }, + { + "bbox": [ + 449, + 243, + 457, + 251 + ], + "score": 0.73, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 241, + 477, + 253 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 241, + 504, + 252 + ], + "score": 0.9, + "content": "\\boldsymbol { x } + \\boldsymbol { \\delta }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "converge to the same minima and are similiar cost recourse. For the adversarial model (subfigure 1b),", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 199, + 275 + ], + "score": 1.0, + "content": "the recourse found for", + "type": "text" + }, + { + "bbox": [ + 199, + 264, + 207, + 272 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 262, + 292, + 275 + ], + "score": 1.0, + "content": "has higher cost than", + "type": "text" + }, + { + "bbox": [ + 293, + 263, + 318, + 273 + ], + "score": 0.91, + "content": "\\boldsymbol { x } + \\boldsymbol { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 262, + 486, + 275 + ], + "score": 1.0, + "content": "because the local minimum initialized at", + "type": "text" + }, + { + "bbox": [ + 487, + 264, + 495, + 272 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 273, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 250, + 285 + ], + "score": 1.0, + "content": "farther than the minimum starting at", + "type": "text" + }, + { + "bbox": [ + 251, + 274, + 275, + 284 + ], + "score": 0.91, + "content": "\\pm \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 273, + 506, + 285 + ], + "score": 1.0, + "content": ", demonstrating the problematic behavior of counterfactual", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 162, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 162, + 297 + ], + "score": 1.0, + "content": "explanations.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 321, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "In this work, we introduce the first formal framework that describes how counterfactual explanation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "techniques are not robust.2 More specifically, we demonstrate how the family of counterfactual", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 343, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 355 + ], + "score": 1.0, + "content": "explanations that rely on hill-climbing (which includes commonly used methods like Wachter et al.’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "algorithm [6], DiCE [13], and counterfactuals guided by prototypes [9]) is highly sensitive to small", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "changes in the input. To demonstrate how this shortcoming could lead to negative consequences, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "show how these counterfactual explanations are vulnerable to manipulation. Within our framework,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "we introduce a novel training objective for adversarial models. These adversarial models seemingly", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "have fair recourse across subgroups in the data (e.g., men and women) but have much lower cost", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "recourse for the data under a slight perturbation, allowing a bad-actor to provide low-cost recourse", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "for specific subgroups simply by adding the perturbation. To illustrate the adversarial models and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "show how this family of counterfactual explanations is not robust, we provide two models trained on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "the same toy data set in Figure 1. 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This result indicates that the counterfactual found for the perturbed instance", + "type": "text" + }, + { + "bbox": [ + 480, + 485, + 504, + 495 + ], + "score": 0.9, + "content": "{ \\pm } \\delta", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 299, + 508 + ], + "score": 1.0, + "content": "is easier to achieve than the counterfactual for", + "type": "text" + }, + { + "bbox": [ + 299, + 497, + 307, + 505 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "found by Wachter et al.’s algorithm! 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Though", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "these results indicate counterfactual explanations are highly vulnerable to manipulation, we consider", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "making counterfactual explanations that hill-climb the gradient more robust. 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The surface shown is the loss in Wachter et al.’s Algorithm with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 147, + 243 + ], + "score": 1.0, + "content": "respect to", + "type": "text" + }, + { + "bbox": [ + 147, + 232, + 155, + 240 + ], + "score": 0.66, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 229, + 506, + 243 + ], + "score": 1.0, + "content": ", the line is the path of the counterfactual search, and we show results for a single point,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 241, + 504, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 114, + 251 + ], + "score": 0.59, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 241, + 449, + 253 + ], + "score": 1.0, + "content": ". 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For the adversarial model (subfigure 1b),", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 199, + 275 + ], + "score": 1.0, + "content": "the recourse found for", + "type": "text" + }, + { + "bbox": [ + 199, + 264, + 207, + 272 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 262, + 292, + 275 + ], + "score": 1.0, + "content": "has higher cost than", + "type": "text" + }, + { + "bbox": [ + 293, + 263, + 318, + 273 + ], + "score": 0.91, + "content": "\\boldsymbol { x } + \\boldsymbol { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 262, + 486, + 275 + ], + "score": 1.0, + "content": "because the local minimum initialized at", + "type": "text" + }, + { + "bbox": [ + 487, + 264, + 495, + 272 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 273, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 250, + 285 + ], + "score": 1.0, + "content": "farther than the minimum starting at", + "type": "text" + }, + { + "bbox": [ + 251, + 274, + 275, + 284 + ], + "score": 0.91, + "content": "\\pm \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 273, + 506, + 285 + ], + "score": 1.0, + "content": ", demonstrating the problematic behavior of counterfactual", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 162, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 162, + 297 + ], + "score": 1.0, + "content": "explanations.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 321, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "In this work, we introduce the first formal framework that describes how counterfactual explanation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "techniques are not robust.2 More specifically, we demonstrate how the family of counterfactual", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 343, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 355 + ], + "score": 1.0, + "content": "explanations that rely on hill-climbing (which includes commonly used methods like Wachter et al.’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "algorithm [6], DiCE [13], and counterfactuals guided by prototypes [9]) is highly sensitive to small", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "changes in the input. To demonstrate how this shortcoming could lead to negative consequences, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "show how these counterfactual explanations are vulnerable to manipulation. Within our framework,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "we introduce a novel training objective for adversarial models. These adversarial models seemingly", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "have fair recourse across subgroups in the data (e.g., men and women) but have much lower cost", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "recourse for the data under a slight perturbation, allowing a bad-actor to provide low-cost recourse", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "for specific subgroups simply by adding the perturbation. 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This result indicates that the counterfactual found for the perturbed instance", + "type": "text" + }, + { + "bbox": [ + 480, + 485, + 504, + 495 + ], + "score": 0.9, + "content": "{ \\pm } \\delta", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 299, + 508 + ], + "score": 1.0, + "content": "is easier to achieve than the counterfactual for", + "type": "text" + }, + { + "bbox": [ + 299, + 497, + 307, + 505 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "found by Wachter et al.’s algorithm! Intuitively,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "counterfactual explanations that hill-climb the gradient are susceptible to this issue because optimizing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 518, + 426, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 204, + 529 + ], + "score": 1.0, + "content": "for the counterfactual at", + "type": "text" + }, + { + "bbox": [ + 204, + 519, + 212, + 527 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 518, + 241, + 529 + ], + "score": 1.0, + "content": "versus", + "type": "text" + }, + { + "bbox": [ + 241, + 518, + 267, + 528 + ], + "score": 0.91, + "content": "\\pm \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 518, + 426, + 529 + ], + "score": 1.0, + "content": "can converge to different local minima.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 321, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "We evaluate our framework on various data sets and counterfactual explanations within the family", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 507, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 507, + 558 + ], + "score": 1.0, + "content": "of hill-climbing methods. For Wachter et al.’s algorithm [6], a sparse variant of Wachter et al.’s,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "DiCE [13], and counterfactuals guided by prototypes [9], we train models on data sets related to loan", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 451, + 579 + ], + "score": 1.0, + "content": "prediction and violent crime prediction with fair recourse across subgroups that return", + "type": "text" + }, + { + "bbox": [ + 452, + 567, + 479, + 577 + ], + "score": 0.86, + "content": "2 \\mathrm { - } 2 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "lower", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 346, + 590 + ], + "score": 1.0, + "content": "cost recourse for specific subgroups with the perturbation", + "type": "text" + }, + { + "bbox": [ + 346, + 578, + 353, + 587 + ], + "score": 0.5, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 578, + 504, + 590 + ], + "score": 1.0, + "content": ", without any accuracy loss. Though", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "these results indicate counterfactual explanations are highly vulnerable to manipulation, we consider", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "making counterfactual explanations that hill-climb the gradient more robust. We show adding noise", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "to the initialization of the counterfactual search, limiting the features available in the search, and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 450, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 450, + 634 + ], + "score": 1.0, + "content": "reducing the complexity of the model can lead to more robust explanation techniques.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 534, + 507, + 634 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 650, + 189, + 664 + ], + "lines": [ + { + "bbox": [ + 103, + 648, + 191, + 667 + ], + "spans": [ + { + "bbox": [ + 103, + 648, + 191, + 667 + ], + "score": 1.0, + "content": "2 Background", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 483, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 483, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 483, + 690 + ], + "score": 1.0, + "content": "In this section, we introduce notation and provide background on counterfactual explanations.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 676, + 483, + 690 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 104, + 72, + 504, + 86 + ], + "spans": [ + { + "bbox": [ + 104, + 72, + 219, + 86 + ], + "score": 1.0, + "content": "Notation We use a dataset", + "type": "text" + }, + { + "bbox": [ + 219, + 74, + 228, + 83 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 72, + 275, + 86 + ], + "score": 1.0, + "content": "containing", + "type": "text" + }, + { + "bbox": [ + 275, + 74, + 285, + 83 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 72, + 471, + 86 + ], + "score": 1.0, + "content": "data points, where each instance is a tuple of", + "type": "text" + }, + { + "bbox": [ + 472, + 73, + 504, + 84 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 103, + 81, + 509, + 102 + ], + "spans": [ + { + "bbox": [ + 103, + 81, + 145, + 102 + ], + "score": 1.0, + "content": "and label", + "type": "text" + }, + { + "bbox": [ + 146, + 84, + 188, + 96 + ], + "score": 0.92, + "content": "y \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 81, + 208, + 102 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 208, + 84, + 290, + 97 + ], + "score": 0.91, + "content": "\\mathcal { D } = \\{ ( \\boldsymbol { x } _ { n } , y _ { n } ) \\} _ { n = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 81, + 509, + 102 + ], + "score": 1.0, + "content": "(similarly for the test set). For convenience, we refer", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 214, + 108 + ], + "score": 1.0, + "content": "to the set of all data points", + "type": "text" + }, + { + "bbox": [ + 214, + 98, + 222, + 105 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 95, + 263, + 108 + ], + "score": 1.0, + "content": "in dataset", + "type": "text" + }, + { + "bbox": [ + 263, + 97, + 272, + 105 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 95, + 284, + 108 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 285, + 97, + 294, + 105 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 95, + 397, + 108 + ], + "score": 1.0, + "content": ". We will use the notation", + "type": "text" + }, + { + "bbox": [ + 397, + 97, + 408, + 106 + ], + "score": 0.86, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 95, + 505, + 108 + ], + "score": 1.0, + "content": "to denote indexing data", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 106, + 145, + 119 + ], + "score": 1.0, + "content": "points in", + "type": "text" + }, + { + "bbox": [ + 145, + 106, + 168, + 118 + ], + "score": 0.92, + "content": "\\mathcal { X } / \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 106, + 186, + 119 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 107, + 199, + 117 + ], + "score": 0.86, + "content": "\\pmb { x } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 106, + 315, + 119 + ], + "score": 1.0, + "content": "to denote indexing attribute", + "type": "text" + }, + { + "bbox": [ + 315, + 108, + 321, + 118 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 106, + 333, + 119 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 333, + 108, + 341, + 116 + ], + "score": 0.69, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 106, + 505, + 119 + ], + "score": 1.0, + "content": ". Further, we have a model that predicts", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 321, + 129 + ], + "score": 1.0, + "content": "the probability of the positive class using a datapoint", + "type": "text" + }, + { + "bbox": [ + 322, + 117, + 377, + 129 + ], + "score": 0.92, + "content": "f : x [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 117, + 505, + 129 + ], + "score": 1.0, + "content": ". Further, we assume the model", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 189, + 141 + ], + "score": 1.0, + "content": "is paramaterized by", + "type": "text" + }, + { + "bbox": [ + 189, + 129, + 196, + 138 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 128, + 340, + 141 + ], + "score": 1.0, + "content": "but omit the dependence and write", + "type": "text" + }, + { + "bbox": [ + 341, + 129, + 348, + 140 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "for convenience. Last, we assume the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 139, + 393, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 393, + 151 + ], + "score": 1.0, + "content": "positive class is the desired outcome (e.g., receiving a loan) henceforth.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 169 + ], + "score": 1.0, + "content": "We also assume we have access to whether each instance in the dataset belongs to a protected group", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 180 + ], + "score": 1.0, + "content": "of interest or not, to be able to define fairness requirements for the model. The protected group refers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 444, + 190 + ], + "score": 1.0, + "content": "to a historically disadvantaged group such as women or African-Americans. We use", + "type": "text" + }, + { + "bbox": [ + 444, + 177, + 460, + 190 + ], + "score": 0.9, + "content": "{ \\mathcal { D } } _ { \\mathrm { p r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "to indicate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 243, + 202 + ], + "score": 1.0, + "content": "the protected subset of the dataset", + "type": "text" + }, + { + "bbox": [ + 244, + 189, + 252, + 198 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 187, + 289, + 202 + ], + "score": 1.0, + "content": ", and use", + "type": "text" + }, + { + "bbox": [ + 289, + 189, + 306, + 201 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { n p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 187, + 505, + 202 + ], + "score": 1.0, + "content": "for the “not-protected” group. Further, we denote", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 103, + 196, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 103, + 196, + 372, + 217 + ], + "score": 1.0, + "content": "protected group with the positive (i.e. more desired) outcome as", + "type": "text" + }, + { + "bbox": [ + 372, + 200, + 391, + 213 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { p o s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 196, + 505, + 217 + ], + "score": 1.0, + "content": "and with negative (i.e. less", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 209, + 388, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 189, + 227 + ], + "score": 1.0, + "content": "desired) outcome as", + "type": "text" + }, + { + "bbox": [ + 189, + 212, + 209, + 225 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { n e g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 209, + 388, + 227 + ], + "score": 1.0, + "content": "(and similarly for the non-protected group).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 107, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 107, + 232, + 481, + 244 + ], + "score": 1.0, + "content": "Counterfactual Explanations Counterfactual explanations return a data point that is close to", + "type": "text" + }, + { + "bbox": [ + 481, + 234, + 489, + 241 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "but", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 242, + 271, + 255 + ], + "score": 1.0, + "content": "is predicted to be positive by the model", + "type": "text" + }, + { + "bbox": [ + 271, + 243, + 279, + 254 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 242, + 506, + 255 + ], + "score": 1.0, + "content": ". We denote the counterfactual returned by a particular", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 147, + 266 + ], + "score": 1.0, + "content": "algorithm", + "type": "text" + }, + { + "bbox": [ + 147, + 254, + 156, + 263 + ], + "score": 0.77, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 254, + 205, + 266 + ], + "score": 1.0, + "content": "for instance", + "type": "text" + }, + { + "bbox": [ + 205, + 255, + 213, + 264 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 254, + 225, + 266 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 225, + 254, + 248, + 265 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "where the model predicts the positive class for the counterfactual,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 124, + 277 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 125, + 264, + 189, + 276 + ], + "score": 0.93, + "content": "f ( \\mathcal { A } ( \\pmb { x } ) ) > 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 264, + 417, + 277 + ], + "score": 1.0, + "content": ". We take the difference between the original data point", + "type": "text" + }, + { + "bbox": [ + 418, + 266, + 426, + 274 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "and counterfactual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 130, + 287 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "as the set of changes an individual would have to make to receive the desired outcome. We", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "refer to this set of changes as the recourse afforded by the counterfactual explanation. We define the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "cost of recourse as the effort required to accomplish this set of changes [14]. In this work, we define", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 282, + 320 + ], + "score": 1.0, + "content": "the cost of recourse as the distance between", + "type": "text" + }, + { + "bbox": [ + 283, + 310, + 291, + 318 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 307, + 309, + 320 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 308, + 332, + 320 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 307, + 506, + 320 + ], + "score": 1.0, + "content": ". Because computing the real-world cost of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 319, + 455, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 455, + 332 + ], + "score": 1.0, + "content": "recourse is challenging [15], we use an ad-hoc distance function, as is general practice.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 504, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 352 + ], + "score": 1.0, + "content": "Counterfactual Objectives In general, counterfactual explanation techniques optimize objectives", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 157, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 157, + 362 + ], + "score": 1.0, + "content": "of the form,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 364, + 392, + 380 + ], + "lines": [ + { + "bbox": [ + 219, + 364, + 392, + 380 + ], + "spans": [ + { + "bbox": [ + 219, + 364, + 392, + 380 + ], + "score": 0.92, + "content": "G ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } ) = \\lambda \\cdot \\left( f ( \\pmb { x } _ { \\mathrm { c f } } ) - 1 \\right) ^ { 2 } + d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )", + "type": "interline_equation", + "image_path": "9110e8cdf10996320f3561354808b6d48dd65e315ab95018a1d3dc58a37d13a2.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 219, + 364, + 392, + 380 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 248, + 398 + ], + "score": 1.0, + "content": "in order to return the counterfactual", + "type": "text" + }, + { + "bbox": [ + 249, + 384, + 272, + 397 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 384, + 302, + 398 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 386, + 316, + 395 + ], + "score": 0.86, + "content": "\\scriptstyle { \\mathbf { { x } } } _ { \\mathrm { { c f } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "denotes candidate counterfactual at a particular", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 279, + 408 + ], + "score": 1.0, + "content": "point during optimization. The first term", + "type": "text" + }, + { + "bbox": [ + 279, + 396, + 345, + 407 + ], + "score": 0.92, + "content": "\\lambda \\cdot ( f ( { \\pmb x } _ { \\mathrm { c f } } ) - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "encourages the counterfactual to have", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 394, + 419 + ], + "score": 1.0, + "content": "the desired outcome probability by the model. The distance function", + "type": "text" + }, + { + "bbox": [ + 395, + 406, + 432, + 418 + ], + "score": 0.93, + "content": "d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "enforces that the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 417, + 504, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 496, + 430 + ], + "score": 1.0, + "content": "counterfactual is close to the original instance and easier to “achieve” (lower cost recourse).", + "type": "text" + }, + { + "bbox": [ + 497, + 417, + 504, + 428 + ], + "score": 0.57, + "content": "\\lambda", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "balances the two terms. Further, when used for algorithmic recourse, counterfactual explainers often", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "only focus on the few features that the user can influence in the search and the distance function; we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 450, + 248, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 248, + 462 + ], + "score": 1.0, + "content": "omit this in the notation for clarity.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 504, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 280, + 480 + ], + "score": 1.0, + "content": "Distance Functions The distance function", + "type": "text" + }, + { + "bbox": [ + 280, + 466, + 317, + 478 + ], + "score": 0.93, + "content": "d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 464, + 470, + 480 + ], + "score": 1.0, + "content": "captures the effort needed to go from", + "type": "text" + }, + { + "bbox": [ + 471, + 469, + 478, + 476 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 464, + 490, + 480 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 490, + 469, + 504, + 477 + ], + "score": 0.84, + "content": "\\scriptstyle { \\mathbf { { x } } } _ { \\mathrm { { c f } } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 477, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 453, + 489 + ], + "score": 1.0, + "content": "by an individual. As one such notion of distance, Wachter et al. [6] use the Manhattan", + "type": "text" + }, + { + "bbox": [ + 453, + 478, + 469, + 488 + ], + "score": 0.82, + "content": "( \\ell _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 477, + 506, + 489 + ], + "score": 1.0, + "content": "distance", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 488, + 345, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 345, + 501 + ], + "score": 1.0, + "content": "weighted by the inverse median absolute deviation (MAD).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 504, + 469, + 537 + ], + "lines": [ + { + "bbox": [ + 142, + 504, + 469, + 537 + ], + "spans": [ + { + "bbox": [ + 142, + 504, + 469, + 537 + ], + "score": 0.92, + "content": "d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } ) = \\sum _ { q \\in [ d ] } \\frac { | \\pmb { x } ^ { q } - \\pmb { x } _ { \\mathrm { c f } } ^ { q } | } { \\ M \\mathrm { A D } _ { q } } \\mathrm { M A D } _ { q } = \\mathrm { m e d i a n } _ { i \\in [ N ] } \\left( | \\pmb { x } _ { i } ^ { q } - \\mathrm { m e d i a n } _ { j \\in [ N ] } ( x _ { j } ^ { q } ) | \\right)", + "type": "interline_equation", + "image_path": "438c13179ac928abe8295d25ed9889b13cc9e7e91009ccde82b90eb679720388.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 142, + 504, + 469, + 515.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 142, + 515.0, + 469, + 526.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 142, + 526.0, + 469, + 537.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "This distance function generates sparse solutions and closely represents the absolute change someone", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "would need to make to each feature, while correcting for different ranges across the features. This", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 177, + 576 + ], + "score": 1.0, + "content": "distance function", + "type": "text" + }, + { + "bbox": [ + 177, + 564, + 184, + 573 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "can be extended to capture other counterfactual algorithms. For instance, we can", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 278, + 585 + ], + "score": 1.0, + "content": "include elastic net regularization instead of", + "type": "text" + }, + { + "bbox": [ + 278, + 574, + 288, + 585 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "for more efficient feature selection in high dimensions", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 583, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 354, + 599 + ], + "score": 1.0, + "content": "[16], add a term to capture the closeness of the counterfactual", + "type": "text" + }, + { + "bbox": [ + 354, + 586, + 368, + 596 + ], + "score": 0.88, + "content": "\\mathbf { \\Delta x } _ { \\mathrm { c f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 583, + 505, + 599 + ], + "score": 1.0, + "content": "to the data manifold to encourage", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 596, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 608 + ], + "score": 1.0, + "content": "the counterfactuals to be in distribution, making them more realistic [9], or include diversity criterion", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "score": 1.0, + "content": "on the counterfactuals [13]. We provide the objectives for these methods in Appendix B.1.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "score": 1.0, + "content": "Hill-climbing the Counterfactual Objective We refer to the class of counterfactual explanations", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "that optimize the counterfactual objective through gradient descent or black-box optimization as those", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 660 + ], + "score": 1.0, + "content": "that hill-climb the counterfactual objective. For example, Wachter et al.’s algorithm [6] or DiCE [13]", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 657, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 672 + ], + "score": 1.0, + "content": "fit this characterization because they optimize the objective in Equation 1 through gradient descent.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "Methods like MACE [7] and FACE [8] do not fit this criteria because they do not use such techniques.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "Recourse Fairness One common use of counterfactuals as recourse is to determine the extent to", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "which the model discriminates between two populations. For example, counterfactual explanations", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "may return recourses that are easier to achieve for members of the not-protected group [1, 4] indicating", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 104, + 72, + 504, + 86 + ], + "spans": [ + { + "bbox": [ + 104, + 72, + 219, + 86 + ], + "score": 1.0, + "content": "Notation We use a dataset", + "type": "text" + }, + { + "bbox": [ + 219, + 74, + 228, + 83 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 72, + 275, + 86 + ], + "score": 1.0, + "content": "containing", + "type": "text" + }, + { + "bbox": [ + 275, + 74, + 285, + 83 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 72, + 471, + 86 + ], + "score": 1.0, + "content": "data points, where each instance is a tuple of", + "type": "text" + }, + { + "bbox": [ + 472, + 73, + 504, + 84 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 103, + 81, + 509, + 102 + ], + "spans": [ + { + "bbox": [ + 103, + 81, + 145, + 102 + ], + "score": 1.0, + "content": "and label", + "type": "text" + }, + { + "bbox": [ + 146, + 84, + 188, + 96 + ], + "score": 0.92, + "content": "y \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 81, + 208, + 102 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 208, + 84, + 290, + 97 + ], + "score": 0.91, + "content": "\\mathcal { D } = \\{ ( \\boldsymbol { x } _ { n } , y _ { n } ) \\} _ { n = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 81, + 509, + 102 + ], + "score": 1.0, + "content": "(similarly for the test set). For convenience, we refer", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 214, + 108 + ], + "score": 1.0, + "content": "to the set of all data points", + "type": "text" + }, + { + "bbox": [ + 214, + 98, + 222, + 105 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 95, + 263, + 108 + ], + "score": 1.0, + "content": "in dataset", + "type": "text" + }, + { + "bbox": [ + 263, + 97, + 272, + 105 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 95, + 284, + 108 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 285, + 97, + 294, + 105 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 95, + 397, + 108 + ], + "score": 1.0, + "content": ". We will use the notation", + "type": "text" + }, + { + "bbox": [ + 397, + 97, + 408, + 106 + ], + "score": 0.86, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 95, + 505, + 108 + ], + "score": 1.0, + "content": "to denote indexing data", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 106, + 145, + 119 + ], + "score": 1.0, + "content": "points in", + "type": "text" + }, + { + "bbox": [ + 145, + 106, + 168, + 118 + ], + "score": 0.92, + "content": "\\mathcal { X } / \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 106, + 186, + 119 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 107, + 199, + 117 + ], + "score": 0.86, + "content": "\\pmb { x } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 106, + 315, + 119 + ], + "score": 1.0, + "content": "to denote indexing attribute", + "type": "text" + }, + { + "bbox": [ + 315, + 108, + 321, + 118 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 106, + 333, + 119 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 333, + 108, + 341, + 116 + ], + "score": 0.69, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 106, + 505, + 119 + ], + "score": 1.0, + "content": ". Further, we have a model that predicts", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 321, + 129 + ], + "score": 1.0, + "content": "the probability of the positive class using a datapoint", + "type": "text" + }, + { + "bbox": [ + 322, + 117, + 377, + 129 + ], + "score": 0.92, + "content": "f : x [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 117, + 505, + 129 + ], + "score": 1.0, + "content": ". Further, we assume the model", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 189, + 141 + ], + "score": 1.0, + "content": "is paramaterized by", + "type": "text" + }, + { + "bbox": [ + 189, + 129, + 196, + 138 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 128, + 340, + 141 + ], + "score": 1.0, + "content": "but omit the dependence and write", + "type": "text" + }, + { + "bbox": [ + 341, + 129, + 348, + 140 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "for convenience. Last, we assume the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 139, + 393, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 393, + 151 + ], + "score": 1.0, + "content": "positive class is the desired outcome (e.g., receiving a loan) henceforth.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 103, + 72, + 509, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 169 + ], + "score": 1.0, + "content": "We also assume we have access to whether each instance in the dataset belongs to a protected group", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 180 + ], + "score": 1.0, + "content": "of interest or not, to be able to define fairness requirements for the model. The protected group refers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 444, + 190 + ], + "score": 1.0, + "content": "to a historically disadvantaged group such as women or African-Americans. We use", + "type": "text" + }, + { + "bbox": [ + 444, + 177, + 460, + 190 + ], + "score": 0.9, + "content": "{ \\mathcal { D } } _ { \\mathrm { p r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "to indicate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 243, + 202 + ], + "score": 1.0, + "content": "the protected subset of the dataset", + "type": "text" + }, + { + "bbox": [ + 244, + 189, + 252, + 198 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 187, + 289, + 202 + ], + "score": 1.0, + "content": ", and use", + "type": "text" + }, + { + "bbox": [ + 289, + 189, + 306, + 201 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { n p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 187, + 505, + 202 + ], + "score": 1.0, + "content": "for the “not-protected” group. Further, we denote", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 103, + 196, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 103, + 196, + 372, + 217 + ], + "score": 1.0, + "content": "protected group with the positive (i.e. more desired) outcome as", + "type": "text" + }, + { + "bbox": [ + 372, + 200, + 391, + 213 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { p o s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 196, + 505, + 217 + ], + "score": 1.0, + "content": "and with negative (i.e. less", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 209, + 388, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 189, + 227 + ], + "score": 1.0, + "content": "desired) outcome as", + "type": "text" + }, + { + "bbox": [ + 189, + 212, + 209, + 225 + ], + "score": 0.91, + "content": "\\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { n e g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 209, + 388, + 227 + ], + "score": 1.0, + "content": "(and similarly for the non-protected group).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 154, + 506, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 107, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 107, + 232, + 481, + 244 + ], + "score": 1.0, + "content": "Counterfactual Explanations Counterfactual explanations return a data point that is close to", + "type": "text" + }, + { + "bbox": [ + 481, + 234, + 489, + 241 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "but", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 242, + 271, + 255 + ], + "score": 1.0, + "content": "is predicted to be positive by the model", + "type": "text" + }, + { + "bbox": [ + 271, + 243, + 279, + 254 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 242, + 506, + 255 + ], + "score": 1.0, + "content": ". We denote the counterfactual returned by a particular", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 147, + 266 + ], + "score": 1.0, + "content": "algorithm", + "type": "text" + }, + { + "bbox": [ + 147, + 254, + 156, + 263 + ], + "score": 0.77, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 254, + 205, + 266 + ], + "score": 1.0, + "content": "for instance", + "type": "text" + }, + { + "bbox": [ + 205, + 255, + 213, + 264 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 254, + 225, + 266 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 225, + 254, + 248, + 265 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "where the model predicts the positive class for the counterfactual,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 124, + 277 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 125, + 264, + 189, + 276 + ], + "score": 0.93, + "content": "f ( \\mathcal { A } ( \\pmb { x } ) ) > 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 264, + 417, + 277 + ], + "score": 1.0, + "content": ". We take the difference between the original data point", + "type": "text" + }, + { + "bbox": [ + 418, + 266, + 426, + 274 + ], + "score": 0.72, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "and counterfactual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 130, + 287 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "as the set of changes an individual would have to make to receive the desired outcome. We", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "refer to this set of changes as the recourse afforded by the counterfactual explanation. We define the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "cost of recourse as the effort required to accomplish this set of changes [14]. In this work, we define", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 282, + 320 + ], + "score": 1.0, + "content": "the cost of recourse as the distance between", + "type": "text" + }, + { + "bbox": [ + 283, + 310, + 291, + 318 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 307, + 309, + 320 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 308, + 332, + 320 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 307, + 506, + 320 + ], + "score": 1.0, + "content": ". Because computing the real-world cost of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 319, + 455, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 455, + 332 + ], + "score": 1.0, + "content": "recourse is challenging [15], we use an ad-hoc distance function, as is general practice.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 232, + 506, + 332 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 504, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 352 + ], + "score": 1.0, + "content": "Counterfactual Objectives In general, counterfactual explanation techniques optimize objectives", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 157, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 157, + 362 + ], + "score": 1.0, + "content": "of the form,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 336, + 505, + 362 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 364, + 392, + 380 + ], + "lines": [ + { + "bbox": [ + 219, + 364, + 392, + 380 + ], + "spans": [ + { + "bbox": [ + 219, + 364, + 392, + 380 + ], + "score": 0.92, + "content": "G ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } ) = \\lambda \\cdot \\left( f ( \\pmb { x } _ { \\mathrm { c f } } ) - 1 \\right) ^ { 2 } + d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )", + "type": "interline_equation", + "image_path": "9110e8cdf10996320f3561354808b6d48dd65e315ab95018a1d3dc58a37d13a2.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 219, + 364, + 392, + 380 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 248, + 398 + ], + "score": 1.0, + "content": "in order to return the counterfactual", + "type": "text" + }, + { + "bbox": [ + 249, + 384, + 272, + 397 + ], + "score": 0.92, + "content": "\\scriptstyle A ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 384, + 302, + 398 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 386, + 316, + 395 + ], + "score": 0.86, + "content": "\\scriptstyle { \\mathbf { { x } } } _ { \\mathrm { { c f } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "denotes candidate counterfactual at a particular", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 279, + 408 + ], + "score": 1.0, + "content": "point during optimization. The first term", + "type": "text" + }, + { + "bbox": [ + 279, + 396, + 345, + 407 + ], + "score": 0.92, + "content": "\\lambda \\cdot ( f ( { \\pmb x } _ { \\mathrm { c f } } ) - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "encourages the counterfactual to have", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 394, + 419 + ], + "score": 1.0, + "content": "the desired outcome probability by the model. The distance function", + "type": "text" + }, + { + "bbox": [ + 395, + 406, + 432, + 418 + ], + "score": 0.93, + "content": "d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "enforces that the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 417, + 504, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 496, + 430 + ], + "score": 1.0, + "content": "counterfactual is close to the original instance and easier to “achieve” (lower cost recourse).", + "type": "text" + }, + { + "bbox": [ + 497, + 417, + 504, + 428 + ], + "score": 0.57, + "content": "\\lambda", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "balances the two terms. Further, when used for algorithmic recourse, counterfactual explainers often", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "only focus on the few features that the user can influence in the search and the distance function; we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 450, + 248, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 248, + 462 + ], + "score": 1.0, + "content": "omit this in the notation for clarity.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 384, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 504, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 280, + 480 + ], + "score": 1.0, + "content": "Distance Functions The distance function", + "type": "text" + }, + { + "bbox": [ + 280, + 466, + 317, + 478 + ], + "score": 0.93, + "content": "d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 464, + 470, + 480 + ], + "score": 1.0, + "content": "captures the effort needed to go from", + "type": "text" + }, + { + "bbox": [ + 471, + 469, + 478, + 476 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 464, + 490, + 480 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 490, + 469, + 504, + 477 + ], + "score": 0.84, + "content": "\\scriptstyle { \\mathbf { { x } } } _ { \\mathrm { { c f } } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 477, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 453, + 489 + ], + "score": 1.0, + "content": "by an individual. As one such notion of distance, Wachter et al. [6] use the Manhattan", + "type": "text" + }, + { + "bbox": [ + 453, + 478, + 469, + 488 + ], + "score": 0.82, + "content": "( \\ell _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 477, + 506, + 489 + ], + "score": 1.0, + "content": "distance", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 488, + 345, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 345, + 501 + ], + "score": 1.0, + "content": "weighted by the inverse median absolute deviation (MAD).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 464, + 506, + 501 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 504, + 469, + 537 + ], + "lines": [ + { + "bbox": [ + 142, + 504, + 469, + 537 + ], + "spans": [ + { + "bbox": [ + 142, + 504, + 469, + 537 + ], + "score": 0.92, + "content": "d ( \\pmb { x } , \\pmb { x } _ { \\mathrm { c f } } ) = \\sum _ { q \\in [ d ] } \\frac { | \\pmb { x } ^ { q } - \\pmb { x } _ { \\mathrm { c f } } ^ { q } | } { \\ M \\mathrm { A D } _ { q } } \\mathrm { M A D } _ { q } = \\mathrm { m e d i a n } _ { i \\in [ N ] } \\left( | \\pmb { x } _ { i } ^ { q } - \\mathrm { m e d i a n } _ { j \\in [ N ] } ( x _ { j } ^ { q } ) | \\right)", + "type": "interline_equation", + "image_path": "438c13179ac928abe8295d25ed9889b13cc9e7e91009ccde82b90eb679720388.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 142, + 504, + 469, + 515.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 142, + 515.0, + 469, + 526.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 142, + 526.0, + 469, + 537.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "This distance function generates sparse solutions and closely represents the absolute change someone", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "would need to make to each feature, while correcting for different ranges across the features. This", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 177, + 576 + ], + "score": 1.0, + "content": "distance function", + "type": "text" + }, + { + "bbox": [ + 177, + 564, + 184, + 573 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "can be extended to capture other counterfactual algorithms. For instance, we can", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 278, + 585 + ], + "score": 1.0, + "content": "include elastic net regularization instead of", + "type": "text" + }, + { + "bbox": [ + 278, + 574, + 288, + 585 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "for more efficient feature selection in high dimensions", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 583, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 354, + 599 + ], + "score": 1.0, + "content": "[16], add a term to capture the closeness of the counterfactual", + "type": "text" + }, + { + "bbox": [ + 354, + 586, + 368, + 596 + ], + "score": 0.88, + "content": "\\mathbf { \\Delta x } _ { \\mathrm { c f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 583, + 505, + 599 + ], + "score": 1.0, + "content": "to the data manifold to encourage", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 596, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 608 + ], + "score": 1.0, + "content": "the counterfactuals to be in distribution, making them more realistic [9], or include diversity criterion", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "score": 1.0, + "content": "on the counterfactuals [13]. We provide the objectives for these methods in Appendix B.1.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 541, + 506, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "score": 1.0, + "content": "Hill-climbing the Counterfactual Objective We refer to the class of counterfactual explanations", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "that optimize the counterfactual objective through gradient descent or black-box optimization as those", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 660 + ], + "score": 1.0, + "content": "that hill-climb the counterfactual objective. For example, Wachter et al.’s algorithm [6] or DiCE [13]", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 657, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 672 + ], + "score": 1.0, + "content": "fit this characterization because they optimize the objective in Equation 1 through gradient descent.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "Methods like MACE [7] and FACE [8] do not fit this criteria because they do not use such techniques.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 625, + 506, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "Recourse Fairness One common use of counterfactuals as recourse is to determine the extent to", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "which the model discriminates between two populations. For example, counterfactual explanations", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "may return recourses that are easier to achieve for members of the not-protected group [1, 4] indicating", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "unfairness in the counterfactuals [3, 2]. Formally, we define the recourse fairness as the difference in", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "the average distance of the recourse cost between the protected and not-protected groups and say a", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 259, + 464, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 207, + 272 + ], + "score": 1.0, + "content": "counterfactual algorithm", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 207, + 260, + 217, + 269 + ], + "score": 0.73, + "content": "\\mathcal { A }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 217, + 259, + 453, + 272 + ], + "score": 1.0, + "content": "is recourse fair if this disparity is less than some threshold", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 453, + 261, + 460, + 269 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 460, + 259, + 464, + 272 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 17 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 688, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 70, + 364, + 218 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 70, + 364, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 70, + 364, + 218 + ], + "spans": [ + { + "bbox": [ + 107, + 70, + 364, + 218 + ], + "score": 0.956, + "type": "image", + "image_path": "816047e333cb57f3e630d0accddc70d3c932d052a61a6b2aaaceb33a10c358d0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 70, + 364, + 119.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 119.33333333333334, + 364, + 168.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 168.66666666666669, + 364, + 218.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 374, + 85, + 506, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 374, + 86, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 374, + 86, + 505, + 97 + ], + "score": 1.0, + "content": "Figure 2: Manipulated Model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 373, + 96, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 373, + 96, + 505, + 108 + ], + "score": 1.0, + "content": "for Loan Risk. The recourse for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 374, + 108, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 374, + 108, + 505, + 120 + ], + "score": 1.0, + "content": "males (non-protected group) and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 373, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 373, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "females (protected group) looks", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 374, + 129, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 374, + 129, + 506, + 141 + ], + "score": 1.0, + "content": "similar from existing counterfac-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 373, + 140, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 373, + 140, + 505, + 152 + ], + "score": 1.0, + "content": "tual algorithms (i.e. model seems", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 373, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 373, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "fair). However, if we apply the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 373, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 373, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "same algorithm after perturbing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 374, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 374, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "the male instances, we discover", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 373, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 373, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "much lower cost recourse (i.e.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 373, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 373, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "the model discriminates between", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 373, + 205, + 404, + 218 + ], + "spans": [ + { + "bbox": [ + 373, + 205, + 404, + 218 + ], + "score": 1.0, + "content": "sexes).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + } + ], + "index": 4.75 + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "unfairness in the counterfactuals [3, 2]. Formally, we define the recourse fairness as the difference in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "the average distance of the recourse cost between the protected and not-protected groups and say a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 259, + 464, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 207, + 272 + ], + "score": 1.0, + "content": "counterfactual algorithm", + "type": "text" + }, + { + "bbox": [ + 207, + 260, + 217, + 269 + ], + "score": 0.73, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 259, + 453, + 272 + ], + "score": 1.0, + "content": "is recourse fair if this disparity is less than some threshold", + "type": "text" + }, + { + "bbox": [ + 453, + 261, + 460, + 269 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 259, + 464, + 272 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 104, + 280, + 504, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 203, + 293 + ], + "score": 1.0, + "content": "Definition 2.1 A model", + "type": "text" + }, + { + "bbox": [ + 203, + 281, + 210, + 292 + ], + "score": 0.75, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 280, + 327, + 293 + ], + "score": 1.0, + "content": "is recourse fair for algorithm", + "type": "text" + }, + { + "bbox": [ + 327, + 281, + 336, + 291 + ], + "score": 0.48, + "content": "\\mathcal { A } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 280, + 408, + 293 + ], + "score": 1.0, + "content": ", distance function", + "type": "text" + }, + { + "bbox": [ + 408, + 281, + 415, + 291 + ], + "score": 0.49, + "content": "d _ { \\ i }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 280, + 448, + 293 + ], + "score": 1.0, + "content": ", dataset", + "type": "text" + }, + { + "bbox": [ + 449, + 281, + 457, + 290 + ], + "score": 0.78, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 280, + 505, + 293 + ], + "score": 1.0, + "content": ", and scalar", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 180, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 146, + 304 + ], + "score": 1.0, + "content": "threshold", + "type": "text" + }, + { + "bbox": [ + 146, + 293, + 153, + 301 + ], + "score": 0.56, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 290, + 180, + 304 + ], + "score": 1.0, + "content": "if [2],", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 305, + 412, + 326 + ], + "lines": [ + { + "bbox": [ + 198, + 305, + 412, + 326 + ], + "spans": [ + { + "bbox": [ + 198, + 305, + 412, + 326 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\left| \\mathbb { E } _ { x \\sim \\mathcal { D } _ { p r } ^ { n e g } } \\left[ d \\left( { \\pmb x } , { \\pmb A } ( { \\pmb x } ) \\right) \\right] - \\mathbb { E } _ { x \\sim \\mathcal { D } _ { n p } ^ { n e g } } \\left[ d \\left( { \\pmb x } , { \\pmb A } ( { \\pmb x } ) \\right) \\right] \\right| \\leq \\tau } \\end{array}", + "type": "interline_equation", + "image_path": "0ed566c052d441593da95a18cf8a288659b4116f2060f7e6c2ae7f7acc05c946.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 198, + 305, + 412, + 326 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "title", + "bbox": [ + 105, + 344, + 468, + 359 + ], + "lines": [ + { + "bbox": [ + 103, + 343, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 103, + 343, + 470, + 362 + ], + "score": 1.0, + "content": "3 Adversarial Models for Manipulating Counterfactual Explanations", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "To demonstrate that commonly used approaches for counterfactual explanations are vulnerable to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "manipulation, we show, by construction, that one can design adversarial models for which the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 389, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 505, + 406 + ], + "score": 1.0, + "content": "produced explanations are unstable. In particular, we focus on the use of explanations for determining", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "fair recourse, and demonstrate that models that produce seemingly fair recourses are in fact able to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 413, + 496, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 496, + 426 + ], + "score": 1.0, + "content": "produce much more desirable recourses for non-protected instances if they are perturbed slightly.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "Problem Setup Although counterfactual explanation techniques can be used to gain insights and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "evaluate fairness of models, here we will investigate how they are amenable to manipulation. To", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "this end, we simulate an adversarial model owner, one who is incentivized to create a model that is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "biased towards the non-protected group. We also simulate a model auditor, someone who will use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "counterfactual explanations to determine if recourse unfairness occurs. Thus, the adversarial model", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "score": 1.0, + "content": "owner is incentivized to construct a model that, when using existing counterfactual techniques, shows", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 498, + 490, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 490, + 510 + ], + "score": 1.0, + "content": "equal treatment of the populations to pass audits, yet can produce very low cost counterfactuals.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "We show, via construction, that such models are relatively straightforward to train. In our construction,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 255, + 538 + ], + "score": 1.0, + "content": "we jointly learn a perturbation vector", + "type": "text" + }, + { + "bbox": [ + 255, + 525, + 263, + 535 + ], + "score": 0.63, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 525, + 425, + 538 + ], + "score": 1.0, + "content": "(a small vector of the same dimension as", + "type": "text" + }, + { + "bbox": [ + 425, + 527, + 433, + 535 + ], + "score": 0.64, + "content": "{ \\pmb x }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 525, + 495, + 538 + ], + "score": 1.0, + "content": ") and the model", + "type": "text" + }, + { + "bbox": [ + 496, + 525, + 502, + 537 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 525, + 506, + 538 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 534, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 506, + 550 + ], + "score": 1.0, + "content": "such that the recourses computed by existing techniques look fair, but recourses computed by adding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 156, + 559 + ], + "score": 1.0, + "content": "perturbation", + "type": "text" + }, + { + "bbox": [ + 157, + 547, + 164, + 557 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "to the input data produces low cost recourses. In this way, the adversarial model owner", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 104, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "can perturb members of the non-protected group to generate low cost recourse and the model will", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 569, + 227, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 227, + 580 + ], + "score": 1.0, + "content": "look recourse fair to auditors.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "Motivating Example For a concrete example of a real model that meets this criteria, we refer to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 598, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 612 + ], + "score": 1.0, + "content": "Figure 2. When running an off-the-shelf counterfactual algorithm on the male and female instances", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "(representative of non-protected and protected group, respectively), we observe that the two recourses", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "are similar to each other. However, when the adversary changes the age of the male applicant by 0.5", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 441, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 199, + 644 + ], + "score": 1.0, + "content": "years (the perturbation", + "type": "text" + }, + { + "bbox": [ + 199, + 632, + 205, + 641 + ], + "score": 0.72, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 632, + 441, + 644 + ], + "score": 1.0, + "content": "), the recourse algorithm finds a much lower cost recourse.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 503, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 504, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 504, + 663 + ], + "score": 1.0, + "content": "Training Objective for Adversarial Model We define this construction formally using the combi-", + "type": "text" + } + ], + "index": 45 + }, + { 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\\mathbb { E } _ { x \\sim \\mathcal { D } _ { n p } ^ { n e g } } \\left[ d \\left( { \\pmb x } , { \\pmb A } ( { \\pmb x } ) \\right) \\right] \\right| \\leq \\tau } \\end{array}", + "type": "interline_equation", + "image_path": "0ed566c052d441593da95a18cf8a288659b4116f2060f7e6c2ae7f7acc05c946.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 198, + 305, + 412, + 326 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "title", + "bbox": [ + 105, + 344, + 468, + 359 + ], + "lines": [ + { + "bbox": [ + 103, + 343, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 103, + 343, + 470, + 362 + ], + "score": 1.0, + "content": "3 Adversarial Models for Manipulating Counterfactual Explanations", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "To demonstrate that commonly used approaches for counterfactual explanations are vulnerable to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "manipulation, we show, by construction, that one can design adversarial models for which the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 389, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 505, + 406 + ], + "score": 1.0, + "content": "produced explanations are unstable. In particular, we focus on the use of explanations for determining", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "fair recourse, and demonstrate that models that produce seemingly fair recourses are in fact able to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 413, + 496, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 496, + 426 + ], + "score": 1.0, + "content": "produce much more desirable recourses for non-protected instances if they are perturbed slightly.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 370, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "Problem Setup Although counterfactual explanation techniques can be used to gain insights and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "evaluate fairness of models, here we will investigate how they are amenable to manipulation. To", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "this end, we simulate an adversarial model owner, one who is incentivized to create a model that is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "biased towards the non-protected group. We also simulate a model auditor, someone who will use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "counterfactual explanations to determine if recourse unfairness occurs. Thus, the adversarial model", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "score": 1.0, + "content": "owner is incentivized to construct a model that, when using existing counterfactual techniques, shows", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 498, + 490, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 490, + 510 + ], + "score": 1.0, + "content": "equal treatment of the populations to pass audits, yet can produce very low cost counterfactuals.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 431, + 506, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "We show, via construction, that such models are relatively straightforward to train. In our construction,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 255, + 538 + ], + "score": 1.0, + "content": "we jointly learn a perturbation vector", + "type": "text" + }, + { + "bbox": [ + 255, + 525, + 263, + 535 + ], + "score": 0.63, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 525, + 425, + 538 + ], + "score": 1.0, + "content": "(a small vector of the same dimension as", + "type": "text" + }, + { + "bbox": [ + 425, + 527, + 433, + 535 + ], + "score": 0.64, + "content": "{ \\pmb x }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 525, + 495, + 538 + ], + "score": 1.0, + "content": ") and the model", + "type": "text" + }, + { + "bbox": [ + 496, + 525, + 502, + 537 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 525, + 506, + 538 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 534, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 506, + 550 + ], + "score": 1.0, + "content": "such that the recourses computed by existing techniques look fair, but recourses computed by adding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 156, + 559 + ], + "score": 1.0, + "content": "perturbation", + "type": "text" + }, + { + "bbox": [ + 157, + 547, + 164, + 557 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "to the input data produces low cost recourses. In this way, the adversarial model owner", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 104, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "can perturb members of the non-protected group to generate low cost recourse and the model will", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 569, + 227, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 227, + 580 + ], + "score": 1.0, + "content": "look recourse fair to auditors.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 514, + 506, + 580 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "Motivating Example For a concrete example of a real model that meets this criteria, we refer to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 598, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 612 + ], + "score": 1.0, + "content": "Figure 2. When running an off-the-shelf counterfactual algorithm on the male and female instances", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "(representative of non-protected and protected group, respectively), we observe that the two recourses", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "are similar to each other. However, when the adversary changes the age of the male applicant by 0.5", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 441, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 199, + 644 + ], + "score": 1.0, + "content": "years (the perturbation", + "type": "text" + }, + { + "bbox": [ + 199, + 632, + 205, + 641 + ], + "score": 0.72, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 632, + 441, + 644 + ], + "score": 1.0, + "content": "), the recourse algorithm finds a much lower cost recourse.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 587, + 506, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 503, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 504, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 504, + 663 + ], + "score": 1.0, + "content": "Training Objective for Adversarial Model We define this construction formally using the combi-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 661, + 304, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 304, + 674 + ], + "score": 1.0, + "content": "nation of the following terms in the training loss:", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 106, + 650, + 504, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 677, + 506, + 724 + ], + "lines": [ + { + "bbox": [ + 108, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 108, + 676, + 318, + 690 + ], + "score": 1.0, + "content": "• Fairness: We want the counterfactual algorithm", + "type": "text" + }, + { + "bbox": [ + 318, + 678, + 327, + 688 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 676, + 412, + 690 + ], + "score": 1.0, + "content": "to be fair for model", + "type": "text" + }, + { + "bbox": [ + 412, + 678, + 420, + 690 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "according to Defini-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 116, + 688, + 475, + 701 + ], + "spans": [ + { + "bbox": [ + 116, + 688, + 475, + 701 + ], + "score": 1.0, + "content": "tion 2.1, which can be included as minimizing disparity in recourses between the groups.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 109, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 109, + 699, + 256, + 712 + ], + "score": 1.0, + "content": "• Unfairness: A perturbation vector", + "type": "text" + }, + { + "bbox": [ + 257, + 700, + 286, + 710 + ], + "score": 0.91, + "content": "\\pmb { \\delta } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "should lead to lower cost recourse when added to non-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 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{ + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 109, + 73, + 506, + 121 + ], + "lines": [ + { + "bbox": [ + 108, + 71, + 507, + 87 + ], + "spans": [ + { + "bbox": [ + 108, + 71, + 249, + 87 + ], + "score": 1.0, + "content": "• Small perturbation: Perturbation", + "type": "text" + }, + { + "bbox": [ + 249, + 74, + 256, + 83 + ], + "score": 0.77, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 71, + 422, + 87 + ], + "score": 1.0, + "content": "should be small. i.e. we need to minimize", + "type": "text" + }, + { + "bbox": [ + 423, + 73, + 503, + 87 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { { \\pmb x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } d ( { \\pmb x } , { \\pmb x } + \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 71, + 507, + 87 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 108, + 85, + 504, + 98 + ], + "spans": [ + { + "bbox": [ + 108, + 85, + 334, + 98 + ], + "score": 1.0, + "content": "• Accuracy: We should minimize the classification loss", + "type": "text" + }, + { + "bbox": [ + 334, + 86, + 343, + 96 + ], + "score": 0.77, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 85, + 490, + 98 + ], + "score": 1.0, + "content": "(such as cross entropy) of the model", + "type": "text" + }, + { + "bbox": [ + 491, + 87, + 498, + 98 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 85, + 504, + 98 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 108, + 97, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 108, + 97, + 185, + 109 + ], + "score": 1.0, + "content": "• Counterfactual:", + "type": "text" + }, + { + "bbox": [ + 186, + 97, + 218, + 109 + ], + "score": 0.9, + "content": "( { \\pmb x } + { \\pmb \\delta } )", + "type": "inline_equation" + }, + { 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_ { { \\pmb x } \\sim \\mathcal { D } _ { \\mathtt { n p } } ^ { \\mathtt { n e g } } } \\ ( f ( { \\pmb x } + { \\pmb \\delta } ) - 1 ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 106, + 382, + 122 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 437, + 138 + ], + "score": 1.0, + "content": "This combined training objective is defined over both the parameters of the model", + "type": "text" + }, + { + "bbox": [ + 437, + 127, + 444, + 136 + ], + "score": 0.73, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "and the pertur-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 163, + 150 + ], + "score": 1.0, + "content": "bation vector", + "type": "text" + }, + { + "bbox": [ + 163, + 137, + 169, + 147 + ], + "score": 0.62, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 136, + 505, + 150 + ], + "score": 1.0, + "content": ". Apart from requiring dual optimization over these two variables, the objective is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 243, + 160 + ], + "score": 1.0, + "content": "further complicated as it involves", + "type": "text" + }, + { + "bbox": [ + 243, + 149, + 252, + 158 + ], + "score": 0.75, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 148, + 505, + 160 + ], + "score": 1.0, + "content": ", a black-box counterfactual explanation approach. We address", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 251, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 251, + 171 + ], + "score": 1.0, + "content": "these challenges in the next section.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 506, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 507, + 195 + ], + "score": 1.0, + "content": "Training Adversarial Models Our optimization proceeds in two parts, dividing the terms depend-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 193, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 480, + 204 + ], + "score": 1.0, + "content": "ing on whether they involve the counterfactual terms or not. First, we optimize the perturbation", + "type": "text" + }, + { + "bbox": [ + 481, + 193, + 487, + 203 + ], + "score": 0.79, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 194, + 506, + 204 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 204, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 178, + 216 + ], + "score": 1.0, + "content": "model parameters", + "type": "text" + }, + { + "bbox": [ + 179, + 204, + 186, + 214 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 204, + 506, + 216 + ], + "score": 1.0, + "content": "on the subset of the terms that do not depend on the counterfactual algorithm, i.e.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 346, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 346, + 227 + ], + "score": 1.0, + "content": "optimizing accuracy, counterfactual, and perturbation size3:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 230, + 467, + 252 + ], + "lines": [ + { + "bbox": [ + 144, + 230, + 467, + 252 + ], + "spans": [ + { + "bbox": [ + 144, + 230, + 467, + 252 + ], + "score": 0.88, + "content": "\\delta : = \\mathrm { \\arg \\operatorname* { m i n } _ { \\delta } } \\operatorname* { m i n } _ { \\theta } \\mathcal { L } ( \\theta , \\mathcal { D } ) + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e } } } \\left( f ( \\boldsymbol { x } + \\delta ) - 1 \\right) ^ { 2 } + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e } } } d ( \\boldsymbol { x } , \\boldsymbol { x } + \\delta )", + "type": "interline_equation", + "image_path": "1ca7155313d0628d191f46fb60ff01ef999a526eb322fc94ff087ebe5b799926.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 144, + 230, + 467, + 252 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 506, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 239, + 269 + ], + "score": 1.0, + "content": "Second, we optimize parameters", + "type": "text" + }, + { + "bbox": [ + 239, + 258, + 246, + 267 + ], + "score": 0.76, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 258, + 342, + 269 + ], + "score": 1.0, + "content": ", fixing the perturbation", + "type": "text" + }, + { + "bbox": [ + 342, + 258, + 348, + 267 + ], + "score": 0.69, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 258, + 505, + 269 + ], + "score": 1.0, + "content": ". We still include the classification loss", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 372, + 281 + ], + "score": 1.0, + "content": "so that the model will be accurate, but also terms that depend on", + "type": "text" + }, + { + "bbox": [ + 373, + 269, + 383, + 279 + ], + "score": 0.77, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 268, + 418, + 281 + ], + "score": 1.0, + "content": "(we use", + "type": "text" + }, + { + "bbox": [ + 418, + 268, + 432, + 280 + ], + "score": 0.89, + "content": "\\scriptstyle { \\mathcal { A } } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 268, + 474, + 281 + ], + "score": 1.0, + "content": "to denote", + "type": "text" + }, + { + "bbox": [ + 474, + 269, + 484, + 279 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "uses", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 148, + 291 + ], + "score": 1.0, + "content": "the model", + "type": "text" + }, + { + "bbox": [ + 149, + 280, + 156, + 291 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 279, + 229, + 291 + ], + "score": 1.0, + "content": "parameterized by", + "type": "text" + }, + { + "bbox": [ + 229, + 280, + 236, + 289 + ], + "score": 0.72, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "). In particular, we add the two competing recourse fairness related", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "terms: reduced disparity between subgroups for the recourses on the original data and increasing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "disparity between subgroups by generating lower cost counterfactuals for the protected group when", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 353, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 172, + 324 + ], + "score": 1.0, + "content": "the perturbation", + "type": "text" + }, + { + "bbox": [ + 173, + 312, + 179, + 322 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 313, + 353, + 324 + ], + "score": 1.0, + "content": "is added to the instances. This objective is,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 330, + 514, + 352 + ], + "lines": [ + { + "bbox": [ + 111, + 330, + 514, + 352 + ], + "spans": [ + { + "bbox": [ + 111, + 330, + 514, + 352 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\boldsymbol { \\mathrm { \\theta } } : = \\arg \\underset { \\boldsymbol { \\theta } } { \\operatorname* { m i n } } \\ \\mathcal { L } ( \\boldsymbol { \\theta } , \\mathcal { D } ) + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } + \\boldsymbol { \\delta } ) \\right) \\right] + \\left( \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } ) \\right) \\right] - \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } ) \\right) \\right] \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "3b36a36cb970a3cc07a5e803f800c81a3284fb0935aff9a4250c718ca7bc77e1.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 111, + 330, + 514, + 352 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 367, + 425, + 383 + ], + "lines": [ + { + "bbox": [ + 185, + 367, + 425, + 383 + ], + "spans": [ + { + "bbox": [ + 185, + 367, + 425, + 383 + ], + "score": 0.87, + "content": "\\begin{array} { r l } { \\mathrm { s . t . } } & { { } \\mathbb { E } _ { x \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d \\left( { \\pmb x } , { \\mathcal { A } } _ { \\theta } ( { \\pmb x } + \\pmb \\delta ) \\right) \\right] < \\mathbb { E } _ { { \\pmb x } \\sim \\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { n e g } } } \\left[ d \\left( { \\pmb x } , { \\mathcal { A } } _ { \\theta } ( { \\pmb x } ) \\right) \\right] } \\end{array}", + "type": "interline_equation", + "image_path": "59a3a78f8fce33e8d8789928dd23096f7b2f350f703e9eed78bbc8968dac5266.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 185, + 367, + 425, + 383 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 387, + 506, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 506, + 400 + ], + "score": 1.0, + "content": "Optimizing this objective requires computing the derivative (Jacobian) of the counterfactual explana-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 394, + 509, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 124, + 418 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 125, + 400, + 139, + 411 + ], + "score": 0.88, + "content": "\\scriptstyle { \\mathcal { A } } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 394, + 201, + 418 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 201, + 400, + 208, + 410 + ], + "score": 0.61, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 394, + 212, + 418 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 213, + 398, + 252, + 413 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial \\pmb { \\theta } } \\mathcal { A } _ { \\pmb { \\theta } } ( \\pmb { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 394, + 509, + 418 + ], + "score": 1.0, + "content": ". Because counterfactual explanations use a variety of different", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "optimization strategies, computing this Jacobian would require access to the internal optimization", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "details of the implementation. For instance, some techniques use black box optimization while others", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "score": 1.0, + "content": "require gradient access. These details may vary by implementation or even be unavailable. Instead,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "we consider a solution based on implicit differentiation that decouples the Jacobian from choice of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 466 + ], + "score": 1.0, + "content": "optimization strategy for counterfactual explanations that follow the form in Eq. (1). We calculate the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 189, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 189, + 477 + ], + "score": 1.0, + "content": "Jacobian as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 329, + 501 + ], + "score": 1.0, + "content": "Lemma 3.1 Assuming the counterfactual explanation", + "type": "text" + }, + { + "bbox": [ + 330, + 488, + 358, + 500 + ], + "score": 0.92, + "content": "\\scriptstyle A _ { \\theta } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 487, + 505, + 501 + ], + "score": 1.0, + "content": "follows the form of the objective in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 497, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 156, + 514 + ], + "score": 1.0, + "content": "Equation 1,", + "type": "text" + }, + { + "bbox": [ + 157, + 497, + 506, + 516 + ], + "score": 1.0, + "content": "∂∂x G(x, Aθ(x)) = 0, and m is the number of parameters in the model, we can write", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 284, + 527 + ], + "score": 1.0, + "content": "the derivative of counterfactual explanation", + "type": "text" + }, + { + "bbox": [ + 284, + 515, + 293, + 524 + ], + "score": 0.76, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 513, + 429, + 527 + ], + "score": 1.0, + "content": "with respect to model parameters", + "type": "text" + }, + { + "bbox": [ + 429, + 515, + 437, + 524 + ], + "score": 0.48, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "as the Jacobian,", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 530, + 498, + 566 + ], + "lines": [ + { + "bbox": [ + 113, + 530, + 498, + 566 + ], + "spans": [ + { + "bbox": [ + 113, + 530, + 498, + 566 + ], + "score": 0.95, + "content": "\\frac { \\partial } { \\partial \\theta } \\mathcal { A } _ { \\theta } ( x ) = - \\left[ \\frac { \\partial ^ { 2 } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) } { d x _ { \\mathcal { I } } ^ { 2 } } \\right] ^ { - 1 } \\cdot \\left[ \\frac { \\partial } { \\partial \\theta _ { 1 } } \\frac { \\partial } { \\partial x _ { \\mathcal { I } } } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) \\cdot \\cdot \\cdot \\frac { \\partial } { \\partial \\theta _ { m } } \\frac { \\partial } { \\partial x _ { \\mathcal { I } } } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) \\right] .", + "type": "interline_equation", + "image_path": "efd0552342e9250763b404cbdf9a4e87e186df52faf82cbd656f6ba2914c9061.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 113, + 530, + 498, + 542.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 113, + 542.0, + 498, + 554.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 113, + 554.0, + 498, + 566.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 588, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 504, + 600 + ], + "score": 1.0, + "content": "We provide a proof in Appendix A. Critically, this objective does not depend on the implementation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 599, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 262, + 611 + ], + "score": 1.0, + "content": "details of counterfactual explanation", + "type": "text" + }, + { + "bbox": [ + 262, + 600, + 271, + 609 + ], + "score": 0.72, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 599, + 504, + 611 + ], + "score": 1.0, + "content": ", but only needs black box access to the counterfactual", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "explanation. One potential issue is the matrix inversion of the Hessian. Because we consider tabular", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 620, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 635 + ], + "score": 1.0, + "content": "data sets with relatively small feature sizes, this is not much of an issue. For larger feature sets, taking", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "the diagonal approximation of the Hessian has been shown to be a reasonable approximation [17, 18].", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "score": 1.0, + "content": "To provide an intuition as to how this objective exploits counterfactual explanations to train manipula-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 411, + 672 + ], + "score": 1.0, + "content": "tive models, we refer again to Figure 1. Because the counterfactual objective", + "type": "text" + }, + { + "bbox": [ + 411, + 660, + 420, + 670 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "relies on an arbitrary", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 142, + 683 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 671, + 149, + 682 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 671, + 395, + 683 + ], + "score": 1.0, + "content": ", this objective can be non-convex. As a result, we can design", + "type": "text" + }, + { + "bbox": [ + 396, + 671, + 403, + 682 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 671, + 442, + 683 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 442, + 671, + 452, + 681 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 681, + 491, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 290, + 694 + ], + "score": 1.0, + "content": "higher cost local minimums for all datapoints", + "type": "text" + }, + { + "bbox": [ + 290, + 682, + 318, + 692 + ], + "score": 0.91, + "content": "\\pmb { x } \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 681, + 362, + 694 + ], + "score": 1.0, + "content": "than those", + "type": "text" + 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181, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 181, + 723 + ], + "score": 1.0, + "content": "set everywhere else.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 109, + 73, + 506, + 121 + ], + "lines": [ + { + "bbox": [ + 108, + 71, + 507, + 87 + ], + "spans": [ + { + "bbox": [ + 108, + 71, + 249, + 87 + ], + "score": 1.0, + "content": "• Small perturbation: Perturbation", + "type": "text" + }, + { + "bbox": [ + 249, + 74, + 256, + 83 + ], + "score": 0.77, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 71, + 422, + 87 + ], + "score": 1.0, + "content": "should be small. i.e. we need to minimize", + "type": "text" + }, 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Apart from requiring dual optimization over these two variables, the objective is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 243, + 160 + ], + "score": 1.0, + "content": "further complicated as it involves", + "type": "text" + }, + { + "bbox": [ + 243, + 149, + 252, + 158 + ], + "score": 0.75, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 148, + 505, + 160 + ], + "score": 1.0, + "content": ", a black-box counterfactual explanation approach. We address", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 251, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 251, + 171 + ], + "score": 1.0, + "content": "these challenges in the next section.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 126, + 506, + 171 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 506, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 507, + 195 + ], + "score": 1.0, + "content": "Training Adversarial Models Our optimization proceeds in two parts, dividing the terms depend-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 193, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 480, + 204 + ], + "score": 1.0, + "content": "ing on whether they involve the counterfactual terms or not. First, we optimize the perturbation", + "type": "text" + }, + { + "bbox": [ + 481, + 193, + 487, + 203 + ], + "score": 0.79, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 194, + 506, + 204 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 204, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 178, + 216 + ], + "score": 1.0, + "content": "model parameters", + "type": "text" + }, + { + "bbox": [ + 179, + 204, + 186, + 214 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 204, + 506, + 216 + ], + "score": 1.0, + "content": "on the subset of the terms that do not depend on the counterfactual algorithm, i.e.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 346, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 346, + 227 + ], + "score": 1.0, + "content": "optimizing accuracy, counterfactual, and perturbation size3:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 182, + 507, + 227 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 230, + 467, + 252 + ], + "lines": [ + { + "bbox": [ + 144, + 230, + 467, + 252 + ], + "spans": [ + { + "bbox": [ + 144, + 230, + 467, + 252 + ], + "score": 0.88, + "content": "\\delta : = \\mathrm { \\arg \\operatorname* { m i n } _ { \\delta } } \\operatorname* { m i n } _ { \\theta } \\mathcal { L } ( \\theta , \\mathcal { D } ) + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e } } } \\left( f ( \\boldsymbol { x } + \\delta ) - 1 \\right) ^ { 2 } + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e } } } d ( \\boldsymbol { x } , \\boldsymbol { x } + \\delta )", + "type": "interline_equation", + "image_path": "1ca7155313d0628d191f46fb60ff01ef999a526eb322fc94ff087ebe5b799926.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 144, + 230, + 467, + 252 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 506, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 239, + 269 + ], + "score": 1.0, + "content": "Second, we optimize parameters", + "type": "text" + }, + { + "bbox": [ + 239, + 258, + 246, + 267 + ], + "score": 0.76, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 258, + 342, + 269 + ], + "score": 1.0, + "content": ", fixing the perturbation", + "type": "text" + }, + { + "bbox": [ + 342, + 258, + 348, + 267 + ], + "score": 0.69, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 258, + 505, + 269 + ], + "score": 1.0, + "content": ". We still include the classification loss", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 372, + 281 + ], + "score": 1.0, + "content": "so that the model will be accurate, but also terms that depend on", + "type": "text" + }, + { + "bbox": [ + 373, + 269, + 383, + 279 + ], + "score": 0.77, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 268, + 418, + 281 + ], + "score": 1.0, + "content": "(we use", + "type": "text" + }, + { + "bbox": [ + 418, + 268, + 432, + 280 + ], + "score": 0.89, + "content": "\\scriptstyle { \\mathcal { A } } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 268, + 474, + 281 + ], + "score": 1.0, + "content": "to denote", + "type": "text" + }, + { + "bbox": [ + 474, + 269, + 484, + 279 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "uses", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 148, + 291 + ], + "score": 1.0, + "content": "the model", + "type": "text" + }, + { + "bbox": [ + 149, + 280, + 156, + 291 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 279, + 229, + 291 + ], + "score": 1.0, + "content": "parameterized by", + "type": "text" + }, + { + "bbox": [ + 229, + 280, + 236, + 289 + ], + "score": 0.72, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "). In particular, we add the two competing recourse fairness related", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "terms: reduced disparity between subgroups for the recourses on the original data and increasing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "disparity between subgroups by generating lower cost counterfactuals for the protected group when", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 353, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 172, + 324 + ], + "score": 1.0, + "content": "the perturbation", + "type": "text" + }, + { + "bbox": [ + 173, + 312, + 179, + 322 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 313, + 353, + 324 + ], + "score": 1.0, + "content": "is added to the instances. This objective is,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 258, + 506, + 324 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 330, + 514, + 352 + ], + "lines": [ + { + "bbox": [ + 111, + 330, + 514, + 352 + ], + "spans": [ + { + "bbox": [ + 111, + 330, + 514, + 352 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\boldsymbol { \\mathrm { \\theta } } : = \\arg \\underset { \\boldsymbol { \\theta } } { \\operatorname* { m i n } } \\ \\mathcal { L } ( \\boldsymbol { \\theta } , \\mathcal { D } ) + \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } + \\boldsymbol { \\delta } ) \\right) \\right] + \\left( \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } ) \\right) \\right] - \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathfrak { w } } ^ { \\mathrm { w e } } } \\left[ d \\left( \\boldsymbol { x } , \\mathcal { A } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } ) \\right) \\right] \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "3b36a36cb970a3cc07a5e803f800c81a3284fb0935aff9a4250c718ca7bc77e1.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 111, + 330, + 514, + 352 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 367, + 425, + 383 + ], + "lines": [ + { + "bbox": [ + 185, + 367, + 425, + 383 + ], + "spans": [ + { + "bbox": [ + 185, + 367, + 425, + 383 + ], + "score": 0.87, + "content": "\\begin{array} { r l } { \\mathrm { s . t . } } & { { } \\mathbb { E } _ { x \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d \\left( { \\pmb x } , { \\mathcal { A } } _ { \\theta } ( { \\pmb x } + \\pmb \\delta ) \\right) \\right] < \\mathbb { E } _ { { \\pmb x } \\sim \\mathcal { D } _ { \\mathrm { p r } } ^ { \\mathrm { n e g } } } \\left[ d \\left( { \\pmb x } , { \\mathcal { A } } _ { \\theta } ( { \\pmb x } ) \\right) \\right] } \\end{array}", + "type": "interline_equation", + "image_path": "59a3a78f8fce33e8d8789928dd23096f7b2f350f703e9eed78bbc8968dac5266.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 185, + 367, + 425, + 383 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 387, + 506, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 506, + 400 + ], + "score": 1.0, + "content": "Optimizing this objective requires computing the derivative (Jacobian) of the counterfactual explana-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 394, + 509, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 124, + 418 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 125, + 400, + 139, + 411 + ], + "score": 0.88, + "content": "\\scriptstyle { \\mathcal { A } } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 394, + 201, + 418 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 201, + 400, + 208, + 410 + ], + "score": 0.61, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 394, + 212, + 418 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 213, + 398, + 252, + 413 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial \\pmb { \\theta } } \\mathcal { A } _ { \\pmb { \\theta } } ( \\pmb { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 394, + 509, + 418 + ], + "score": 1.0, + "content": ". Because counterfactual explanations use a variety of different", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "optimization strategies, computing this Jacobian would require access to the internal optimization", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "details of the implementation. For instance, some techniques use black box optimization while others", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "score": 1.0, + "content": "require gradient access. These details may vary by implementation or even be unavailable. Instead,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "we consider a solution based on implicit differentiation that decouples the Jacobian from choice of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 466 + ], + "score": 1.0, + "content": "optimization strategy for counterfactual explanations that follow the form in Eq. (1). We calculate the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 189, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 189, + 477 + ], + "score": 1.0, + "content": "Jacobian as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 388, + 509, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 329, + 501 + ], + "score": 1.0, + "content": "Lemma 3.1 Assuming the counterfactual explanation", + "type": "text" + }, + { + "bbox": [ + 330, + 488, + 358, + 500 + ], + "score": 0.92, + "content": "\\scriptstyle A _ { \\theta } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 487, + 505, + 501 + ], + "score": 1.0, + "content": "follows the form of the objective in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 497, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 156, + 514 + ], + "score": 1.0, + "content": "Equation 1,", + "type": "text" + }, + { + "bbox": [ + 157, + 497, + 506, + 516 + ], + "score": 1.0, + "content": "∂∂x G(x, Aθ(x)) = 0, and m is the number of parameters in the model, we can write", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 284, + 527 + ], + "score": 1.0, + "content": "the derivative of counterfactual explanation", + "type": "text" + }, + { + "bbox": [ + 284, + 515, + 293, + 524 + ], + "score": 0.76, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 513, + 429, + 527 + ], + "score": 1.0, + "content": "with respect to model parameters", + "type": "text" + }, + { + "bbox": [ + 429, + 515, + 437, + 524 + ], + "score": 0.48, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "as the Jacobian,", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 487, + 506, + 527 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 530, + 498, + 566 + ], + "lines": [ + { + "bbox": [ + 113, + 530, + 498, + 566 + ], + "spans": [ + { + "bbox": [ + 113, + 530, + 498, + 566 + ], + "score": 0.95, + "content": "\\frac { \\partial } { \\partial \\theta } \\mathcal { A } _ { \\theta } ( x ) = - \\left[ \\frac { \\partial ^ { 2 } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) } { d x _ { \\mathcal { I } } ^ { 2 } } \\right] ^ { - 1 } \\cdot \\left[ \\frac { \\partial } { \\partial \\theta _ { 1 } } \\frac { \\partial } { \\partial x _ { \\mathcal { I } } } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) \\cdot \\cdot \\cdot \\frac { \\partial } { \\partial \\theta _ { m } } \\frac { \\partial } { \\partial x _ { \\mathcal { I } } } G \\left( x , \\mathcal { A } _ { \\theta } ( x ) \\right) \\right] .", + "type": "interline_equation", + "image_path": "efd0552342e9250763b404cbdf9a4e87e186df52faf82cbd656f6ba2914c9061.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 113, + 530, + 498, + 542.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 113, + 542.0, + 498, + 554.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 113, + 554.0, + 498, + 566.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 588, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 504, + 600 + ], + "score": 1.0, + "content": "We provide a proof in Appendix A. Critically, this objective does not depend on the implementation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 599, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 262, + 611 + ], + "score": 1.0, + "content": "details of counterfactual explanation", + "type": "text" + }, + { + "bbox": [ + 262, + 600, + 271, + 609 + ], + "score": 0.72, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 599, + 504, + 611 + ], + "score": 1.0, + "content": ", but only needs black box access to the counterfactual", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "explanation. One potential issue is the matrix inversion of the Hessian. 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For larger feature sets, taking", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "the diagonal approximation of the Hessian has been shown to be a reasonable approximation [17, 18].", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 588, + 506, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "score": 1.0, + "content": "To provide an intuition as to how this objective exploits counterfactual explanations to train manipula-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 411, + 672 + ], + "score": 1.0, + "content": "tive models, we refer again to Figure 1. 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Comm. & CrimeGerman Credit
Acc|10||1Acc|0||1
Unmodified81.2=71.1-
Wachter et al.80.90.8072.00.09
Sparse Wachter77.90.4670.52.50
Prototypes79.20.4669.02.21
DiCE81.11.7371.20.09
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Note,", + "type": "text" + }, + { + "bbox": [ + 413, + 147, + 420, + 157 + ], + "score": 0.56, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 145, + 506, + 159 + ], + "score": 1.0, + "content": "is comparable across", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 333, + 156, + 482, + 170 + ], + "spans": [ + { + "bbox": [ + 333, + 156, + 482, + 170 + ], + "score": 1.0, + "content": "datasets due to unit variance scaling.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 204, + 218, + 218 + ], + "lines": [ + { + "bbox": [ + 104, + 201, + 221, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 201, + 221, + 222 + ], + "score": 1.0, + "content": "4 Experiment Setup", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 252 + ], + "score": 1.0, + "content": "We use the following setup, including multiple counterfactual explanation techniques on two datasets,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 249, + 336, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 336, + 261 + ], + "score": 1.0, + "content": "to evaluate the proposed approach of training the models.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 267, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "Counterfactual Explanations We consider four different counterfactual explanation algorithms as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 277, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 167, + 292 + ], + "score": 1.0, + "content": "the choices for", + "type": "text" + }, + { + "bbox": [ + 167, + 279, + 176, + 289 + ], + "score": 0.69, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 277, + 506, + 292 + ], + "score": 1.0, + "content": "that hill-climb the counterfactual objective. We use Wachter et al.’s Algorithm [6],", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 289, + 507, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 507, + 303 + ], + "score": 1.0, + "content": "Wachter et al.’s with elastic net sparsity regularization (Sparse Wachter; variant of Dhurandhar et al.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "[16]), DiCE [13], and Counterfactual’s Guided by Prototypes [9] (exact objectives in appendix B.1).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "These counterfactual explanations are widely used to compute recourse and assess the fairness of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 214, + 336 + ], + "score": 1.0, + "content": "models [3, 19, 20]. 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DiCE is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "the only approach that computes multiple counterfactual explanations; we generate 4 counterfactuals", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 354, + 483, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 314, + 368 + ], + "score": 1.0, + "content": "and take the closest one to the original point (as per", + "type": "text" + }, + { + "bbox": [ + 314, + 356, + 324, + 366 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 354, + 483, + 368 + ], + "score": 1.0, + "content": "distance) to get a single counterfactual.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "Data sets We use two data sets: Communities and Crime and the German Credit datasets [21], as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 386, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 397 + ], + "score": 1.0, + "content": "they are commonly used benchmarks in both the counterfactual explanation and fairness literature", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "score": 1.0, + "content": "[19, 22]. 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In communities and crime, communities assessed at higher risks for crime could be subject to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "reduced funding for desirable programs, incentivizing being predicted at low risk of violent crime", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "[23], while in German credit, it is more desirable to receive a loan. We preprocess the data as in Slack", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 431, + 496 + ], + "score": 1.0, + "content": "et al. [24], and apply 0 mean, unit variance scaling to the features and perform an", + "type": "text" + }, + { + "bbox": [ + 431, + 483, + 457, + 495 + ], + "score": 0.3, + "content": "8 0 / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "split on the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "data to create training and testing sets. 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In German Credit, we use Gender as the sensitive attribute (Female as the protected class)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 524, + 507, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 507, + 542 + ], + "score": 1.0, + "content": "and treat low credit risk as the positive outcome. We compute counterfactuals on each data set using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "the numerical features. The numerical features include all 99 features for Communities and Crime", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "and 7 of 27 total features for German Credit. 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Comm. & CrimeGerman Credit
Acc|10||1Acc|0||1
Unmodified81.2=71.1-
Wachter et al.80.90.8072.00.09
Sparse Wachter77.90.4670.52.50
Prototypes79.20.4669.02.21
DiCE81.11.7371.20.09
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Note,", + "type": "text" + }, + { + "bbox": [ + 413, + 147, + 420, + 157 + ], + "score": 0.56, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 145, + 506, + 159 + ], + "score": 1.0, + "content": "is comparable across", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 333, + 156, + 482, + 170 + ], + "spans": [ + { + "bbox": [ + 333, + 156, + 482, + 170 + ], + "score": 1.0, + "content": "datasets due to unit variance scaling.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 333, + 70, + 506, + 170 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 204, + 218, + 218 + ], + "lines": [ + { + "bbox": [ + 104, + 201, + 221, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 201, + 221, + 222 + ], + "score": 1.0, + "content": "4 Experiment Setup", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 252 + ], + "score": 1.0, + "content": "We use the following setup, including multiple counterfactual explanation techniques on two datasets,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 249, + 336, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 336, + 261 + ], + "score": 1.0, + "content": "to evaluate the proposed approach of training the models.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 235, + 506, + 261 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 267, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "Counterfactual Explanations We consider four different counterfactual explanation algorithms as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 277, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 167, + 292 + ], + "score": 1.0, + "content": "the choices for", + "type": "text" + }, + { + "bbox": [ + 167, + 279, + 176, + 289 + ], + "score": 0.69, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 277, + 506, + 292 + ], + "score": 1.0, + "content": "that hill-climb the counterfactual objective. We use Wachter et al.’s Algorithm [6],", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 289, + 507, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 507, + 303 + ], + "score": 1.0, + "content": "Wachter et al.’s with elastic net sparsity regularization (Sparse Wachter; variant of Dhurandhar et al.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "[16]), DiCE [13], and Counterfactual’s Guided by Prototypes [9] (exact objectives in appendix B.1).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "These counterfactual explanations are widely used to compute recourse and assess the fairness of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 214, + 336 + ], + "score": 1.0, + "content": "models [3, 19, 20]. We use", + "type": "text" + }, + { + "bbox": [ + 214, + 323, + 221, + 333 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 321, + 505, + 336 + ], + "score": 1.0, + "content": "to compute the cost of a recourse discovered by counterfactuals. We use", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 332, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 346 + ], + "score": 1.0, + "content": "the official DiCE implementation4, and reimplement the others (details in Appendix B.2). DiCE is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "the only approach that computes multiple counterfactual explanations; we generate 4 counterfactuals", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 354, + 483, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 314, + 368 + ], + "score": 1.0, + "content": "and take the closest one to the original point (as per", + "type": "text" + }, + { + "bbox": [ + 314, + 356, + 324, + 366 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 354, + 483, + 368 + ], + "score": 1.0, + "content": "distance) to get a single counterfactual.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 267, + 507, + 368 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "Data sets We use two data sets: Communities and Crime and the German Credit datasets [21], as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 386, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 397 + ], + "score": 1.0, + "content": "they are commonly used benchmarks in both the counterfactual explanation and fairness literature", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "score": 1.0, + "content": "[19, 22]. Both these datasets are in the public domain. Communities and Crime contains demographic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "and economic information about communities across the United States, with the goal to predict", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "whether there is violent crime in the community. The German credit dataset includes financial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "information about individuals, and we predict whether the person is of high credit risk. There are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "strong incentives to “game the system” in both these datasets, making them good choices for this", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "attack. In communities and crime, communities assessed at higher risks for crime could be subject to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "reduced funding for desirable programs, incentivizing being predicted at low risk of violent crime", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "[23], while in German credit, it is more desirable to receive a loan. We preprocess the data as in Slack", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 431, + 496 + ], + "score": 1.0, + "content": "et al. [24], and apply 0 mean, unit variance scaling to the features and perform an", + "type": "text" + }, + { + "bbox": [ + 431, + 483, + 457, + 495 + ], + "score": 0.3, + "content": "8 0 / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "split on the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "data to create training and testing sets. 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In German Credit, we use Gender as the sensitive attribute (Female as the protected class)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 524, + 507, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 507, + 542 + ], + "score": 1.0, + "content": "and treat low credit risk as the positive outcome. We compute counterfactuals on each data set using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "the numerical features. The numerical features include all 99 features for Communities and Crime", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "and 7 of 27 total features for German Credit. 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It is common to use neural networks when requiring counterfactuals since they are", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "differentiable, enabling counterfactual discovery via gradient descent [13]. We perform the first", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "part of optimization for 10, 000 steps for Communities and Crime and German Credit. We train the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "second part of the optimization for 15 steps. 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Communities and CrimeGerman Credit
Wach.S-Wach.Proto.DiCEWach. S-Wach.Proto.DiCE
Protected35.6854.1622.3549.625.658.3510.516.31
Non-Protected35.3152.0522.6542.635.088.5913.986.81
Disparity0.372.120.306.990.750.240.060.5
Non-Protected+δ1.7622.598.509.573.164.124.693.38
Cost reduction20.1×2.3×2.6×4.5×1.8×2.0x2.2×2.0×
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We also measure the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 189, + 353 + ], + "score": 1.0, + "content": "average costs (using", + "type": "text" + }, + { + "bbox": [ + 189, + 342, + 196, + 351 + ], + "score": 0.52, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 341, + 478, + 353 + ], + "score": 1.0, + "content": ") for the non-protected group and the non-protected group perturbed by", + "type": "text" + }, + { + "bbox": [ + 479, + 341, + 485, + 351 + ], + "score": 0.64, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 341, + 505, + 353 + ], + "score": 1.0, + "content": ". 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The disparity in counterfactual cost on the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "unperturbed data is very small in most cases, indicating the models would appear counterfactual fair", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "to the auditors. 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The adversarial model is considerably more", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "effective applied on Wachter et al.’s algorithm in Communities and Crime. The success of the model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "in this setting could be attributed to the simplicity of the objective. The Wachter et al. objective", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 309, + 520 + ], + "score": 1.0, + "content": "only considers the squared loss (i.e., Eq (1)) and", + "type": "text" + }, + { + "bbox": [ + 309, + 507, + 320, + 518 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "distance, whereas counterfactuals guided by", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 519, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 530 + ], + "score": 1.0, + "content": "prototypes takes into account closeness to the data manifold. 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The relative success is likely", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 539, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 553 + ], + "score": 1.0, + "content": "due to Communities and Crime having a larger number of features than German Credit (99 versus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "7), making it easier to learn a successful adversarial model due to the higher dimensional space.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "Overall, these results demonstrate the adversarial models work quite successfully at manipulating the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 221, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 221, + 585 + ], + "score": 1.0, + "content": "counterfactual explanations.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 596, + 273, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 275, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 275, + 609 + ], + "score": 1.0, + "content": "5.2 Outlier Factor of Counterfactuals", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 617, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "One potential concern is that the manipulated models returns counterfactuals that are out of distribu-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "tion, resulting in unrealistic recourses. 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Communities and CrimeGerman Credit
Wach.S-Wach.Proto.DiCEWach. S-Wach.Proto.DiCE
Protected35.6854.1622.3549.625.658.3510.516.31
Non-Protected35.3152.0522.6542.635.088.5913.986.81
Disparity0.372.120.306.990.750.240.060.5
Non-Protected+δ1.7622.598.509.573.164.124.693.38
Cost reduction20.1×2.3×2.6×4.5×1.8×2.0x2.2×2.0×
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We", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 351, + 392, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 392, + 364 + ], + "score": 1.0, + "content": "use the ratio between these costs as metric for success of manipulation,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 307, + 506, + 364 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 368, + 398, + 398 + ], + "lines": [ + { + "bbox": [ + 212, + 368, + 398, + 398 + ], + "spans": [ + { + "bbox": [ + 212, + 368, + 398, + 398 + ], + "score": 0.93, + "content": "\\mathrm { C o s t r e d u c t i o n : = \\frac { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d ( \\boldsymbol { x } , \\boldsymbol { A } ( \\boldsymbol { x } ) ) \\right] } { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathcal { D } _ { \\mathrm { n p } } ^ { \\mathrm { n e g } } } \\left[ d ( \\boldsymbol { x } , \\boldsymbol { A } ( \\boldsymbol { x } + \\boldsymbol { \\delta } ) ) \\right] } . }", + "type": "interline_equation", + "image_path": "3ca2a57aea0939d03ffb43f857968bb104b6c50853b90ff8176b4c82c25a29fd.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 368, + 398, + 383.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 212, + 383.0, + 398, + 398.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 503, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "If the manipulation is successful, we expect the non-protected group to have much lower cost with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 414, + 389, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 172, + 427 + ], + "score": 1.0, + "content": "the perturbation", + "type": "text" + }, + { + "bbox": [ + 172, + 415, + 179, + 424 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 414, + 389, + 427 + ], + "score": 1.0, + "content": "than without, and thus the cost reduction to be high.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 403, + 505, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 506, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 431, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 442 + ], + "score": 1.0, + "content": "We provide the results for both datasets in Table 2. The disparity in counterfactual cost on the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "unperturbed data is very small in most cases, indicating the models would appear counterfactual fair", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "to the auditors. At the same time, we observe that the cost reduction in the counterfactual distances", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 342, + 476 + ], + "score": 1.0, + "content": "for the non-protected groups after applying the perturbation", + "type": "text" + }, + { + "bbox": [ + 343, + 464, + 349, + 473 + ], + "score": 0.76, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "is quite high, indicating that lower cost", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "recourses are easy to compute for non-protected groups. The adversarial model is considerably more", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "effective applied on Wachter et al.’s algorithm in Communities and Crime. The success of the model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "in this setting could be attributed to the simplicity of the objective. The Wachter et al. objective", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 309, + 520 + ], + "score": 1.0, + "content": "only considers the squared loss (i.e., Eq (1)) and", + "type": "text" + }, + { + "bbox": [ + 309, + 507, + 320, + 518 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "distance, whereas counterfactuals guided by", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 519, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 530 + ], + "score": 1.0, + "content": "prototypes takes into account closeness to the data manifold. Also, all adversarial models are more", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "successful applied to Communities and Crime than German Credit. The relative success is likely", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 539, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 553 + ], + "score": 1.0, + "content": "due to Communities and Crime having a larger number of features than German Credit (99 versus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "7), making it easier to learn a successful adversarial model due to the higher dimensional space.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "Overall, these results demonstrate the adversarial models work quite successfully at manipulating the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 221, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 221, + 585 + ], + "score": 1.0, + "content": "counterfactual explanations.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 431, + 506, + 585 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 596, + 273, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 275, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 275, + 609 + ], + "score": 1.0, + "content": "5.2 Outlier Factor of Counterfactuals", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 617, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "One potential concern is that the manipulated models returns counterfactuals that are out of distribu-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "tion, resulting in unrealistic recourses. To evaluate whether this is the case, we follow Pawelczyk", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 638, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 652 + ], + "score": 1.0, + "content": "et al. [25], and compute the local outlier factor of the counterfactuals with respect to the positively", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 649, + 403, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 327, + 663 + ], + "score": 1.0, + "content": "classified data [26]. 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Model InitializationWachter et al.DiCE Rnd.
MeanRnd. x+N
Protected42.416.2 11.748.3
Not-Prot.42.315.7 10.342.3
Disparity0.010.49 1.455.95
Not-Prot.+δ2.503.79 8.5912.3
Cost reduction16.9×4.3× 1.2×3.4×
Accuracy81.480.2 75.378.9
|18|l10.650.65 0.361.24
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Each provides a potentially viable defense, with different trade-offs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "Communities and Crime, in Figure 3 (results from additional datasets/methods in Appendix E).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "We see the counterfactuals of the adversarial models appear more in-distribution than those of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "unmodified model. These results demonstrate the manipulated models do not produce counterfactuals", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 451, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 451, + 528 + ], + "score": 1.0, + "content": "that are unrealistic due to training on the manipulative objective, as may be a concern.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 542, + 260, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 262, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 262, + 557 + ], + "score": 1.0, + "content": "5.3 Potential Mitigation Strategies", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 105, + 563, + 496, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 497, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 497, + 576 + ], + "score": 1.0, + "content": "In this section, we explore a number of constraints that could lead to more robust counterfactuals.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 594 + ], + "score": 1.0, + "content": "Search Initialization Strategies Our analysis assumes that the search for the counterfactual ex-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 317, + 606 + ], + "score": 1.0, + "content": "planation initializes at the original data point (i.e.,", + "type": "text" + }, + { + "bbox": [ + 317, + 595, + 325, + 604 + ], + "score": 0.63, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 593, + 338, + 606 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 339, + 594, + 366, + 604 + ], + "score": 0.89, + "content": "{ \\pmb x } + \\delta _ { \\mathrm { { \\scriptscriptstyle f } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "), as is common in counterfactual", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "score": 1.0, + "content": "explanations. Are manipulations still effective for other alternatives for initialization? We consider", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "score": 1.0, + "content": "three different initialization schemes and examine the effectiveness of the Wachter et al. and DiCE", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 346, + 638 + ], + "score": 1.0, + "content": "Communities and Crime Adversarial Model: (1) Randomly", + "type": "text" + }, + { + "bbox": [ + 347, + 626, + 371, + 637 + ], + "score": 0.86, + "content": "( \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 626, + 505, + 638 + ], + "score": 1.0, + "content": ", (2) at the Mean of the positively", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 352, + 649 + ], + "score": 1.0, + "content": "predicted data, and (3) at a perturbation of the data point using", + "type": "text" + }, + { + "bbox": [ + 352, + 637, + 384, + 649 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "noise. 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From the results in Figure", + "type": "text" + }, + { + "bbox": [ + 325, + 671, + 336, + 680 + ], + "score": 0.28, + "content": "4 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 670, + 505, + 682 + ], + "score": 1.0, + "content": ", we see perturbing the data before search", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "reduces the cost reduction most effectively. 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Model InitializationWachter et al.DiCE Rnd.
MeanRnd. x+N
Protected42.416.2 11.748.3
Not-Prot.42.315.7 10.342.3
Disparity0.010.49 1.455.95
Not-Prot.+δ2.503.79 8.5912.3
Cost reduction16.9×4.3× 1.2×3.4×
Accuracy81.480.2 75.378.9
|18|l10.650.65 0.361.24
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Each provides a potentially viable defense, with different trade-offs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 427, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "Communities and Crime, in Figure 3 (results from additional datasets/methods in Appendix E).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "We see the counterfactuals of the adversarial models appear more in-distribution than those of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "unmodified model. These results demonstrate the manipulated models do not produce counterfactuals", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 451, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 451, + 528 + ], + "score": 1.0, + "content": "that are unrealistic due to training on the manipulative objective, as may be a concern.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 483, + 506, + 528 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 542, + 260, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 262, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 262, + 557 + ], + "score": 1.0, + "content": "5.3 Potential Mitigation Strategies", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 105, + 563, + 496, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 497, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 497, + 576 + ], + "score": 1.0, + "content": "In this section, we explore a number of constraints that could lead to more robust counterfactuals.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 562, + 497, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 594 + ], + "score": 1.0, + "content": "Search Initialization Strategies Our analysis assumes that the search for the counterfactual ex-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 317, + 606 + ], + "score": 1.0, + "content": "planation initializes at the original data point (i.e.,", + "type": "text" + }, + { + "bbox": [ + 317, + 595, + 325, + 604 + ], + "score": 0.63, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 593, + 338, + 606 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 339, + 594, + 366, + 604 + ], + "score": 0.89, + "content": "{ \\pmb x } + \\delta _ { \\mathrm { { \\scriptscriptstyle f } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "), as is common in counterfactual", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "score": 1.0, + "content": "explanations. Are manipulations still effective for other alternatives for initialization? We consider", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 626 + ], + "score": 1.0, + "content": "three different initialization schemes and examine the effectiveness of the Wachter et al. and DiCE", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 346, + 638 + ], + "score": 1.0, + "content": "Communities and Crime Adversarial Model: (1) Randomly", + "type": "text" + }, + { + "bbox": [ + 347, + 626, + 371, + 637 + ], + "score": 0.86, + "content": "( \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 626, + 505, + 638 + ], + "score": 1.0, + "content": ", (2) at the Mean of the positively", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 352, + 649 + ], + "score": 1.0, + "content": "predicted data, and (3) at a perturbation of the data point using", + "type": "text" + }, + { + "bbox": [ + 352, + 637, + 384, + 649 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "noise. To initialize Wachter et", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "al. randomly, we follow Mothilal et al. [13] and draw a random instance from a uniform distribution", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 659, + 504, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 504, + 671 + ], + "score": 1.0, + "content": "on the maximum and minimum of each feature (DiCE provides an option to initialize randomly, we", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 325, + 682 + ], + "score": 1.0, + "content": "use just this initialization). From the results in Figure", + "type": "text" + }, + { + "bbox": [ + 325, + 671, + 336, + 680 + ], + "score": 0.28, + "content": "4 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 670, + 505, + 682 + ], + "score": 1.0, + "content": ", we see perturbing the data before search", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "reduces the cost reduction most effectively. We find similar results for German Credit in appendix E.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 583, + 506, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 504, + 711 + ], + "score": 1.0, + "content": "Number of Attributes We consider reducing the number of attributes used to find counterfactuals", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "and evaluate the success of the adversarial model on Wachter et al.’s algorithm for the Communities", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 74, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 74, + 505, + 86 + ], + "score": 1.0, + "content": "and Crime dataset. Starting with the original number of attributes, 99, we randomly select 10", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 85, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 85, + 505, + 97 + ], + "score": 1.0, + "content": "attributes, remove them from the set of attributes used by the counterfactual algorithm, and train", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 96, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 506, + 108 + ], + "score": 1.0, + "content": "an adversarial model. We repeat this process until we have 59 attributes left. We report the cost", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 174, + 118 + ], + "score": 1.0, + "content": "reduction due to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 174, + 106, + 181, + 117 + ], + "score": 0.57, + "content": "\\pmb { \\delta }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 181, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "(Eq (5)) for each model, averaged over 5 runs. We observe that we are unable to", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 118, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 129 + ], + "score": 1.0, + "content": "find low cost recourses for adversarial model as we reduce the number of attributes, with minimal", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 129, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 141 + ], + "score": 1.0, + "content": "impact on accuracy (not in figure). This suggests the counterfactual explanations are more robust", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "when they are constrained. In safety concerned settings, we thus recommend using a minimal number", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 159, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 159, + 162 + ], + "score": 1.0, + "content": "of attributes.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 49.5, + "bbox_fs": [ + 106, + 700, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 74, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 74, + 505, + 86 + ], + "score": 1.0, + "content": "and Crime dataset. Starting with the original number of attributes, 99, we randomly select 10", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 85, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 85, + 505, + 97 + ], + "score": 1.0, + "content": "attributes, remove them from the set of attributes used by the counterfactual algorithm, and train", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 96, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 506, + 108 + ], + "score": 1.0, + "content": "an adversarial model. We repeat this process until we have 59 attributes left. We report the cost", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 174, + 118 + ], + "score": 1.0, + "content": "reduction due to", + "type": "text" + }, + { + "bbox": [ + 174, + 106, + 181, + 117 + ], + "score": 0.57, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "(Eq (5)) for each model, averaged over 5 runs. We observe that we are unable to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 118, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 129 + ], + "score": 1.0, + "content": "find low cost recourses for adversarial model as we reduce the number of attributes, with minimal", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 129, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 141 + ], + "score": 1.0, + "content": "impact on accuracy (not in figure). This suggests the counterfactual explanations are more robust", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "when they are constrained. In safety concerned settings, we thus recommend using a minimal number", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 159, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 159, + 162 + ], + "score": 1.0, + "content": "of attributes.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 169, + 505, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "score": 1.0, + "content": "Size of the Model To further characterize the manipulation, we train a number of models (on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "Communities and Crime for Wachter et al.’s) that vary in their size. We show that as we increase the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 338, + 203 + ], + "score": 1.0, + "content": "model size, we gain an even higher cost reduction, i.e. an", + "type": "text" + }, + { + "bbox": [ + 339, + 191, + 360, + 202 + ], + "score": 0.88, + "content": "1 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 192, + 505, + 203 + ], + "score": 1.0, + "content": "increase in the cost reduction when", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "the similar additional parameters are added. This is not surprising, since more parameters provide", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "further the flexibility to distort the decision surface as needed. As we reduce the size of the model, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 382, + 236 + ], + "score": 1.0, + "content": "see the opposite trend; the cost reduction reduces substantially when", + "type": "text" + }, + { + "bbox": [ + 382, + 224, + 396, + 234 + ], + "score": 0.86, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "fewer parameters are used.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "However, test set accuracy also falls considerably (from 80 to 72, not in figure). These results suggest", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 245, + 484, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 484, + 258 + ], + "score": 1.0, + "content": "it is safest to use as compact of a model as meets the accuracy requirements of the application.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 108, + 264, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "Takeaways These results provide three main options to increase the robustness of counterfactual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "explanations to manipulation: add a random perturbation to the counterfactual search, use a minimal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 287, + 480, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 480, + 299 + ], + "score": 1.0, + "content": "number of attributes in the counterfactual search, or enforce the use of a less complex model.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 314, + 197, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 198, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 198, + 329 + ], + "score": 1.0, + "content": "6 Related Work", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "Recourse Methods A variety of methods have been proposed to generate recourse for affected", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "individuals [6, 1, 7–9]. Wachter et al. [6] propose gradient search for the closest counterfactual,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "while Ustun et al. [1] introduce the notion of actionable recourse for linear classifiers and propose", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "techniques to find such recourse using linear programming. Because counterfactuals generated by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "these techniques may produce unrealistic recommendations, Van Looveren and Klaise [9] incorporate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 398, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 408 + ], + "score": 1.0, + "content": "constraints in the counterfactual search to encourage them to be in-distribution. Similarly, other", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "approaches incorporate causality in order to avoid such spurious counterfactuals [27, 12, 15]. Further", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "works introduce notions of fairness associated with recourse. For instance, Ustun et al. [1] demonstrate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 507, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 507, + 442 + ], + "score": 1.0, + "content": "disparities in the cost of recourse between groups, which Sharma et al. [4] use to evaluate fairness.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "score": 1.0, + "content": "Gupta et al. [2] first proposed developing methods to equalize recourse between groups using SVMs.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "Karimi et al. [3] establish the notion of fairness of recourse and demonstrate it is distinct from fairness", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 462, + 495, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 495, + 474 + ], + "score": 1.0, + "content": "of predictions. Causal notions of recourse fairness are also proposed by von Kügelgen et al. [28].", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "Shortcomings of Explanations Pawelczyk et al. [11] discuss counterfactuals under predictive", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "multiplicity [29] and demonstrate counterfactuals may not transfer across equally good models.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "score": 1.0, + "content": "Rawal et al. [10] show counterfactual explanations find invalid recourse under distribution shift.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "Kasirzadeh and Smart [30] consider how counterfactual explanations are currently misused and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "propose tenents to better guide their use. Work on strategic behavior considers how individuals might", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "score": 1.0, + "content": "behave with access to either model transparency [31, 32] or counterfactual explanations [33], resulting", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "in potentially sub-optimal outcomes. Though these works highlight shortcomings of counterfactual", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 558, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 506, + 569 + ], + "score": 1.0, + "content": "explanations, they do not indicate how these methods are not robust and vulnerable to manipulation.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "Related studies show that post hoc explanations techniques like LIME [34] and SHAP [35] can also", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "hide the biases of the models [24], and so can gradient-based explanations [36, 37]. Aivodji et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 590, + 435, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 435, + 603 + ], + "score": 1.0, + "content": "[38] and Anders et al. [39] show explanations can make unfair models appear fair.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 618, + 216, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 217, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 217, + 634 + ], + "score": 1.0, + "content": "7 Potential Impacts", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 504, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 641, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 657 + ], + "score": 1.0, + "content": "In this section, we discuss potential impacts of developing adversarial models and evaluating on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 654, + 199, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 199, + 666 + ], + "score": 1.0, + "content": "crime prediction tasks.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "Impacts of Developing Adversarial Models Our goal in designing adversarial models is to demon-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "strate how counterfactual explanations can be misused, and in this way, prevent such occurrences", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "in the real world, either by informing practitioners of the risks associated with their use or moti-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "vating the development of more robust counterfactual explanations. However, there are some risks", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 161 + ], + "lines": [], + "index": 3.5, + "bbox_fs": [ + 105, + 74, + 506, + 162 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 169, + 505, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 505, + 181 + ], + "score": 1.0, + "content": "Size of the Model To further characterize the manipulation, we train a number of models (on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "Communities and Crime for Wachter et al.’s) that vary in their size. We show that as we increase the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 338, + 203 + ], + "score": 1.0, + "content": "model size, we gain an even higher cost reduction, i.e. an", + "type": "text" + }, + { + "bbox": [ + 339, + 191, + 360, + 202 + ], + "score": 0.88, + "content": "1 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 192, + 505, + 203 + ], + "score": 1.0, + "content": "increase in the cost reduction when", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "the similar additional parameters are added. This is not surprising, since more parameters provide", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "further the flexibility to distort the decision surface as needed. As we reduce the size of the model, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 382, + 236 + ], + "score": 1.0, + "content": "see the opposite trend; the cost reduction reduces substantially when", + "type": "text" + }, + { + "bbox": [ + 382, + 224, + 396, + 234 + ], + "score": 0.86, + "content": "4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "fewer parameters are used.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "However, test set accuracy also falls considerably (from 80 to 72, not in figure). These results suggest", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 245, + 484, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 484, + 258 + ], + "score": 1.0, + "content": "it is safest to use as compact of a model as meets the accuracy requirements of the application.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 169, + 506, + 258 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 264, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "Takeaways These results provide three main options to increase the robustness of counterfactual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "explanations to manipulation: add a random perturbation to the counterfactual search, use a minimal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 287, + 480, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 480, + 299 + ], + "score": 1.0, + "content": "number of attributes in the counterfactual search, or enforce the use of a less complex model.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 264, + 505, + 299 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 314, + 197, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 198, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 198, + 329 + ], + "score": 1.0, + "content": "6 Related Work", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "Recourse Methods A variety of methods have been proposed to generate recourse for affected", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "individuals [6, 1, 7–9]. Wachter et al. [6] propose gradient search for the closest counterfactual,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "while Ustun et al. [1] introduce the notion of actionable recourse for linear classifiers and propose", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "techniques to find such recourse using linear programming. Because counterfactuals generated by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "these techniques may produce unrealistic recommendations, Van Looveren and Klaise [9] incorporate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 398, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 408 + ], + "score": 1.0, + "content": "constraints in the counterfactual search to encourage them to be in-distribution. Similarly, other", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "approaches incorporate causality in order to avoid such spurious counterfactuals [27, 12, 15]. Further", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "works introduce notions of fairness associated with recourse. For instance, Ustun et al. [1] demonstrate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 507, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 507, + 442 + ], + "score": 1.0, + "content": "disparities in the cost of recourse between groups, which Sharma et al. [4] use to evaluate fairness.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "score": 1.0, + "content": "Gupta et al. [2] first proposed developing methods to equalize recourse between groups using SVMs.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "Karimi et al. [3] establish the notion of fairness of recourse and demonstrate it is distinct from fairness", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 462, + 495, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 495, + 474 + ], + "score": 1.0, + "content": "of predictions. Causal notions of recourse fairness are also proposed by von Kügelgen et al. [28].", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 341, + 507, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "Shortcomings of Explanations Pawelczyk et al. [11] discuss counterfactuals under predictive", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "multiplicity [29] and demonstrate counterfactuals may not transfer across equally good models.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 515 + ], + "score": 1.0, + "content": "Rawal et al. [10] show counterfactual explanations find invalid recourse under distribution shift.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "Kasirzadeh and Smart [30] consider how counterfactual explanations are currently misused and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "propose tenents to better guide their use. Work on strategic behavior considers how individuals might", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "score": 1.0, + "content": "behave with access to either model transparency [31, 32] or counterfactual explanations [33], resulting", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "in potentially sub-optimal outcomes. Though these works highlight shortcomings of counterfactual", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 558, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 506, + 569 + ], + "score": 1.0, + "content": "explanations, they do not indicate how these methods are not robust and vulnerable to manipulation.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "Related studies show that post hoc explanations techniques like LIME [34] and SHAP [35] can also", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "hide the biases of the models [24], and so can gradient-based explanations [36, 37]. Aivodji et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 590, + 435, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 435, + 603 + ], + "score": 1.0, + "content": "[38] and Anders et al. [39] show explanations can make unfair models appear fair.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 481, + 506, + 603 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 618, + 216, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 217, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 217, + 634 + ], + "score": 1.0, + "content": "7 Potential Impacts", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 504, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 641, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 657 + ], + "score": 1.0, + "content": "In this section, we discuss potential impacts of developing adversarial models and evaluating on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 654, + 199, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 199, + 666 + ], + "score": 1.0, + "content": "crime prediction tasks.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 641, + 505, + 666 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "Impacts of Developing Adversarial Models Our goal in designing adversarial models is to demon-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "strate how counterfactual explanations can be misused, and in this way, prevent such occurrences", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "in the real world, either by informing practitioners of the risks associated with their use or moti-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "vating the development of more robust counterfactual explanations. However, there are some risks", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "that the proposed techniques could be applied to generate manipulative models that are used for", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 98 + ], + "score": 1.0, + "content": "harmful purposes. This could come in the form of applying the techniques discussed in the paper", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "score": 1.0, + "content": "to train manipulative models or modifying the objectives in other ways to train harmful models.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "However, exposing such manipulations is one of the key ways to make designers of recourse systems", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "score": 1.0, + "content": "aware of risks so that they can ensure that they place appropriate checks in place and design robust", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 258, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 258, + 141 + ], + "score": 1.0, + "content": "counterfactual generation algorithms.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "that the proposed techniques could be applied to generate manipulative models that are used for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 98 + ], + "score": 1.0, + "content": "harmful purposes. This could come in the form of applying the techniques discussed in the paper", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "score": 1.0, + "content": "to train manipulative models or modifying the objectives in other ways to train harmful models.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "However, exposing such manipulations is one of the key ways to make designers of recourse systems", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "score": 1.0, + "content": "aware of risks so that they can ensure that they place appropriate checks in place and design robust", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 258, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 258, + 141 + ], + "score": 1.0, + "content": "counterfactual generation algorithms.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "Critiques of Crime Prediction Tasks In the paper, we include the Communities and Crime data", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "set. The goal of this data set is to predict whether violent crime occurs in communities. Using machine", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "learning in the contexts of criminal justice and crime prediction has been extensively critiqued by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 185, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 195 + ], + "score": 1.0, + "content": "the fairness community [40–42]. By including this data set, we do not advocate for the use of crime", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "prediction models, which have been shown to have considerable negative impacts. Instead, our goal", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "is to demonstrate how counterfactual explanations might be misused in such a setting to demonstrate", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 217, + 213, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 213, + 229 + ], + "score": 1.0, + "content": "how they are problematic.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 242, + 252, + 256 + ], + "lines": [ + { + "bbox": [ + 104, + 241, + 254, + 258 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 254, + 258 + ], + "score": 1.0, + "content": "8 Discussion & Conclusion", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 267, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "score": 1.0, + "content": "In this paper, we demonstrate a critical vulnerability in counterfactual explanations and show that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "score": 1.0, + "content": "they can be manipulated, raising questions about their reliability. We show such manipulations are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "possible across a variety of commonly-used counterfactual explanations, including Wachter [6], a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "sparse version of Wachter, Counterfactuals guided by prototypes [9], and DiCE [13]. These results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "bring into the question the trustworthiness of counterfactual explanations as a tool to recommend", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "recourse to algorithm stakeholders. We also propose three strategies to mitigate such threats: adding", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "noise to the initialization of the counterfactual search, reducing the set of features used to compute", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 318, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 318, + 356 + ], + "score": 1.0, + "content": "counterfactuals, and reducing the model complexity.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "One consideration with the adversarial training procedure is that it assumes the counterfactual", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 382 + ], + "score": 1.0, + "content": "explanation is known. In some cases, it might be reasonable to assume the counterfactual explanation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "score": 1.0, + "content": "is private, such as those where an auditor wishes to keep this information away from those under audit.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "score": 1.0, + "content": "However, the assumption that the counterfactual explanation is known is still valuable in many cases.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "To ensure transparency, accountability, and more clearly defined compliance with regulations, tests", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "score": 1.0, + "content": "performed by auditing agencies are often public information. As one real-world example, the EPA in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "the USA publishes standard tests they perform [43]. These tests are detailed, reference the academic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "literature, and are freely available online. Fairness audits may likely be public information as well,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "and thus, it could be reasonable to assume the used methods are generally known. This discussion", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "also motivates the need to understand how well the manipulation transfers between explanations. For", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "instance, in cases where the adversarial model designer does not know the counterfactual explanation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "used by the auditor, could they train with a different counterfactual explanation and still be successful?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "Our results also motivate several futher research directions. First, it would be useful to evaluate", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 508, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 519 + ], + "score": 1.0, + "content": "if model families beyond neural networks can be attacked, such as decision trees or rule lists. In", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "this work, we consider neural networks because they provide the capacity to optimize the objec-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "score": 1.0, + "content": "tives in Equations (3) and (4) as well as the (over) expressiveness necessary to make the attack", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "successful. However, because model families besides neural networks are frequently used in high-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "stakes applications, it would be useful to evaluate if they can be manipulated. Second, there is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 574 + ], + "score": 1.0, + "content": "a need for constructing counterfactual explanations that are robust to small changes in the input.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "Robust counterfactuals could prevent counterfactual explanations from producing drastically different", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "counterfactuals under small perturbations. Third, this work motivates need for explanations with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "optimality guarantees, which could lead to more trust in the counterfactuals. Last, it could be useful", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "to study when practitioners should use simpler models, such as in consequential domains, to have", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 445, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 445, + 630 + ], + "score": 1.0, + "content": "more knowledge about their decision boundaries, even if it is at the cost of accuracy.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 107, + 642, + 219, + 656 + ], + "lines": [ + { + "bbox": [ + 104, + 641, + 221, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 221, + 659 + ], + "score": 1.0, + "content": "9 Acknowledgments", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "We would like to thank the anonymous reviewers for their insightful feedback. This work is supported", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "in part by the NSF awards #IIS-2008461, #IIS-2008956, and #IIS-2040989, and research awards", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "from the Harvard Data Science Institute, Amazon, Bayer, Google, and the HPI Research Center in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Machine Learning and Data Science at UC Irvine. The views expressed are those of the authors and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 711, + 378, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 378, + 723 + ], + "score": 1.0, + "content": "do not reflect the official policy or position of the funding agencies.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 139 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 73, + 507, + 141 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "Critiques of Crime Prediction Tasks In the paper, we include the Communities and Crime data", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "set. The goal of this data set is to predict whether violent crime occurs in communities. Using machine", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "learning in the contexts of criminal justice and crime prediction has been extensively critiqued by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 185, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 195 + ], + "score": 1.0, + "content": "the fairness community [40–42]. By including this data set, we do not advocate for the use of crime", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "prediction models, which have been shown to have considerable negative impacts. Instead, our goal", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "is to demonstrate how counterfactual explanations might be misused in such a setting to demonstrate", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 217, + 213, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 213, + 229 + ], + "score": 1.0, + "content": "how they are problematic.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 151, + 505, + 229 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 242, + 252, + 256 + ], + "lines": [ + { + "bbox": [ + 104, + 241, + 254, + 258 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 254, + 258 + ], + "score": 1.0, + "content": "8 Discussion & Conclusion", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 267, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "score": 1.0, + "content": "In this paper, we demonstrate a critical vulnerability in counterfactual explanations and show that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "score": 1.0, + "content": "they can be manipulated, raising questions about their reliability. We show such manipulations are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "possible across a variety of commonly-used counterfactual explanations, including Wachter [6], a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "sparse version of Wachter, Counterfactuals guided by prototypes [9], and DiCE [13]. These results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "bring into the question the trustworthiness of counterfactual explanations as a tool to recommend", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "recourse to algorithm stakeholders. We also propose three strategies to mitigate such threats: adding", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "noise to the initialization of the counterfactual search, reducing the set of features used to compute", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 318, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 318, + 356 + ], + "score": 1.0, + "content": "counterfactuals, and reducing the model complexity.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 266, + 506, + 356 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "One consideration with the adversarial training procedure is that it assumes the counterfactual", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 382 + ], + "score": 1.0, + "content": "explanation is known. In some cases, it might be reasonable to assume the counterfactual explanation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "score": 1.0, + "content": "is private, such as those where an auditor wishes to keep this information away from those under audit.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "score": 1.0, + "content": "However, the assumption that the counterfactual explanation is known is still valuable in many cases.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "To ensure transparency, accountability, and more clearly defined compliance with regulations, tests", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "score": 1.0, + "content": "performed by auditing agencies are often public information. As one real-world example, the EPA in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "the USA publishes standard tests they perform [43]. These tests are detailed, reference the academic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "literature, and are freely available online. Fairness audits may likely be public information as well,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "and thus, it could be reasonable to assume the used methods are generally known. This discussion", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "also motivates the need to understand how well the manipulation transfers between explanations. For", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "instance, in cases where the adversarial model designer does not know the counterfactual explanation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "used by the auditor, could they train with a different counterfactual explanation and still be successful?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 360, + 507, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "Our results also motivate several futher research directions. First, it would be useful to evaluate", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 508, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 519 + ], + "score": 1.0, + "content": "if model families beyond neural networks can be attacked, such as decision trees or rule lists. In", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "this work, we consider neural networks because they provide the capacity to optimize the objec-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 506, + 541 + ], + "score": 1.0, + "content": "tives in Equations (3) and (4) as well as the (over) expressiveness necessary to make the attack", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "successful. However, because model families besides neural networks are frequently used in high-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "stakes applications, it would be useful to evaluate if they can be manipulated. Second, there is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 574 + ], + "score": 1.0, + "content": "a need for constructing counterfactual explanations that are robust to small changes in the input.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "Robust counterfactuals could prevent counterfactual explanations from producing drastically different", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "counterfactuals under small perturbations. Third, this work motivates need for explanations with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "optimality guarantees, which could lead to more trust in the counterfactuals. Last, it could be useful", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "to study when practitioners should use simpler models, such as in consequential domains, to have", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 445, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 445, + 630 + ], + "score": 1.0, + "content": "more knowledge about their decision boundaries, even if it is at the cost of accuracy.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 496, + 506, + 630 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 642, + 219, + 656 + ], + "lines": [ + { + "bbox": [ + 104, + 641, + 221, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 221, + 659 + ], + "score": 1.0, + "content": "9 Acknowledgments", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "We would like to thank the anonymous reviewers for their insightful feedback. This work is supported", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "in part by the NSF awards #IIS-2008461, #IIS-2008956, and #IIS-2040989, and research awards", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "from the Harvard Data Science Institute, Amazon, Bayer, Google, and the HPI Research Center in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Machine Learning and Data Science at UC Irvine. 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Comm. & CrimeGerman Credit
Acc|10||1Acc|0||1
Unmodified81.2=71.1-
Wachter et al.80.90.8072.00.09
Sparse Wachter77.90.4670.52.50
Prototypes79.20.4669.02.21
DiCE81.11.7371.20.09
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Communities and CrimeGerman Credit
Wach.S-Wach.Proto.DiCEWach. S-Wach.Proto.DiCE
Protected35.6854.1622.3549.625.658.3510.516.31
Non-Protected35.3152.0522.6542.635.088.5913.986.81
Disparity0.372.120.306.990.750.240.060.5
Non-Protected+δ1.7622.598.509.573.164.124.693.38
Cost reduction20.1×2.3×2.6×4.5×1.8×2.0x2.2×2.0×
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Model InitializationWachter et al.DiCE Rnd.
MeanRnd. x+N
Protected42.416.2 11.748.3
Not-Prot.42.315.7 10.342.3
Disparity0.010.49 1.455.95
Not-Prot.+δ2.503.79 8.5912.3
Cost reduction16.9×4.3× 1.2×3.4×
Accuracy81.480.2 75.378.9
|18|l10.650.65 0.361.24
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